• .I. BIBLIOTEEK
University Free State
HIEROlE EKSEMPLAAR MAG ONDER \1~~I~~~~~ml~~~~~
(~EEN OMSTANOIGHEDEUIT DIE 34300000426670Universiteit Vrystaat
I~. l.'lll ti EEK vi:ó{WYDfR WORD N!£_..
PARAMETRIC AND NONPARAMETRIC BAYESIAN
STATISTICAL INFERENCE IN ANIMAL SCIENCE
by
ALBERTUS LODEWIKUS PRETORIUS
THESIS
Submitted infu/fillment of the requirements for the degree
PHILOSOPHIAE DOCTOR
111
THE FACUL TY OF NATURAL AND AGRICULTURAL SCIENCES
Department of Mathematical Statistics
University of the Free State
Bloemfontein: November 2000
PROMOTER: PROFESSOR Abrie 1. VAN DER MERWE
-----__.-
""l"el~nel: '/00 Il e
0:-("' '. 'v0''' \..Y.lt
\ pL.-'-~f\.. Ti.il:
\\ 3 \ t4~S ZOOl
uovS s .?'" ItlL.lOiEE~·
Acknowledgements and Contents
ACKNOWLEDGEMENTS
It is imperative that I begin by thanking Prof Abrie for setting such a patient and supportive example
for me that I try to emulate. Shared visions of utilizing Bayesian Inference in Animal Breeding
turned our ideas into the present PhD thesis. I know that the road to success is neither easy nor
straight. ..
"~CI¥e-- CfM"Vey called. Ye{eree1t; ..
Loop.\'~~ ...
SPeed-b-ump.\'~L~ ..
Y~waLhaNe--fl.cw,r~ci4app~ ..
'BUT
If y~ have: Ct< drtve« ccdled. God: ..
I~CIA'\.Ce'~F~ ..
CCL«t'wrv~~F~ CLf'\d,FY~ ..
AI'\I~~Pe-v~e-vClA'\.Ce'~ ..
sp~~Vet"~wrvCLf'\d,pe-v~
Y~ waL make: (;(;1;0- Ct< plcLce.-ccdled. Succew"
Indeed, I know now that nothing in the world can take the place of persistence. Talent will not;
nothing is more common than unsuccessful men with talent. Genius will not; unrewarded
genius is almost a proverb. Education will not; the world is full of educated derelicts;
Determination and Persistence alone are Omnipotent.
I r\I "-1emcry of 110/ UU-e,- "-10»1/
(1942 -1993)
Acknowledgements and Contents
CONTENTS
CHAPTERl BAYESIAN STATISTICS AND ANIMAL BREEDING
THEORY 1
1.1 Prologue 1
1.2 The Mixed Model Methodology 3
1.2.1 Background 3
1.2.2 Notation 3
1.3 The Classical Solution 5
1.4 The Traditional Bayesian Solution 7
1.5 Prior Distributions 10
1.6 Joint and Full Conditional Posterior Distributions 11
1.7 The Gibbs Sampler 14
1.7.1 Background 14
1.7.2 Illustrating the Gibbs Sampler 15
1.8 An Example 19
1.8.1 The Data 19
1.8.2 Analysis of Variance Components 21
1.8.3 Analysis of Random Effects 25
1.8.4 Analysis of Fixed Effects 31
1.9 Chapter Summary 36
CHAPTER2 BAYESIAN METHOD OF MOMENTS 37
2.1 Prologue 37
2.2 Review of the BMOM approach 40
2.3 Extension of the BMOM to the Mixed Linear Model 42
2.4 The Gibbs Sampler 51
-111-
Acknowledgements and Contents
2.5 Another Bayesian Method of Moments Approach
for the Mixed Linear Model 52
2.6 An Example 54
2.6.1 The Data 54
2.6.2 Analysis of Variance Components 55
2.6.3 Analysis of Random Effects 59
2.6.4 Analysis of Fixed Effects 64
2.7 Chapter Summary 68
CHAPTERJ THE nIRICHLET PROCESS 70
3.1 Prologue 70
3.2 The Classical Perspective 71
3.3 The Bayesian Perspective 72
3.4 The Mixture ofDirichlet Process (MDP) 73
3.4.1 Background 73
3.4.2 The Model Structure 74
3.4.3 The Dirichlet Process Prior in the case of
The Mixed Linear Model 75
3.4.4 The Uniform Prior for jJ and (j/ 81
3.4.5 Prior for (j/ 82
3.4.6 Simulation of the Precision Parameter M 84
3.4.7 The Gibbs Sampler 88
3.5 An Example 89
3.5.1 The Data 89
3.5.2 Analysis of Variance Components 90
3.5.3 Analysis of Random Effects 98
3.5.4 Analysis of Fixed Effects 113
3.6 An Experimental Design - Model Validation 119
3.7 Chapter Summary 123
-lV-
Acknowledgements and Contents
CHAPTER4 THE DIRICHLET PROCESS IN VETERINARY MEDICINE
RESEARCH 125
4.1 Prologue 125
4.2 The Experiment and Model Structure 126
4.3 Priors and Conditional Posterior Distributions 131
4.3.1 The Uniform Prior for zi and c," 131
4.3.2 Prior for D 133
4.3.3 Dirich1et Process Prior for 'Y; 134
4.4 Priors and Conditional Posterior Distributions for the
Non-Conjugate MDP Model 139
4.4.1 The Uniform Prior for,8 and (j/ 140
4.4.2 Prior and Conditional Posterior for 1] 140
4.4.3 Prior for D 142
4.4.4 Dirichlet Process Prior for 'Y; 143
4.5 A Veterinary Medicine Example 145
4.5.1 REML Solution 145
4.5.2 Nonparametrie Bayesian Solution 149
4.6 Chapter Summary 162
CHAPTERS REFERENCE AND PROBABILITY-MATCHING
PRIORS 163
5.1 Prologue 163
5.2 The Mixed Linear Model 164
5.3 Reference Priors and Probability-Matching Priors 167
5.3.1 Background 167
5.3.2 The Fisher Information Matrix 167
5.3.3 Reference Prior for ,8, (j/ and v 172
5.3.4 Reference Prior for the Intraclass Correlation
Coefficient, p 178
5.3.5 Probability-Matching Priors 182
-v-
Acknowledgements and Contents
5.4 Reference Posterior Distributions 186
5.4.1 Background 186
5.4.2 Calculation of the Posterior Density
p(a;,vIY) 189
5.4.3 Calculation of the Posterior Density
pea; I v, Y) 190
5.4.4 Calculation of the Posterior Density
p(v IY) 191
5.4.5 Posterior Distribution of () = Xf3 + Zy and Predictive
Distribution of Yf = x~f3 + z~y + ef 193
5.5 An Example 196
5.5.1 The Data 196
5.5.2 Estimation and Prediction using Uniform and
Reference Priors 197
5.5.3 Analysis of Variance Components 197
5.5.4 Analysis of Random Effects 202
5.5.5 Analysis of Fixed Effects 204
5.5.6 Predictive Density of a Future Observation 207
5.6 Priors for the Mixed Linear Model in the Case of Three
Variance Components 209
5.6.1 Reference Prior for the Variance Components 209
5.6.2 Proper Prior for the Variance Components 216
5.7 Joint and Conditional Posterior Densities for the
Mixed Linear Model in the Case of Three
Variance Components 217
5.8 An Example 220
5.8.1 The Data 220
5.8.2 Analysis of Variance Components 221
5.9 Chapter Summary 225
-vi-
Acknowledgements and Contents
CHAPTER6 CONCLUSION AND SUMMARY 226
REFERENCES 231
APPENDIX A - F 247 - 311
-vn-
Animal Breeding Theory and Mixed Model Methodology
CHAPTER 1
«Bayesian Statistics and Animal Breeding Theory»
Introductory words: Harville, (1990), (see also Gianola, (1990)) stated UAmore extensive use of Bayesian
ideas by animal breeders and other practitioners is desirable and is more feasible from a computational
standpoint than commonly thought. The Bayesian approach can be used to devise prediction procedures that
are more sensible - from both a Bayesian and Frequentist perspective - than those in current use ", The.
Bayesian approach is also conceptually more appealing than the Classical approach.
1.1 Prologue
Animal breeding theory deals with the formulation and validation of mathematical (primarily
statistical) models aimed at developing procedures for selecting and mating individuals so that
performance is optimal in some sense. The primary goal of such selection experiments in animal
breeding is to identify animals to use for producing the next generation of progeny in order to
maximize genetic progress with respect to traits of interest. Examples of traits that have been subject
to such selection experiments are milk yield in diary cows, rapid weight gain in pigs, and weaning
weights in lambs.
A bold improvement 111 genetic evaluation occurred when mixed model methodology was
introduced". Familial relationships between sires enhanced the accuracy at which breeding values
-1-
Animal Breeding Theory and Mixed Model Methodology
were estimated because the effective separation of genetic merit from environment effects became
possible. In animal breeding experiments, the observed trait values, or phenotypes, are mode led as
the sum of a number of effects, including individual breeding values. Also, the breeding values are
mode led as correlated random effects, with the correlation arising due to known genetic relationships.
To maximize future progress of a population, the goal is to identify the animals with the highest
breeding values.
Genetic evaluation of South African flocks of sheep started with the analysis of the experimental
Merino flock at Klerefontein near Carnavon. This was followed by single flock evaluation as part of
a variety of postgraduate studies and the evaluation of progeny groups of rams for the industry (Van
Wyk, 1992). The Dormer sheep stud, started at the Elsenburg College of Agriculture in 1940, also
represents such a flock for animal breeding experiments. The main object in developing the Dormer
was the establishment of a mutton sheep breed which would be well adapted to the conditions
prevailing in the Western Cape (winter rainfall) and which could produce the desired type of ram for
crossbreeding purposes (Swart 1967).
Only a small example from this Dormer stud data was used. It was not our attention to reanalyze
the data from a genetic point of view; rather, we used it to illustrate how the Bayesian approach and
Gibbs sampling could be applied to real animal breeding problems. This data form an integrated part
of the statistical methods introduced in the thesis and can be found in APPENDIX B.
-2-
Animal Breeding Theory and Mixed Model Methodology
1.2 The Mixed Model Methodology
1.2.1 Background
The mixed model methodology was first developed for animal genetics and breeding research. In
resent years, however, the mixed model has also been introduced in variety of other disciplines (e.g.
sociology and education) to analyze experiments with more complex data structures. These mixed
models are also called Hierarchical Models, Random Effects Models or Variance Components
Models. As the mixed model methodology is heavily based on matrix notation, it is important that a
clear notation is used in the development of the theory in the present thesis.
1.2.2 Notation
§ A matrix is always put in a bold letter, and a vector in a bold underlined letter, e.g. Y is the
matrix of observations, whereas Y presents the vector of observations. Greek letters are used
for ,the fixed and random effect vectors and the letters are not underlined, e.g. jJ is a vector of
fixed effects, and r the vector of random effects.
§ The transpose of a matrix X or vector Y is denoted by Xl or yl, respectively.
§ The inverse of a matrix X is denoted by X -I .
§ The generalized inverse of a matrix X is denoted by X-.
-3-
Animal Breeding Theory and Mixed Model Methodology
The mixed linear model postulates that the observable random vector Y is a linear combination of
the fixed effects and random effects plus a random error (residual). In its simplest form the
univariate mixed linear model can be written in matrix notation as
y = xp +Zy +s . (1.1 )
X (n x l ) is a vector of observed values for the trait on which selection is desired. Only a single
trait per animal is considered for most of the analysis, although the model and analysis described can
be modified to accommodate multiple traits.
jJ Cp xl) is a vector of fixed effects uniquely defined so that the corresponding design matrix X
(n x p) has full column rank, p. Loosely speaking, a fixed effect, in a Bayesian sense, is a random
variable on which prior knowledge is diffuse or vague, i.e. a priori the investigator is indifferent to its
likely value.
Furthermore, y (q x 1) is a vector of unobservable random effects with y - N(Q,Acr:) and
design matrix Z (n x q). a/ is an unknown scalar and A (q x q) is called a relationship (genetic
covariance) matrix. Its elements reflect the genetic relationship among the sires. The random effects
in the present case, are the breeding values, which account for the variation in Y due to genetic merit.
Note that, in the case of a Sire Model, a breeding value refers to a sire's value as a parent in a
breeding program, and it is a measure of the animal's progeny performance relative to the mean value
of its breed. Genetic evaluation is heavily dependent on the genetic correlation among individuals,
both for higher accuracy and for unbiased results. Therefore, the genetic relationship among
individuals is of fundamental importance in the prediction of breeding values.
-4-
Animal Breeding Theory and Mixed Model Methodology
For the unobservable vector of random errors s (n xl), statistical independent of r, it is common
to assume independent normal distributions with mean vector Q and variance-covariance matrix er/In
In. In represents an n x n identity matrix and er/ an unknown scalar.
As mentioned, er/ and er/ are unknown scalar-value parameters called variance components.
1.3 The Classical Solution
Data from animal breeding experiments are commonly analyzed using a mixed linear model in
order to estimate or predict the breeding values of individual animals. When the values of the
variance components of the model are not known, the Classical Approach to the problem of
predicting linear combinations of the different effects has been to estimate the variance components
and to proceed thereafter as if these estimates were the true values of the components.
Patterson and Thompson (1971) developed a method to derive unbiased estimates of the unknown
variance components based on the maximum likelihood principle; called the Restricted Maximum
Likelihood Estimation (REML). This method is based on the likelihood of a vector whose
components are independent linear combinations of the observations. The basic idea is to end up
with a random vector that contains all the information on the variance components but no longer
contains information on the fixed effects parameters. However, there are several problems with this
(classical) approach.
1. The properties of the predictors are hard to assess. This is particularly the case when
estimates of variance components are substituted for their true values.
-5-
Animal Breeding Theory and Mixed Model Methodology
2. When the values of the variance components are estimated from the data, the sampling errors
are generally not taken into account in the subsequent analysis. Therefore, the variance of
the prediction error will generally be underestimated.
3. Depending upon the size and characteristics of the data, point estimates of the variance
components can be highly variable. For certain values of the components estimates, the
predictors obtained by substituting these values in the "Best Linear Unbiased Predictor" are
intuitively unappealing.
The classical frequentist solution of the mixed linear model (1.1) can be obtained from
Henderson's mixed model equations. Henderson, in Henderson et al. (1959), developed a set of
equations that simultaneously yielded best linear unbiased predictors of the random effects and best
linear unbiased estimators of the fixed effects. They were derived by maximizing the joint density of
Y and. r ,Le.
n
f(Y,y) = (2:,,;r exp{ - 2~;(Y:- Xp - Zy !(Y - Xp - Zy l}
J~ ( 1.2)x ( -- 1 JAJ2_.!. exp{I--r'A-Ir. }2~a2 2r 2a r
Equating to zero the partial derivation of (1.2) with respect to elements, first of jJ and then of r
give the mixed model equations, written more compactly as
X'X
[Z'X X' ~ ][/J] [XIY]Z'Z+ :: A-I r = Z'Y , . (1.3)
-6-
Animal Breeding Theory and Mixed Model Methodology
where jJ and r denote the solutions of jJ and y. By substituting the REML estimates of CT; and
CT: in equation (1.3) give the classical frequentist solution to the mixed linear model.
Hence, the objective is now to propose the Bayesian Approach as a conceptional strategy to solve
problems arising in animal breeding theory, to illustrate how well known results can be retrieved
from the Bayesian perspective, and to suggest possible areas of research in which Bayesian
Approach and Mixed Linear Model Methodology can lead to fruitful results.
1.4 The Traditional Bayesian Solution
Harville (1990), (see also Gianola, (1990)) stated, "A more extensive use of Bayesian ideas by
animal breeders and other practitioners is desirable and is more feasible from a computational
standpoint than commonly thought. The Bayesian approach can be used to devise prediction
procedures that are more sensible - from both a Bayesian and Frequentist perspective - than those in
current use". The Bayesian approach is also conceptually more appealing than the Classical
approach with the following advantages:
I. The Bayesian practitioner does not need to commit himself to a point estimate of the variance
components in order to obtain a point predictor for the variables of interest, and credibility
intervals can easily be obtained.
2. Uncertainty about the true values of the variancecomponents is formally incorporated into the
analysis through the choice of an appropriate prior distribution.
-7-
Animal Breeding Theory and Mixed Model Methodology
3. .Given the data, the prior information about the unknown parameters, and a well-defined loss
function, there exists an optimal Bayes predictor.
4. All the available information about the random variable to be predicted is contained in the
posterior distribution of the random variable. Therefore, the practitioner can base all of his
inferences on this distribution.
5. The Bayesian approach is conceptually more appealing than the classical approach.
Critics of the Bayesian approach have most often cited the following points:
1. The Bayesian practitioner must formally express his prior beliefs about the unknown
parameters in the form of a probability distribution.
2. The Bayesian methodology is computer intensive. In many situations, integrations in
several dimensions are required to obtain the required posterior distributions.
These might have been valid criticisms in the past butby using (a) Non-Informative priors like
Jeffreys and Reference priors and (b) Numerical integration techniques like Markov Chain Monte
Carlo Methods and more specifically Gibbs Sampling, these problems can be overcome.
When analyzing the mixed linear model (or any model) using a Bayesian approach, it only matters
whether a specified quantity is observable or not. In equation (1.1), Y, X and Z are observable
whilst, fJ and rare unobservable. No further classifications are necessary.
-8-
Animal Breeding Theory and Mixed Model Methodology
In the classical approach to analysis of data using a mixed linear model the distinctions of fixed
versus random, known versus unknown, parameter versus statistic, are all-important. These
classifications dictate the type of estimation and inference that are possible. In Bayesian modeling
we treat /3, r and all the variance components in the same way: they are unobservable.
In many Bayesian problems, marginal posterior distributions are often needed to make appropriate
inferences. However, due to the complexity of the joint posterior distribution it is impossible to
obtain these marginal densities analytically and because of the many unknowns, very difficult to
calculate numerically. Instead, a Markov Chain Monte Carlo (MCMC) method, called Gibbs
Sampling, will be implemented to estimate the marginal posterior densities of the different
parameters.
Recently due to the work by Gelfand and Smith (1990), Gelfand et al. (1990), Carlin et al. (1992)
and Gelfand el al. (1992), the Gibbs Sampler has been shown as an useful tool for applied Bayesian
inference in a broad variety of statistical problems. The Gibbs sampler is implicit in the work of
Hastings (1970) and made popular in the image-processing context by Geman and Geman (1984).
The Gibbs sampler is an adaptive Monte Carlo integration technique. The typical objective of the
sampler is to collect a sufficiently large enough number of parameter realizations from conditional
posterior densities in order to obtain accurate estimates of the marginal posterior densities. The
principle requirement of the sampler is that all, conditional densities must be available, in the sense
that random variables can be generated from them. Once the marginal densities are obtained, it is
easy to calculate summary statistics from the posterior distributions.
-9-
Animal Breeding Theory and Mixed Model Methodology
The method is of great appeal on account of its simple logical foundation and reasonable ease of
implementation. The next section elaborates the role of the sampler in relating conditional and
marginal distributions from animal breeding theory.
1.5 .Prlor Distributions
For modelling the hierarchy, the distribution of e gives the sampling distribution, which, in
classical statistics, is the distribution of the data conditional on all the parameters. In a Bayesian
analysis this distribution is called the likelihood function and it is always the first stage in a
traditional Bayesian analysis with prior distributions relegated to other stages.
From (1.1) it follows that the conditional distribution that generates the data (likelihood function)
IS
( 1.4)
where In represents an n x n identity matrix and N(f.J ,£) denotes the n-dimensional multivariate
normal distribution with mean vector f.J and variance-covariance matrix E.
An integral part of Bayesian analysis is now the assignment of a prior distribution to the unknown
parameters in the model. The information contained in the prior distribution is combined with the
information supplied by the data, through the likelihood' function (if it is known), into the conditional
posterior distribution of the parameters given the data, which is known as the posterior distribution,
All inferences about the model parameters are based on the posterior distribution. In the above
-10-
Animal Breeding Theory and Mixed Model Methodology
model, "flat" or uniform prior distributions are assigned to the vector of fixed effects and error
variance, as to represent lack of prior knowledge.
Therefore
p(j3, a/) = p(jJ) p(a/) cc constant. ( 1.5)
Further, the prior distribution of the vector r is given by
r I A , a/ ~ Nq( Q , Aa/ ). ( 1.6)
As mentioned in the case of the sire model the elements of the numerator relationship matrix A
describe the covariance of the sires due to shared genes, and r is the vector of breeding values. Also
. p ( d/ ) IX constant. (1. 7)
1.6 Joint and Full Conditional Posterior Distributions
The joint posterior distribution of the unknowns (jJ, r , a/ ,a/) is proportional to the product of
the likelihood function, and the joint prior distribution is given by
-11-
Animal Breeding Theory and Mixed Model Methodology
p(f3,y,O'; ,O'~ I D) a: (~Jn ~exp{- ~(y - Xf3 - Zy)' (Y - Xf3 - Zy)}
O'e 2O'e .
X -21 J~ { ( 1.8)exp ---21 yl,\. -I( Y } ,O'y 2O'y
where D = (Y , X ) denotes the data.
The required full conditional for the fixed effects, is multivariate normal:
.' fJ lY, 2 2 ~ I -I 2Ue .a, 'D~ Np ( f3 ' (X X) a, ), ( 1.9)
where /3 = (XiX)" I X I (Y - Z y). Note that this distribution does not depend on u/.
The conditional distribution of y is also multivariate normal:
(1.1 0)
where
For the variance components, the conditionals are
n
P(O'e2 1f3,y'O'r,D2)=Ke ( -21 J2 exp { --2 1 (Y-Xf3-Zy)(Y-Xf3-Zy,) }
0'e 20' e
0'; > 0,
(1.11)
-12-
Animal Breeding Theory and Mixed Model Methodology
an Inverse Gamma density where
= {(V - n-2Xp - Zy)'(Y - Xp - ZY)}-2 1
K, 2 r(~)2 '
2
and
rq ,p(a; I ,B,y,a;,D) = K,(;; ex+ 2~;y\\-Iy} (I.I2)
a2r>0
also an,Inverse Gamma where
Moreover, animal breeders are often interested 10 the posterior distributions of functions of
a2
varianc.e components like the intraclass correlation coefficient, p = Y and the variance ratio
a2 +a2Y c
0'2
V = -T' The conditional posterior distributions of these parameters can be obtained by making use
ac
of transformation of random variables. For example, the conditional posterior distribution of p can
be obtained by using the transformation of 0'/ --f P in the conditional posterior density of 0'/,
equation (1.12).
-13-
Animal Breeding Theory and Mixed Model Methodology
aa2 a2
Since the Jacobian of the transformation, __ r = --&-2 ' equation (1.12) can be written as
ap (l-p)
p(pljJ,y,a;,D)=K 1a; Jf-I(l)r ( pf (l-p) q-42 exp {(-l-P2)a; y1I\-ly } (1.13)
O c o c I,
and for the variance ratio,
p(v I jJ .r,o;2 ,D) = Kr ( -21 Jf exp{- --21 Y IA -I r } (1.14)
va& 2va&
G c v c l .
It is clear from equations (1.9) - (1.l4) that account of the genetic covariance matrix has been taken
into the conditional posterior distributions.
1.7 The Gibbs Sampler
1.7.1 Background
The Gibbs sampler enjoyed an initial surge popularity starting in 1984 with Geman and Geman,
who studied image-processing models. Gelfand and Smith (1990) then put the sampler in a new
light, revealing its potential in a wide variety of conventional statistical problems. It is characterized
by always using full conditionals, however, other sets of conditionals may also be used, sets which
are also sufficient to determine joint distributions. The ultimate value of the Gibbs sampler lies in its
-14-
Animal Breeding Theory and Mixed Model Methodology
practical potential. Now that the groundwork has been laid in pioneering research work, the present
research is focused on exploring and expanding the Gibbs sampler using mixed linear model
methodology to animal breeding problems.
The Gibbs sampler is a technique for generating random variables from a marginal distribution
indirectly, without having to calculate the density. In that which follows, it is easy to see that the
Gibbs sampler is iterative and based only on elementary properties of Markov chains. In this respect,
there are two issues of concern: convergence and uniqueness. However, Geman and Geman (1984)
showed that under mild regularity conditions, the Gibbs sampler converges uniquely to the
appropriate marginal distributions. CaselIa and George (1990) discuss numerical means to accelerate
convergence. Another way of speeding up convergence is to integrate out analytically some nuisance
parameters from the joint posterior distribution before running the Gibbs sampler.
1.7.2 Illustrating the Gibbs Sampler
Suppose we are given a joint density fïx.y hY2....• y,J and are interested in obtaining characteristics
of the marginal density
f(x) = IJ f(x.YI.Y2. ···.Y,JdY1d2 ... dy; (1.14)
such as the mean or variance. Probably the most natural and straightforward approach would be to
calculate f(x) and use it to obtain the desired characteristic. However, there are many cases where the
integration in (1.14) is extremely difficult to perform, either analytically or numerically. In such
cases the Gibbs sampler provides an alternative method for obtainingj(x), i.e. to generate a Markov
-15-
Animal Breeding Theory and Mixed Model Methodology
chain of random variables (also called a "Gibbs sequence") that converge to the distribution of
interestf(y) .
Rather than compute or approximate fïx) directly, the Gibbs sampler allows us effectively to
generate a sample XI, X2, ... , Xm - f(x) without requiring fïx). By simulating a large enough sample,
the mean, variance, or any other characteristic of f(x) can be calculated to the desired degree of
accuracy. It is important to note that, in effect, the end results of any calculations, although based on
simulations, are the population quantities. Thus by taking a large enough sample, any population
characteristic, even the density itself, can be obtained by averaging the final conditional densities
from each Gibbs sequence. These estimates are called Rao-Blackwell estimates (Gelfand & Smith,
1990). An alternative form of estimating the marginal posterior densities is by obtaining kernel
density estimators; however, the Rao-Blackwell estimates are more accurate.
To understand the working of the Gibbs sampler, we explore "it in the three-variable case. The
initial values y/O) = y/O), y/O) = yl) and y/O) = y/O) are specified and the rest of the Gibbs sequence of
random variables is obtained iteratively by alternately generating values in the following way:
Draw
then
also draw
and
This completes one iteration of the scheme. Thus, at the kth iteration we draw
-16-
Animal Breeding Theory and Mixed Model Methodology
then
then
then
.Gernan and Geman (1984) have shown that under fairly general conditions, the distribution of X(k)
converges to fix) (the true marginal distribution of x) as k nears infinity. Thus, the value X(k) can be
regarded as a simulated observation fromfix) if k is large enough. By repeating the Gibbs sequence
m times, the Gibbs sampler generates m observations
If the repetitions are independent, using predetermined initial values y/O), y/O) and yl) for each
sequence, the final values will be independent. Thus, by simulating a large enough sample,
characteristics such as the mean and variance of fix) can be determined to the desired degree of
accuracy (Van der Merwe & Botha, 1993). Characteristics of./{y,), ./{yl) and ./{yl) can be obtained
in a similar way. It is important to remember that to generate m random variables with approximate
density fix), we have to generate (2k) x m random variables, where k is the length of each Gibbs
sequence (CaselIa & George, 1990).
In the light of the aforementioned, the Gibbs sampler can thus be thought of as a practical
implementation of the fact that knowledge of the conditional distributions is sufficient to determine a
joint distribution, if it exists. In the Markov Chain Monte Carlo procedure and more specifically
Gibbs sampling, we construct a stochastic process that has the desired posterior distribution as its
-17-
Animal Breeding Theory and Mixed Model Methodology
stationary distribution and then simulate the process. Standard routines are used to generate random
numbers from these required distributions. Selective algorithms of the Gibbs sampler are given in
APPENDIX A.
In the case of the mixed linear model, we begin with a set of arbitrary starting values for the
model parameters, jJ (0), r (0), a/(O), a/ (0) and then successively generate values from the conditional
posterior distribution of each of the parameters, conditioning on the most recently generated values of
the other parameters of each step.
The Gibbs sampler for p(J3, r ,a/, a/ ID) is as follows:
(0) Select starting values for r (0) ,a/ (0) ,a/ (0) . Set i = O.
. (I) SamplejJ(iTl) from (1.9),
(2) Sample a/(i+I) from (1.11),
(3) Sample r"" from (1.10),
(4) Sample ay2(i+I) from (1.12),
(5) Seti=i+ 1 and return to (1).
MATLAB software has been developed to generate the samples that enabled us to obtain the
marginal posterior densities for the model parameter, using the Gibbs sampler. The full conditional
posteriors were updated after every iteration. We ran multiple chains, i.e. m=101 000 of the Gibbs
sampler to obtain draws from the posterior distributions of the model parameters given the data. The
first 1 000 draws of each chain were discarded, and then every 101h draw was saved. By saving every
io" draw, the chain yielded a posterior sample of 1 000 approximately uncorrelated draws. All
posterior analyses were based on these I 000 draws, giving us a full Bayesian solution to all the
mixed linear model parameters.
- I 8-
Animal Breeding Theory and Mixed Model Methodology
1.8 An Example
1.8.1 The Data
This section describes the analysis of data from the Dormer Sheep Stud started at the Elsenburg
College of Agriculture near Stellenbosch. The main objective in developing the Dormer was the
establishment of a mutton sheep breed which would be well adapted to the conditions prevailing in
the Western Cape and which could produce the desired type of ram for crossbreeding purposes.
The origin of the Dormer Sheep breed can be traced back to December 1940 when four imported
Dorset Horn rams were each mated to fourteen registered and thirty-five grade German (presently
s.A. Mutton) Merino ewes. This was a direct consequence of a comprehensive series of
crossbreeding studies carried out at the Elsenburg Agricultural College from 1927 over a period in
excess of ten years. After the initial cross only the two rams with the best progeny results were used
in the next breeding season (December 1941). Each was mated to 20 registered German Merino
ewes. As no further crossbreeding between these parental breeds were practiced after 1941, only two
first-cross rams and 77 first-cross ewe lambs served as basic material for further development of the
new breed. It could therefore be concluded that the Dormer originated from a small number of
animals. From the parental side it descends from only four Dorset Horn rams and because of
selection only 31 registered and 40 high-grade German Merino ewes eventually contributed to the
development of the Dormer breed. Although the Dormer sheep stud originated from a small number
of animals, it can be assumed that, being a cross bet:ween two unrelated breeds; the inbreeding
coefficients of the base animals were zero (Van Wyk, 1992).
-19-
Animal Breeding Theory and Mixed Model Methodology
Sheep used in the analysis were born in the period 1943 - 1950. Single sire mating was practiced
with 25 to 30 ewes allocated to each ram. A spring breeding season (6 weeks duration) was used
throughout the study. A total of n = 879 weaning weight records, from the progeny of q = 17 sires
were available after editing, and p = 17 fixed effects were included in the final model. The data can
be observed in APPENDIX B. The REML estimates were obtained by using the MTDFREML
program developed by Boldman et al. (1995).
The mixed linear model used for this data structure, is the sire model of section (1.2),
y = Xp + Zy + s , where Y (879 x I) vector of weaning weights. In this application, X is a
(879 x 17) design matrix of regressors, with one column corresponding to the overall mean weaning
weight, seven columns corresponding to the season of birth effects, six to the age of dam effects, one
to the sex of lambs effects, and two final columns corresponding to the birth status effects.
f3 (17 x l ) is the vector oï fixed effects. Using this notation, f30 is the average weaning weight of
female lambs born in 1950 if the age of the dam is 8 years or older, and the birth status "triplets".
f31 is the difference in average weaning weight between lambs born in 1943 and those born in 1950.
f32 is the difference in average weaning weight between lambs born in 1944 and those born in 1950.
Whilst f37 is the difference in average weaning weight between lambs born in 1949 and lambs born in
1950, f3s measures the difference in average weaning weight of lambs with dams 2 years of age and
those with dams 8 years and older of age. Further, f3/3 is the difference in average weaning weight of
lambs with dams 7 years of age and those with dams 8 years and older of age. The difference in
average weaning weight between male and female lambs is measured by f31. and f315 measures the
difference in average weaning weight between single births and triplets. Finally, f316 is the difference
in average weaning weight between twins and triplets.
-20-
Animal Breeding Theory and Mixed Model Methodology
The design matrix Z (879 x 17) is a matrix identifying the random effects. Note that r is a
(17 x I) vector of random effects consisting of the breeding values for the 17 sires for which the data
are observed.
1.8.2 Analysis of Variance Components
For the classical analysis, the estimates of the variance components are found by maximizing the
likelihood function as developed by Patterson and Thompson (1971). Given these estimates, the Best
Linear Unbiased Predictions (BLUBs) for r and jJ are then obtained by solving Henderson's mixed
model ,equations (equation (l.3)). Posterior means, and modes of the Traditional Bayesian analysis,
95% credibility intervals, and the REML estimates (along with standard errors) are summarized in
Table l.I. The REML estimates of a/ and a/ are more similar to the modes of the posterior
distributions than the means. This is because the REML estimate represents the mode of the
marginal likelihood and thus might be better compared to the mode of the posterior distribution.
Table 1.1 REML and Traditional Bayesian Estimates (posterior values) for the Variance
Components, along with 95% Credibility Intervals.
REML Traditional Bayes 95% Credibility
Parameter Mean Mode Interval
a.2 2l.1096 2l.2595 2l.2619 19.3130; 23.2531
a2r 3.08 4.9239 3.01 l.2461; 12.1343
-21-
Animal Breeding Theory and Mixed Model Methodology
Using the posterior densities for CF} and CF/, the marginal posterior densities are estimated as the
average of the conditional posterior densities, obtained from the Gibbs sampler, and are depicted in
Figures 1,1 and 1,2, Note that the distribution in Figure 1,2 is quite skew, resulting in a difference
between the posterior mean and posterior mode (quantities will not coincide),
Figure 1.1 Histogram and Estimated Marginal Posterior Density of the Variance Component,
CF} (Error variance),
-22-
Animal Breeding Theory and Mixed Model Methodology
Figure 1.2 Estimated Marginal Posterior Density of the Variance Component (7/ (Sire
variance).
The posterior means and modes of the Traditional Bayesian analysis and 95% credibility intervals
(J2
of functions of variance components Iike the intraclass correlation coefficient, p = 2 r 2' and the
. ~+~
(J2
variance ratio, v = -T are summarized in Table 1.2. It is evident from this table that the credibility
(J.
interval for the intraclass correlation coefficient does not contain 0.5. This result corresponds well to
the statement made by Wang et al. (1992) namely that from a genetic point of view, an intraclass
correlation coefficient of 0.5 is not possible in a sire model.
-23-
Animal Breeding Theory and Mixed Model Methodology
Table 1.2 Traditional Bayesian Estimates of Functions of the Variance Components, along
with 95% Credibility Intervals.
REML Traditional Bayes 95% Credibility
Parameter Mean Mode Interval
p 0.127 0.1789 0.133 0.0550; 0.3658
v 0.146 0.2326 0.140 0.0582 ; 0.5768
The posterior distributions of these functions are illustrated in Figure 1.3 below.
Figure 1.3 The Estimated Marginal Posterior Density of the (a) Intraclass Correlation
(j2 (j2
Coefficient, p = 2 y 2' and the (b) Variance Ratio, v = ---T .
(jy + (jc (jc
-24-
Animal Breeding Theory and Mixed Model Methodology
We can conclude that a Bayesian approach to variance component estimation has several practical
advantages over a classical approach.
Firstly, although the estimate for a variance component is always positive, the REML estimate's
asymptotic distribution can generate interval estimates that include negative values. This potentially
embarrassing phenomenon is often overlooked in the discussions of likelihood-based methods. An
interval estimate such as a highest posterior density region will not include negative values.
Secondly, highest posterior density regions are never empty, whereas confidence intervals for the
ratio of two variances can be empty. One can also report the whole of the posterior probability
distribution, not just a single number, and report some measure of posterior precision. Finally,
classical estimators generally have intractable sampling distributions and standard errors are hard to
calculate (Van der Merwe & Botha, 1993)
1.8.3 Analysis of Random Effects
Table 1.3 contains the BLUPs (with the variance components fixed at the REML estimates) and
posterior means of the random effects (breeding values) for the 17 sires, along with the posterior
ranks of each animal based on its breeding value and 95% credibility intervals. To put these numbers
in perspective, progeny from sire 3 (ranked l" according to its Trad. Bayes and REML estimates)
with an estimated breeding value of 3.4858 will therefore have an estimated average weaning weight
of 3.4858 kilogram more than the progeny from the rest of the sires. The progeny from sire 10
(ranked l7'h according to its Trad. Bayes and REML estimates) on the other hand with an estimated
breeding value of -1.7983 will have an estimated average weaning weight of I.7983 kilogram less
than the average wean ing weight of lambs from the rest of the sires.
-25-
Animal Breeding Theory and Mixed Model Methodology
Further inspection of the credibility intervals in Table 1.3 shows that the lower bound of the 95%
credibil ity interval for the breeding value of sire 3 is 1.2143 whilst the upper bound is 6.1194. Since
this interval does not contain zero, we can be reasonable sure that the average weaning weight of
lambs born from sire 3 will be between 1.21 and 6.12 kilogram more than the average weaning
weight of lambs born from the other sires. Furthermore, by comparing the 95% credibility intervals
of the preeding values in Table 1.3 it is clear that the upper limit of the interval in the case of sire 10
is smaller than the lower limit of the corresponding interval for sire 3. By implication this means that
sire 10 will never (very seldom) produce progeny with greater weaning weights than sire 3.
-26-
Animal Breeding Theory and Mixed Model Methodology
Table 1.3 Estimated Breeding Values for 17 Sires from the Elsenburg Dormer Stud, Posterior
Rankings, 95% Credibility Intervals, and Standard Errors of REML Estimates.
Sire ID Trad. Bayes Rank 95% Credibility Interval REML Rank SE's
41037 0.7350 3 -1.4728; 3.4395 0.5781 3 1.06
41004 0.2478 6 -1.6531 ; 2.6977 0.1396 6 0.92
41019 3.4858 1 1.2143; 6.1194 3.33 1 0.99
43002 -1.1985 14 -3.7586; 1.3778 -1.181 14 1.18
44170 -0.0930 7 -2.7340; 2.7585 -0.17 7 1.18
44174 -0.6524 10 -3.9471 ; 2.3055 -0.5694 10 1.34
44042 -1.3053 15 -3.6029; 0.9157 -1.2565 16 0.95
45070 -1.1460 13 -3.6855; 1.1319 -0.9631 13 0.93
45135 -0.5301 9 -3.2348 ; 1.9326 -0.5371 9 1.1
46015 -1.7983 17 -4.4578; 0.3174 -1.7092 17 0.96
46037 -0.8524 Il ' -3.2205; 1.4669 -0.8423 Il 0.91
48014 -1.0059 12 -3.6639 ; 1.2705 -0.9537 12 0.97
48052 -0.4208 8 -2.8863; 1.9739 -0.3019 8 1
48148 -1.4307 16 -4.0524; 0.9945 -1.256 15 1.1
49053 0.5309 4 -2.6618; 3.7327 0.463 4 1.31
49134 0.9219 2 -2.0470; 4.1511 0.795 2 1.34
49046 . 0.4395 5 -2.9563 ; 3.7575 0.4059 5 1041
It is evident from the table that the Traditional Bayes estimated are quite close to the REML
estimates. This is not surprising to us, since as showed by Harville, (1974) (see also Searle, CaselIa
and McCulloch, (1992)) that when uniform or "tlat" priors are assigned to the vector offixed effects
and variance components, the modes of the marginal posterior distributions are very close to the
REML estimates.
-27-
Animal Breeding Theory and Mixed Model Methodology
If on the other hand proper priors were assigned to the unknown parameters and if the sample size
was quite small, the differences between Bayesian and non-Bayesian results could have been quite
substantial. The assignment of a proper prior to a specific parameter must however be justifiable
from a practical point of view.
The marginal posterior densities for the breeding values of sire 3 and 10, and the difference in
breeding values for these two sires, are displayed in Figures 1.4 - 1.6 respectively.
Figure 1.4 The Estimated Marginal Posterior Density ofthe Breeding Value for Sire 3 (Y3),
ID41019.
-28-
Animal Breeding Theory and Mixed Model Methodology
Figure 1.5 The Estimated Marginal Posterior Density of the Breeding Value for Sire 10 (YIO),
ID46015.
-29-
Animal Breeding Theory and Mixed Model Methodology
Figure 1.6 The Estimated Marginal Posterior Density of the Difference In Breeding Value
between Sire 3 and Sire 10 (Y3 - YIO).
A key difference between REMLlBLUP predictions and Bayesian inference is the treatment of the
variance components. To obtain the BLUP estimates, the variance components are fixed at a single
value, ignoring uncertainty associated with estimating their values. The Bayesian analysis
incorporates this uncertainty by averaging over the plausible values of the variance components,
making it a more feasible method of analysis, since these components are very important in
evaluating the breeding potential of the sires in the model.
-30-
Animal Breeding Theory and Mixed Model Methodology
1.8.4 Analysis of Fixed Effects
Duchateau et al. (1998) stated that the emphasis in breeding experiments is on the variance
components and on the prediction of particular random effects, but estimation of the fixed effects is
also important, thus Table 1.4 summarizes the estimated fixed effect for the mixed linear model
given the data along with selected joint marginal posterior densities presented in Figures 1.8 - 1.12
(obtained from the Gibbs sampler).
Table 1.4 Estimated Values of Selected Fixed Effects, 95% Credibility Intervals, and REML
Estimates.
Trad.
Fixed Effect 95% CredibilityBayes Interval REML
/30 22.9655 19.2315 ; 26.9031 21.50
/37 5.3523 4.1515 ; 6.5310 5.25
/3u 3.6690 2.9835 ; 4.3353 3.54
/315 9.4874 7.1923; 11.7688 9.35
/316 2.9621 0.6574 ; 5.2308 2.88
As described in section (1.9.1), /30 is the average weaning weight of female lambs born in 1950 if
the age of the dam is 8 years or older, and the birth status "triplets". /37 measures the expected
difference in average weaning weight between lambs born in 1949 and lambs born in 1950. It is
therefore clear that lambs born in 1949 had an average weaning weight of 5.3523 kilogram more than
lambs born in 1950.
-31-
Animal Breeding Theory and Mixed Model Methodology
Figure 1.8' Estimated Marginal Posterior Density of /30, the expected average weaning weight of
, female lambs born in 1950 if the age of the dam is 8 years or older, and the birth
status "triplets".
Figure 1.9 Estimated Marginal Posterior Density of /37, the expected difference in average
weaning weight between lambs born in 1949 and in 1950.
-32-
Animal Breeding Theory and Mixed Model Methodology
/3/4 measures the expected difference in average weaning weight between male and female lambs.
It can therefore be concluded that male lambs will have an average weaning weight of 3.6690
kilogram more than female lambs. From the 95% credibility interval it can be seen that the
difference in average weaning weight between male and female lambs can be as large as 4.3353
kilogram.
Figure 1.10 Estimated Marginal Posterior Density of /314, the expected difference III average
weaning weight between male and female lambs.
-33-
Animal Breeding Theory and Mixed Model Methodology
[JIJ measures the expected difference in average wean 109 weight between single births and
triplets. It can therefore be expected that single births will have an average weaning weight of
9.4874 kilogram more than triplets.
Figure 1.11 Estimated Marginal Posterior Density of [JH, the expected difference 10 average
weaning weight between single births and triplets.
-34-
Animal Breeding Theory and Mixed Model Methodology
/3/6 measures the expected difference in average weaning weight between a pare of twins at birth
and triplets. Therefore, a pair of twins at birth will have an average weaning weight of 2.9621
kilogram more than triplets .. It is therefore clear that birth status can dramatically affect the expected
weaning weight of a sire's progeny, thus affecting its breeding value. According to van Wyk (1992)
single born lambs constitute only 36.02% of all lambs born whilst twins and triplets make up 59.93%
and 4.05% respectively.
Figure 1.12 Estimated Marginal Posterior Density of /3/6, the expected difference In average
weaning weight between a pare of twins at birth and triplets.
One appealing future of the Gibbs simulation approach to the Bayesian data analysis is that we
obtain an approximate sample from the joint distribution of all the unknown parameters given the
data. This sample provides adequate information to estimate any quantity of interest.
-35-
Animal Breeding Theory and Mixed Model Methodology
1.9 Chapter Summary
The present chapter illustrated an extension of the Gibbs sampler to solve problems arising in
animal breeding theory. Formulae were derived and presented to implement the Gibbs sampler in a
more general mixed linear model. With this extension, a full Bayesian solution to the problem of
inference about variance components, functions thereof, and random effects in such a mixed linear
model was possible. Once the marginal densities were obtained from the Gibbs sampler, it was easy
to calculate summary statistics from the posterior distributions, e.g. posterior means, modes and
credibility intervals. Moreover, as mentioned before, the similarities between the Bayesian and
REML estimates were not surprising to us because of the assignment of uniform or "flat" priors to
the vector of fixed effects and variance components. If on the other hand proper priors were assigned
to the unknown parameters and if the sample size was quite small, the differences between Bayesian
and non-Bayesian results could have been more substantial.
Arguing from a Bayesian viewpoint, the Gibbs sampling turned an analytically intractable
multidimensional integration problem into a feasible numerical one, and IS conceptually more
appealing than the classical approach.
Since the Gibbs sampler is now established in animal breeding problems, the objective of the
thesis will be to extent some selected issues regarding The Mixed Linear Model in animal breeding,
e.g. The Bayesian Method of Moments (BMOM) approach to the full Bayesian solution, Reference -,
Probability-Matching -, and Oirichlet Process Priors for the random effects.
© Parts of this chapter have been published in the South African Statistical Journal.
(See Van der Merwe et al. 2000)
-36-
Bayesian Method of Moments and the Mixed Linear Model
CHAPTER2'
«Bayesian Method of Moments»
Introdll:ctory words: After reviewing the purposes and. basic principles of the BMOM approach previously
presented and applied by Zellner and eo-workers to multiple and multivariate regression models as well as
simultaneous equation problems, a new application of the approach is presented. In this section the BMOM
procedure is extended to the Mixed Linear Model with illustrative examples from the animal breeding theory.
2.1 Prologue
On the BMOM and the capability of comparing BMOM and Traditional Bayes models, Barnard
(1997) has written:
"And above all any method is welcome which, unlike nonparametrics, remains fully quantifiable without
paying obeisance to the model which one knows is false. And your proposal to compare BMOM results
with a model based one should achieve the best of both worlds. "
.In addition, Laskey (1997) comments:
"When prior knowledge about the form of the likelihood function is extremely weak, standard Bayesian
analysis can be 'brittle' in the sense of (l) producing absurd conclusions given not-obvious-absurd inputs
and (2) being extremely sensitive to minor variation in inputs. On the other hand, BMOM gives good
-37-
Bayesian Method of Moments and the Mixed Linear Model
answers for the questions it addresses while not purporting to go beyond the information that is really there
in the prior and the data ...
Another view of BMOM is provided by Soofi (1997) in the following words:
"l consider the BMOM as an ingenious contribution to the entire field of statistics. The BMOM is elegant
and easily applicable because it is free from the strong UNVERIFIABLE assumptions that we usually make
just in order to enable us to handle a problem ."
(see Zellner et al. (1999) for quotes)
In the. traditional likelihood and Bayesian approaches, it IS usually assumed that enough
information is available to formulate a likelihood function and, in the Bayesian approach, a prior
density for the parameters of the selected likelihood function. However, if not enough information is
available to specify a form for the likelihood function, then clearly there will be problems in both the
traditional likelihood and Bayesian approaches. In situations like this, some resort to non-likelihood
based methods is proposed, e.g. the Bayesian Method of Moments (BMOM), first introduced by
Arnold Zellner in 1994. Given the data, BMOM enables researchers to compute post data densities
for parameters and future observations if the form of the likelihood function is unknown. The
BMOM approach provides a solution to the famous inver~e problem proposed by Bayes (1763) and
hence the name Bayesian Method of Moments.
As illustrated in Chapter 1, an essential element of the Bayesian approach is Bayes' theorem, also
referred to in the literature as the principle of inverse probability. In problems involving "inverse
probability" we have given data and from the information in the data try to infer what random
process generated them. On the other hand, in problems of "direct probability" we know the random
process, including values of its parameters and from this knowledge make probability statements
-38-
Bayesian Method of Moments and the Mixed Linear Model
about outcomes or data produced by the known random process. Problems of statistical estimation
are thus seen to be problems of "inverse probability", whereas many gambling problems are
problems in direct probability.
In the BMOM approach the posterior and predictive moments, based on a few relatively weak
assumptions are used to obtain maximum entropy densities for the parameters, realized error terms
and future values of the variables. Shannon (1948) defines entropy (or uncertainty) as
w = - Jp(y) log pry) dy (2.1)
where pry) is a probability density function. Maximizing W subject to various side conditions is well
known' in the literature as a method for deriving the forms of minimal information distributions.
Shannon (1948) has also indicated how maximum entropy (ME) distributions can be derived by a
straightforward application of calculus of variation techniques. In particular he has shown that the
ME distribution that maximizes entropy subject to a normalization condition is just the uniform
distribution. By adding additional side conditions given the first two moments of the distribution are
imposed, the ME distribution is the normal distribution. On the other hand if just the side conditions
on the zero" and first moments are utilized, the maxent density is an exponential density. For
discussion and application of maximum entropy, see for example Jaynes (1982, 1988); Shore and
Johns~n (1980); Cover and Thomas (1991); Zellner and Highfield (1988) and Zellner (1997).
In the sections to come, the theory and results derived by Zellner (1997) will be extended to the
mixed linear model, with an appropriate example from an animal breeding experiment. We will also
discuss coherent procedures of updating BMOM maxent post-data densities for parameters and future
observations.
-39-
Bayesian Method of Moments and the Mixed Linear Model
2.2 Review of the BMOM approach
In Table 2.1, the inputs and outputs of the Traditional Bayesian (TB) and the BMOM approaches
are summarized. In both approaches, given the data is an important input along with an entertained
model for the given data, say the mixed linear model. In the TB approach sampling assumptions for
the data or the model's error terms are introduced in order to obtain a likelihood function. This
likelihood function and the prior density for its parameters are inputs to Bayes' theorem and the
outputs are posterior and predictive densities for parameters, realized error terms and future
observations .
.It is evident that in the BMOM approach no sampling assumptions about the given observed data
are made. Rather certain assumptions are made about the realized error terms' properties. Given
these assumptions, posterior moments of the parameters are derived that incorporate the information
in the' given data. These moments are then used as side conditions in the derivation of maxent
probability density functions for the parameters, realized error terms and future observations.
-40-
Bayesian Method of Moments and the Mixed Linear Model
Table 2.1 Inputs and Outputs of Traditional Bayesian and BMOM Approaches.
A. Traditional Bayesian Approach
INPUTS OUTPUTS
1. Data, D 1. Posterior Density
2. Prior information, Il 2. Predictive Density
3. Sampling Assumptions 3. Point & Interval Estimates
4. Data Density & Likelihood Function 4. Point and Interval Predictions .
5. Prior Density
6. Bayes' Theorem
B. BMOM Approach
INPUTS OUTPUTS
1. Data, D SAME AS ABOVE
2. Prior Information, 12
3. Mathematical Form of the model
. 4. . Moments of parameters and future values
5. Maxent Principle
-41-
Bayesian Method of Moments and the Mixed Linear Model
2.3 Extension of the BMOM tó the Mixed Linear Model
In section 1.2 the mixed linear model in its simples form was defined as
Y = XfJ + Zy + e . (2.2)
In the introductory paragraph it was stated that the BMOM approach is particular useful where there
is difficulty in formulating an appropriate likelihood function. Without a likelihood function, it is not
possible to pursue traditional likelihood and Bayesian approaches to estimation and hypothesis
testing. In the next section only the mathematical form of the model as defined in (2.2) will be used,
i.e. no specific distribution will be assigned to the vector B. The likelihood function will therefore be
considered as unknown. This is different 'from the assumption in the previous sections. Let us for
the time being assume that y is given, (2,2) can then be written as
y -Zy = XfJ+& (2.3)
i.e,
y* = XfJ+&· (2.4)
For given y, we will take Y * as our new dependent variable. Equation (2.3) is now the usual
multiple regression model. To assume that the model in (2.3) is adequate implies, among other
things, that there are no systematic elements in the realized error term vector e , correlated with
variables in X. This assumption is formalized as Assumption 1 in the BMOM approach as follows:
-42-
Bayesian Method of Moments and the Mixed Linear Model
Assumption 1:
X'E(&ID,y)=O
where E( BID, r) denotes the post-data mean of the realized error vector B, given the data and y;
that is, the given, unknown values of the elements of the realized error vector are considered
subjectively random, just as in Bayesian analysis of the realized error terms (Chaloner & Brant,
1988; Chaloner, 1994; Zellner & Moulton, 1985). Thus, the assumption indicates that the columns of
X are orthogonal to the vector E(& ID, y ), Further, from (2.4) by taking the posterior expectation, it
follows that
y* = XE(f31 D,y) + E(& I D,y) .
and Assumption I implies that the observation vector Y * is the sum of two orthogonal vectors. Note
also since the first column of X is a n x 1 vector of ones, denoted by l ,we have from the assumption,
(E(&ID,y)=O (2.5)
or
(2.6)
Thus, given that we assume that we have an appropriate form and an adequate number of terms
included in (2.4), the expectation of the mean of the realized error terms is assumed equal to zero,
i.e. there is no systematic component in the realized error vector.
-43-
Bayesian Method of Moments and the Mixed Linear Model
Proof:
Ifwe multiply both sides of(2.4) by (X/X)"1 x', we obtain
jJ = (X'X)-I X'y* = /3 + (X'X)-IX's . (2.7)
~
Now take the post-data expectation of both sides in (2.7), noting that E(/31 D) = /3 , we have
jJ = £(/31 D,y) + (X'Xrl X'E(s 1D,y) (2.8)
and from Assumption 1
E(/31 D,y) = jJ = (X'Xrl x'v *. (2.9)
That is, the post-data expectation of the regression coefficient vector is equal to the least squares
estimate. Further, the post data mean of the realized error vector in (2.4) is
E(G ID, y) = y * - XjJ = i (2.10)
where i is the least squares residual vector that satisfies X'i = o. Note also that from (2.7) and
(2.10),
-44-
Bayesian Method of Moments and the Mixed Linear Model
e - & = Y * - Xf3 - (Y * -XP)
= Y * -Xf3 - {Y * -X(f3 + (X'Xrl X'e)}
= X(X'X)·I X' e
= X(X'X)"I X'(s - i)
(2.11)
where the last step follows from the orthogonality condition mentioned above, X'i = o. We can
thus write
. Var(e I D,y) = E{(e - i)(e - &)'1 D,y}
(2.12)
= X(X'X)"I X'E{(e - i)(e - i)' I D,y}X(X'X)"1 X'
which defines a functional equation that the post-data covariance matrix for e , Var(e I D,y) must
satisfy. Since there are only p free elements of s in the n equations in (2.4), Var(e I D,y) must be of
rank p. Thus we introduce the following assumption that fixes the form of the realized error vector
up to a multiplicative positive scalar multiplier.
Assumption 2:
Var(e Ia; ,D,y) = a;X(X'Xrl X'
where a/ is a variance parameter to be defined below. We use Assumption 2 to evaluate the post-
data eovariance matrix of jJ as follows
Var(f31 a;, D,y) = E{(f3 - P)(f3 - P), I D,~}
= (X'X)"I X'EKf3 - P)(f3 - P'I D,y}X(X'X)"1 (2.13)
= a; (X'X)"I
-45-
Bayesian Method of Moments and the Mixed Linear Model
ac2 1~ 2 1,where = - L..,.£; = -££.
n ,=1 n
It is seen. that the parameter a/ is represented as an average of the sum of squared deviations of .
-
the realized error terms from their expected mean of zero, E( e ID, r) = 0 which follows from
Assumption 1.
Proof:
Using the definition of a/ we have
E(ac·21 D;y)·1=· E -(£'£ I D,y)
n
= E.!_ {(V * -Xf3)'(Y * -Xf3) I D,y}
n
= E ±[ky * -XP) - (Xf3 - XP)} {(V * -XP) - (Xf3 - XP)}I D,y]
= E.!_ [kv * -XP)'(Y * -XP) - 2(Y * -XP)'(Xf3 - XP) + (13- P)(X'X)(f3 - P)}I D,y}
n
Since. the middle term is equal to zero, Y *, X - 13'X' X = 0, it follows that
E(o-; I D,y) = .!_[i'i + E{(f3 - P)'(X'X)(f3 - P) I D,y}]
n
=.!_ {i'i + pE(a; I D,y)}
n
=
n-p
-46-
Bayesian Method of Moments and the Mixed Linear Model
~I~
?
That is, E(a; I D,y) = --& & = ?s, (2.14)
n-p
Note that this post-data expectation differs from the post-data mean of <:5/ in a diffuse prior-
ks2 .
normal likelihood traditional Bayesian approach (TB), namely ETB(a; I D,y) = --. For small
k-2
values' of k = n - p, the last expression is much larger than i. As pointed out in Zellner (1996),
Tobias and Zellner (1999) and mentioned in the introduction, the proper maxent density for jJ given
<:5/, D and y can now be derived from the above assumptions.
Corollary 1: The proper maxent density for <:5/ using the first moment of <:5/ given in (2.14) is an
exponential density,
(2.15)
which will be called BMOM 1.
Corollary 2: The proper maxent density for jJ given <:5/, D and y, using the first two moments is a
normal distribution
. (2.16)
From (2.15) and (2.16) it follows that
p(p,a; I D,y) = fN CP I a; ,D,y)heCa; I,D,y). (2.17)
-47-
Bayesian Method of Moments and the Mixed Linear Model
Also, as shown in Tobias and Zellner (1999) and Zellner (1997), higher order post-data moments of
0/ can be evaluated and used as moment side conditions in deriving maxent densities.
Corollary 3: The proper maxent density for a/ using the first four moments of a/ can be
approximated by a Pearson density. This approach will be called BMOM 2
By extending the method of Tobias and Zellner (1999), these higher order post-data moments of
a/ can be obtained in the following way. From the definition of a/ it follows that
CJ; = _!_ e'e = _!_ (ks 2 + (fJ - jJ'(X'X)(fJ - jJ))
n n
= _!_ (ks 2 + CJ; Q)
n
(2.18)
. 1 ~ ~
and from (2.16) it follows that Q = -2 (fJ - fJ)'(X'X)(fJ - fJ) has a chi-square density with p
CJe
degrees of freedom.
As shown by Tobias and Zellner (1979) this fact can be employed to evaluate the moments of a/ as
illustrated below. From (2.18) it follows that
for j = 1,2, ...... (2.19)
By using the binomial expansion and known moments of X/, the following recursive formulae can
be derived:
-48-
Bayesian Method of Moments and the Mixed Linear Model
(ks2)J + f(~'l(ks2 )J-I E(cr;i I D)[P(p + 2) ...... (p + 2(i -1))]
E(cr;J I D,y) = I=I} . (2.20)
n - [Pep + 2) ...... (p + 2(J -1))]
Thus, from expression (2.20) the first two moments above zero and the variance of cr/ are given by
E(cr4 ID) = 2S4 (k + 2kp)
. e ,y n 2 - p ( p+ 2)'
Var(cr& 2 I D. ,y) = S 4( 2 2p 1•
n - pep + 2)
(2.21 )
A similar approach as just described can be used to obtain the post-data densities of y and cr/.
For the mixed linear model defined in (2.2) we can also assume that jJ instead of y is known.
Equation (2.2) can then be written as
y -xp = Zy+&
i.e.
X=Zy+&. (2.22)
-49-
Bayesian Method of Moments and the Mi.xed Linear Model
Using the same arguments as given in (2.7) - (2.16) it follows that the maxent density for r is
normal. wi•th mean, rA = (Z'Z)-I Z'Y_- an d vana.nee a-2y = (z,z)-I?a;. To I.mp Iement the norma I
prior for r (which is an integral part of mixed linear model analysis) the likelihood of r given a/
and jJ 'is considered to be proportional to the maxent density.
Multiplying the likelihood with the prior (1.6) gives
exp{- ~ (r - r)' (Z' Z)(r - r)} x exp{- .'. r' A -Ir} (2.23)
2a E 2ay
which implies that the posterior distribution of r is normal with density
PN(Y I ,8,0"; ,O";,D - N, {r,( Z'Z + A -i~ri0"; }
(2.24)
where r = ( Z'Z +A -t :1rZ'(Y - X,8).
Equation (2.24) is identical to the conditional posterior derived in the traditional Bayesian case
(equation (1.10)). Also, the conditional posterior density for a/ in the BMOM case is identical to
equation (1.12), this follows from the normal prior density (equation (1.6)).
-50-
Bayesian Method of Moments and the Mixed Linear Model
Finally, we note that the post-data moments for Cf/ can be employed to compute post-data
densities for the realized error terms and functions of the realized errors that are often useful for
diagnostic purposes as has been recognized in the traditional Bayesian analysis; see Chapter I.
Having derived a range of post-data densities for the mixed linear model and indicating how
BMOM analysis can be performed, we now turn to implement the Gibbs sampler to obtain the
posterior densities for the model parameters.
2.4 The Gibbs Sampler
The Gibbs sampler is once again employed to obtain finite sample post-data parameter densities
as described for the traditional Bayesian approach (Chapter I) with one exemption that in the BMOM
case, Cf/ would be sampled from different maxent densities, i.e.
0< a; < 00 (see also equation (2.16»
for the BMOM I, using one moment, and for BMOM 2 using the recursive equation
(ks2)i + Ï(~J(ks2)j-i E(a;i I D)[P(p + 2) ......(p + 2(i -1))]
E( a;i I D,r) = ,_=1-'----'-_. ----;:-r _----------:;-] ----
nl -tp(P.-!- 2)......(p + 2(j -1))
(see also equation (2.20»
and a Pearson curve approximation with four moments.
-51- Lt '1 I
Bayesian Method of Moments and the Mixed Linear Model
Thus, the Gibbs sampler for p(jJ, r. <7/, <7/ I D) IS:
(0) Select starting values for y (0) ,<7/ (0) ,<7/ (0) . Set i = O.
(I) Sample jJ (i-/) from (1.9),
(2) Sample <7/ (i-/) from (2.16; 2.20 or other densities),
(3) Sample y(i-I) from (1.10),
(4) Sample c;y2 {ir l] from (1.12),
, (5) Seti=i+l and return to (I).
2.5 Another Bayesian Method of Moments Approach for the Mixed Linear Model
In this section another approach to the BMOM analysis will be given which is "more distribution
free" or "less likelihood" than the previous one.
As in section 2.3 the proper maxent densities of jJ and <7/ will be obtained by using Assumptions
1 and 2. These densities are given in equations (2.15) and (2.16). Higher order post-data moments of
<7/ can also be calculated and used as moment side conditions in deriving other maxent densities.
The derivations that follow will however be different from those given in equations (2.22) and (2.24).
Substitute starting values y(O) and 13(0) in equation (2.2), to calculate
&(0) = X - Xp(O) _ Zy(O). (2,25)
Also for given y(O), draw a;(I) and 13(1) from (2.15) and (2.16). A new y which will be called
-52-
Bayesian Method of Moments and the Mixed Linear Model
r (I) = (Z'Z)-I Z,(V - xs» - &(0))post (2.26)
can now be calculated as well as
2(1) _ q1 ( r (I)' -I (I) )ar - post A r post (2.27)
where "post" means "posterior". To implement the normal prior assumption for r (equation (1.6»
draw a Y~~st from the normal distribution N(Q, a:(I) A) and calculate
&(1) = Y_ - XjJ(l) - Zr(l)pflor (2.28)
to complete the first iteration. After k iterations in which the conditional distributions were updated
at each iteration, the Gibbs sampler has generated the values jJ(k), r~:~ta' ;(k) and a:(k). The
process is then repeated m times.
For our practical problem the BMOM posterior densities using in this section and those derived
from the previous section were for all purposes the same. It is therefore clear that in the case of the
BMOM analysis the posterior moments, based on a few relatively weak assumptions can be used to
obtain post data densities (maximum entropy densities) for parameters and realized error terms
without the use of the likelihood function or prior den~ity. The assumption of no prior information
has as consequence different types of derivations (post data densities) that differ from those obtained
using the traditional Bayesian approach where prior information was assigned to the unknown
parameters jJ , r , a; and a:.
-53-
Bayesian Method of Moments and the Mixed Linear Model
As mentioned no pnor densities are necessary for the BMOM procedure but if some prior
information is available it can be built into the BMOM procedure. Since the assumption of a normal
prior for the random effects r is an integral part of the mixed linear model, the prior density
r ~N(Q,a}A) is also used in the BMOM analysis.
2.6 An Example
2.6.1 The Data
Consider the Dormer sheep stud of Elsenburg (see section 1.8.1). Recall that the sheep used in
the analysis were born in the period 1943 -;-1950. A total of n = 879 weaning weight records, from
the progeny of q = 17 sires were available after editing, and p = 17 fixed effects were included in the
final model.
The mixed linear model used for this data structure, is the sire model of section (l.2),
X = Xp + Zy + s , where Y (879 xl) vector of weaning weights. fJ (17 xl) is the vector of fixed
effects, and X a (879 x 17) design matrix of regressors, with one column corresponding to the overall
mean weaning weight, seven columns corresponding to the season of birth effects, six to the age of
dam effects, one to the sex of lambs effects, and two final columns corresponding to the birth status
effects. Furthermore, Z is a (879 x 17) matrix identifying the (17 x 1) vector of random effects r
consisting of the breeding values for the 17 sires for which the data are observed.
-54-
Bayesian Method of Moments and the Mixed Linear Model
Finally, s is an unobservable vector of random residuals (879 xl) such that the distribution of e
is assumed to be independent normal with mean vector Q and varianee-co variance matrix a/ In. In
represents an identity matrix (879 x 879).
MATLAB software has once again been developed to generate the samples that enabled us to
obtain "the finite sample post-data parameter densities, using the Gibbs sampler. The full conditional
posteriors are updated after every iteration. The first I 000 draws of each chain are discarded, and
then every to" draw is saved. By saving every io" draw, the chain yielded a posterior sample of
1 000 approximately uncorrelated draws. All posterior analyses are based on these m = 1 000 draws.
2.6.2 Analysis of Variance Components
Posterior modes of the Traditional Bayesian analysis from Chapter 1, post-data estimates obtained
from the BMOM approach, drawing first from an exponential distribution (BMOM 1) and then from
a Pearson Type 4 curve (BMOM 2), as well as the 95% credibility intervals for the variance
components are summarized in Table 2.1. Functions of the variance components (p and v) are given
in Table 2.2. The post-data densities for the variance components are provided in Figures 2.1 and
2.2, and for p and v in Figure 2.3.
Table 2.1 Traditional Bayesian Estimates (posterior modes) and Estimates from the BMOM
Analysis of the Variance Components, along with 95% Credibility Intervals.
" "
95 % Credibility 95 % Credibility
Parameters Trad. Bayes BMOMl Interval (BMOM1) BMOM2 Interval(BM()~\
a/ 21.2595 21.1125 0.0000 ; 64.5875 20.2873 19.9979; 20.6214
a/ 3.01 3.86 1.5651 ; 14.3427 3.16 1.2464; 13.1652
-55-
Bayesian Method of Moments and the Mixed Linear Model
From Table 2.1 it is evident that there is not much difference between the traditional Bayesian
estimates and the BMOM estimates for the error variance, 0-/. However, the 95% credibility
interval for 0-/ in the case of the BMOM1 differs substantially from the corresponding interval for
BMOM 2 (and the Traditional Bayes). This is as expected because the exponential density is quite
skewed. By imposing additional side conditions in the case of BMOM 2 and thus reducing entropy,
the Pearson Type 4 curve is more informative than the exponential density. Note that the distribution
in Figure 2.2 is quite skewed, resulting once again in a difference between the posterior means and·
posterior modes and discrepancies in credibility intervals.
Figure 2.1 Estimated Marginal Post-data Densities for 0-/ in the case of BMOM 1 (a) and
BMOM 2 (b). Note that BMOM I is a proper maxent density (exponential)
which has mean E(a/) = i, while BMOM 2 is a proper Pearson type 4 curve
density.
-56-
Bayesian Method of Moments and the Mixed Linear Model
Figure 2.2 Estimated Marginal Post-data Densities for a/ in the case of BMOM 1 (short
dashed line), Mean = 5.6778; BMOM 2 (long dashed line), Mean = 5.0361 and
the Traditional Bayesian Density (solid line), Mean = 4.9239.
We also observed the same type of parameter behavior for functions of the variance components in
Table 2.2.
-57-
Bayesian Method of Moments and the Mixed Linear Model
Table 2.2 Traditional Bayesian Estimates (posterior modes) and Estimates from the BMOM
Analysis of Functions of the Variance Components, along with 95% Credibility
Intervals.
95 % Credibility 95 % Credibility
Parameters Trad. Bayes BMOMl Interval (BMOMP BMOM2 Interval (BMOM2\
p 0.133 0.102 0.0417 ; 0.8351 . 0.155 0.0577 ; 0.3944
v 0.140 0.082 0.0435; 10.2976 0.162 0.0613; 0.6513
Except for BMOM 1, the 95% credibility interval for the intraclass correlation coefficient does
not contain 0.5. This result corresponds to the statement made by Wang, et al. (1993), namely that
from a genetic point of view, an intraclass correlation of 0.5 is not possible in a sire model.
Moreover, the 95% credibility of p in the case of BMOM 1 differs substantially from the
corresponding intervals for BMOM 2 and Traditional Bayes. This was expected because the
exponential density (the maxent using only one moment) is quite skewed. In practice usually two or
more moments are available. The credibility intervals for BMOM 2 and Traditional Bayes on the
other hand are for all purposes the same. Indeed, if proper priors were assigned to the variance
components, and if the sample size was quite small, the difference between the BMOM and
Traditional Bayes results could have been quite substantial. The assignment of a proper prior to the
variance components must however be justifiable from a practical point of view. In some animal
breeding. experiments for example it is known that a2r < ! 23ae • This information can, if necessary,
be use? to formulate a proper prior on the interval [0;!]' for the variance ratio v = a~..
3 ac
-58-
Bayesian Method of Moments and the Mixed Linear Model
-- .. _- .. - .. - ...
Figure 2.3 The Estimated Marginal Post-data Density of the (a) Intraclass Correlation
(J2 (J2
Coefficient, p = 2 Y 2' and the (b) Variance Ratio, v = -T.
(Jy + (Je (Je
2.6.3 Analysis of Random Effects
The posterior distributions of the random effects can be obtained directly from the Gibbs sampler.
Table 2.3 contains the post-data means and corresponding post-data rankings based on the mean
values of the random effects (breeding values) for the 17 sires. It is evident from the table that the
estimates using the different procedures are quite close to each other. The Traditional Bayes and
BMOM 2 estimates are for all practical purposes the same.
-59-
Bayesian Method of Moments and the Mixed Linear Model
Rather than to comment on results for all 17 sires, we will focus our discussion on the two
animals ranked highest using the Traditional Bayes and BMOM analysis.
Table ,2.3. .Estimated Breeding Values for 17 Sires from the Elsenburg Donner Stud, and Post-data
Rankings using BMOM and Traditional Bayesian approaches. REML estimates along
with Standard Errors are also included.
Sire ID Trad Bayes Rank BMOM I Rank BMOM2 Rank REML Rank SE's
41037 0.7350 3 0.8889 3 0.7098 3 0.5781 3 1.06
41004 0.2478 6 0.3397 6 0.2415 6 0.1396 6 0,92
41019 3.4858 1 3.6370 1 3.4931 1 3.3300 1 0.99
43002 -1.1985 14 -1.3089 14 -1.246 14 -1.1810 14 1.18
44170 -0,0930 7 -0.0387 7 -0.0943 7 -0.1700 7 1.18
44174 -0.6524 10 -0.7942 10 -0.6847 10 -0.5694 10 1.34
44042 -1.3053 15 -IJ338 15 -1.356 15 -1.2565 16 0.95
45070 -1.1460 13 -1.1793 13 -1.1833 13 -0.9631 13 0.93
45135 -0.5301 9 -0.5069 9 -0.5984 9 -0.5371 9 1.10
46015 -1.7983 17 -1.8758 17 -1.8861 17 -1.7092 17 0.96
46037 -0.8524 Il -0.8960 II -0.9098 II -0.8423 11 0.91
48014 -1.0059 12 -1.100 I 12 -1.0759 12 -0.9537 12 0.97
48052 -0.4208 8 -0.4541 8 -0.4708 8 -0.3019 8 1.00
48148 -1.4307 16 -1.5019 16 -1.475 16 -1.2560 15 1.10
49053 0.5309 4 0.8152 4 0.4479 4 0.4630 4 1.31
49134 0.9219 2 1.3176 2 0.9641 2 0.7950 2 1.34
49046 0.4395 5 0.6804 5 0.3379 5 0.4059 5 1.41
The post-data densities for the top three sires and the sire ranked lowest (ID460 15) are included in
Figures 2.4 - 2.7
-60-
Bayesian Method of Moments and the Mixed Linear Model
Consider the results of the BMOM. 2 analysis for the discussion. As might be expected, the best
two sires from the two analysis overlap, with the progeny from Sire 3 ranked 1st according to its
Traditional Bayes, BMOM and REML estimates. With an estimated breeding value of 3.4931, the
progeny from this sire will therefore have an estimated average weaning weight of 3.49 kilogram
more than the progeny from the rest of the sires. Also, the progeny from Sire 16 (ID49l34 and
ranked 2nd), with an estimated breeding value of 0.7950 will have an estimated average weaning
weight of 0.8 kilogram more than the average weaning weight of lambs from the rest of the sires.
The rest of the estimates can be interpreted in the same fashion.
Another appealing feature of the proposed simulation approaches to BMOM and Traditional
Bayes data analysis is that there are only minor disagreements on post-data rankings in the next
fifteen sires. In comparing the REML and BMOM estimates, the only difference in posterior
rankings is reported for Sire 7 (ID44042) and Sire 14 (ID48148). In the REML analysis, Sire 17 is
ranked 16th and Sire 14 ranked is", whereas a visa versa ranking is evident from the Traditional
Bayes and BMOM analysis. Once again this is not surprising to us, since as showed by Harville,
(1974) (see .also Searle, CaselIa and McCulloch, (1992» that when uniform or "flat" priors are
assigned to the vector of fixed effects and variance components and normal priors for the random
effects, the modes of the marginal posterior distributions are very close to the Traditional Bayes
estimates.
Thus, this sample provides adequate information to estimate any other quantities of interest, e.g.
we can also address the question of how well we can detect the best animal, as well as probability
distributions of rank positions for the top sires in the stud. Indeed, the estimates in Table 2.3 indicate
that there is minor uncertainty about the exact breeding value of individual sires, and it likely
indicates considerable certainty about the best selection.
-61-
Bayesian Method of Moments and the Mixed Linear Model
Figure 2.4 Estimated Marginal Post-data Densities for Sire 3 (ID41 019) ranked 1st according to
Traditional Bayes, BMOM and REML Estimates.
Figure 2.5 Estimated Marginal Post-data Densities for Sire 16 (ID49134) ranked 2nd according
to Traditional Bayes, BMOM and REML Estimates.
-62-
Bayesian Method of Moments and the Mixed Linear Model
Figure 2.6 Estimated Marginal Post-data Densities for Sire 1 (LD41037) ranked 3rd according to
Traditional Bayes, BMOM and REML Estimates.
Figure 2.7 Estimated Marginal Post-data Densities for Sire 10 (1046015) ranked 17'h according
to Traditional Bayes, BMOM and REML Estimates.
-63-
Bayesian Method of Moments and the Mixed Linear Model
2.6.4 Analysis of Fixed Effects
The object of interest in this present section may not only be the values of the fixed effects,
Table 2.4a - c, but also the post-data densities thereof. With respect to the values of the estimates,
we have previously demonstrated how these values must be interpreted numerically. Following the
results of the Traditional Bayes and BMOM analysis, we now focus upon the post-data densities of
the fixed effects that are depicted in Figures 2.8 - 2.11.
Table 2.4 Traditional Bayesian Estimates (a), Estimates from the BMOM 1 (b) and Estimated
from the BMOM 2 analysis (c).
(a)
Fixed Effect Trad. 95% Credibility
Bayes Interval
f30 22.9655 19.2315 ; 26.9031
f37 5.3523 4.1515; 6.5310
f3u 3.6690 2.9835 ; 4.3353
f3J5 9.4874 7.1923; 11.7688
f3J6 2.9621 0.6574; 5.2308
(b)
Fixed Effect BMOMl 95% Credibility
Interval
f30 23.0226 19.1496; 27.0333
f37 5.2995 4.1210; 6.3710
f3u 3.6757 3.0873 ; 4.3134
f3J5 9.5505 7.5584; 11.6362
f3J6 3.0469 1.0836; 5.0166
-64-
Bayesian Method of Moments and the Mixed Linear Model
(c)
Fixed Effect BMOM2 95% Credibility
Interval
/30 22.947 19.3359 ; 26.6482
/37 5.3037 4.1175 ; 6.4561
/3/. 3.6854 3.0037 ; 4.2923
/3/5 9.4580 7.2886 ; 11.5086
/3/6 2.9574 0.7831 ; 5.0853
An inspection of the post-data densities of the selected fixed effects shows that the difference
between traditional Bayesian and BMOM results can be quite substantial, especially in the case of
BMOM I. 'Since no likelihood was assumed for the BMOM analysis, these post-data densities
depended on the form of the maxent densities that in turn depended on the number of moments used.
We observed that by imposing only one side condition (exponential density with one moment), the
post-data density for BMOM I is more "spiked" than for traditional Bayes or BMOM 2. The double
exponential effect of the marginal post-data densities of the fixed effects in the case of BMOM I can
easily be recognized from Figures 2.8 - 2.11 below.
-65-
Bayesian Method of Moments and the Mixed Linear Model
Figure 2.8 Estimated Marginal Post-data Densities for the Expected Difference in Average
Weaning Weight between lambs born in 1949 and in 1950 as measured by {J7-
Figure 2.9 Estimated Marginal Post-data Densities for the Expected Difference 111 Average
Weaning Weight between male and female lambs as measured by {JI.-
-66-
Bayesian Method of Moments and the Mixed Linear Model
Figure 2.10 Estimated Marginal Post-data Densities for the Expected Difference In Average
Weaning Weight between single births and triplets as measured by /315.
Figure 2.11 Estimated Marginal Post-data Densities for the Expected Difference in Average
Weaning Weight between a pare of twins at birth and triplets as measured by /316'
-67-
Bayesian Method of Moments and the Mixed Linear Model
2.7 Chapter Summary
In this chapter we have indicated how to apply the Bayesian Method of Moments procedure in the
analysis of the mixed linear model when information in not available to formulate a likelihood
function. On introducing and proving simple assumptions relating to the moments of the realized
error terms and the future, as yet unobserved error terms, we derived post-data moments of
parameters and future values of the dependent variable. Using these moments as side conditions,
proper maxent densities for the model parameters were derived and could easily be computed for the
Dormer data set.
Further, it was evident that in the proposed BMOM approach, no sampling assumptions about the
given observed data were made. Rather certain assumptions were made about the realized error
terms' properties. Given these assumptions, posterior moments of the parameters were derived that
incorporated the information in the given data. These moments were then used as side conditions in
the derivation of maxent probability density functions for the parameters, realized error terms and
future observations.
It was also shown that in the computed example, where use is made of the Gibbs sampler to
compute finite sample post-data parameter densities, some BMOM maxent densities are very similar
to the traditional Bayesian densities, whilst others are not. As mentioned several times before, this is
expected, since as showed by Harville, (1974) (see also Searle, CaselIa and McCulloch, (1992)) that
when uniform or "flat" priors are assigned to the vectol: of fixed effects and variance components and
normal priors for the random effects, the modes of the marginal posterior distributions are very close
to the Traditional Bayes estimates.
-68-
Bayesian Method of Moments and the Mixed Linear Model
From the aforementioned, it should be appreciated that the BMOM approach yields useful inverse
inferences without using assurried likelihood functions, prior densities for their parameters and
Bayes' theorem. Hence, it is the case that the BMOM techniques extended in the present thesis to the
mixed linear model provide valuable and significant solutions in applying traditional likelihood or .
Bayesian analysis in animal breeding problems.
"Finally, the BMOM is elegant and easily applicable because it is free from the strong
UNVERIFIABLE assumptions that we usually make just in order to enable us to handle a problem."
© Parts of this chapter have been published in the South African Statistical Journal.
(See Van der Merwe et al. 2000)
© Parts of this chapter have been accepted for publication in the 'Collection of Refereed Articles' -ISBA20001•
(See Van der Merwe and Pretorius 200 I)
I International Society for Bayesian Analysis
-69-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
CHAPTER3
«The Dirichlet Process»
Introductory 'words: It is very important to accurately model the distribution of the random effects when
predictions of future observations from a given subject are desired. From the Bayesian perspective, inferential
interest in the present chapter focuses on the posterior distribution of the random effects. Allowing
distributions other than the normal for the random effects may more accurately model our prior beliefs, or it
may allow us to better express our uncertainty about the true distribution of the random effects.
3.1 .Prologue
In his '1972 review of Bayesian statistics, Dennis Lindley identified as a success story for
Bayesian ideas the advances made in problems of many parameters and the growth of what is now
referred to as Bayesian Hierarchical Modeling. He also identified non-parametries as an area
notable for lack of Bayesian progress, bemoaning the fact that non-parametric statistics was a
'subject about which the Bayesian method is embarrassingly silent'. However, it is undoubtedly the
case that the wide application of hierarchical models is one of the major success stories of modern
Bayesian statistics since the early nineties, with tremendous growth and substantial contributions via
Markov chain simulations on problems usually referred to as Non-Parametric Density Estimation.
Simultaneously, these computational methods allow development and application of data and prior
models that significantly extend the scope for closer representation of real-world problems.
-70-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Mixture priors, especially Dirichlet Mixtures have opened the way to serious Bayesian
developments in (so-called) Non-parametric Modeling and Density Estimation. It is my purpose in
this chapter to exhibit the Mixed Linear Model for non-parametric modeling and density estimation,
to show how posterior computations via Gibbs sampling simulations can be routinely applied, and to
provide illustrative examples from animal breeding problems. There has been some work towards
this end in the classical setting. In the Bayesian paradigm, it has been accomplished for the repeated
measures (West, Muller, & Escobar, 1994) and for the randomized complete block design (Bush &
MacEachern, 1996).
We provide a general framework for Bayesian analysis of mixed linear models in which a non-.
parametric Dirichlet process prior is specified for the random effects. Only recently have tools
allowing Bayesian analysis to become computationally feasible; here we provide a detailed
exploration of an animal breeding application of interest (see also Kleinman and Ibrahim, (1998)).
3.2 The Classical Perspective
From the classical perspective, the distribution of the random effects has an important effect on
some quantities of interest. Changing the distribution of the random effects will change the estimated
random effect for each individual. This point is important because there are many applications in
which an estimate of the random effect itself is desired. For example, in Tsiatis, DeGruttola, and
Wulfsohn (1995), the estimated random effects are used alternatively as covariates themselves in a
Cox regression model or to create values for time-varying covariates in such a model.
-71-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Similarly, Mori, Woodworth, and Woolson (1992), De Gruttola and Tu (1994), and Wu and
Carroll (1988) all present complex models in which the random effects are both estimated and used
in predicting other pieces of the model. In such applications, unbiased estimation of the random
effects is crucial and the assumption of normality may introduce bias (Kleinmann & Ibrahim, 1998).
Classical non-parametric and semi-parametric methods have a measure of popularity, e.g. the
Kaplan-Meier estimator, kernel density estimation, and Cox regression. No population distributional
assumptions are made in any of these cases, except for the proportional hazards assumption in the
case of Cox regression. We argue that a state of no knowledge at all is hardly, if ever, realistic: we
would typically at least have some ideas concerning location and spread. Such information can· be
incorporated into a Bayesian non-parametric prior.
3.3 The Bayesian Perspective
From the Bayesian perspective, inferential interest focuses on the posterior distribution of the
random and fixed effects. Allowing distributions other than the normal for the random effects, may
more accurately model our prior beliefs, or it may allow us to better express out uncertainty about the
true distribution of the random effects. It is also very important to accurately model the distribution
of the random effects when prediction for a future observation from a given subject is desired.
Another situation in which it would be desirable to relax the assumption of normality is when
inference is to be made about the distribution of the random effects itself.
Another attraction of our approach is that it allows exact Bayesian inference, even in small sample
sizes. This is accomplished through the use of the Gibbs sampler. Computational tools are
-72-
The Dirich/et Process and Non-parametric Mode/ing in Animal Breeding Experiments
developed and demonstrated how the Gibbs sampler can be implemented for the mixed linear model.
It is also showed how to make Bayesian inference for all of the model parameters in the model.
3.4 The Mixture of Dirichlet Process (MDP)
3.4.1 Background
MDP models have become increasingly popular for modeling when conventional parametric
models would impose unreasonably stiff constraints on the distributional assumptions. Examples
include empirical Bayes problems (Escobar, 1994), non-parametric regression (Muller, Erkanli &
West, 1996), density estimation (Escobar & West, 1995; Gasparini, 1996), hierarchical modeling
(MacEachern, 1994; West, Muller & Escobar, 1994) etc. Despite this large variety of applications,
the core of MDP models can basically be thought of as a simple Bayes model given by the likelihood
and prior.
Mixture of Dirichlet process priors can be of great importance in animal breeding experiments
especially in the case of undeclared preferential treatment of animals. As long as genetic evaluation
systems lack information about such preferential treatment, predictions of breeding values of favored
animals obtained with mixed Gaussian models are inflated whereas those of other animals are
deflated. So far, this problem has not yet been solved satisfactorily. Prof. Gianola, well-known
animal breeder, suggested a "robust" mixed effect linear model based on the t - distribution for
"preferential treatment problems". The t - distribution however does not cover departure form
symmetry whilst the Dirichlet process prior will be able to do so.
-73-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
3.4.2 The Model Structure
As mentioned before, an appropriate mixed linear model for a problem arising from animal
breeding experiments is given by
(3.1 )
y. is a n, x I vector of weaning weights for the progeny of the lh sire; f3 (p x I) is a vector of fixed
-I
effects uniquely defined so that the corresponding design matrix X; tn, x p) has full column rank, p.
Also, z, is a vector of n, elements I, y; is the unobservable random effect of sire i, and for the
unobservable vector of random residuals, G; (n, x I), it is common to assume a multivariate normal
distribution with mean vector Q and variance-covariance matrix a/In, where In, represents a n, x n,
identity matrix and a/ an unknown scalar (error variance) .
However, for Y, (q x I), the vector of unobservable random effects which is usually taken to be
normally distributed with mean zero, the normal prior is replaced with a non-parametric prior,
followed by a Dirichlet process prior on the general distribution. In the section to come, it is
illustrated how to apply the Mixture of Dirichlet Process Prior to the mixed linear model.
-74-
The Dirich/et Process and Non-parametric Mode/ing in Anima/ Breeding Experiments
3.4.3 The Dirichlet Process Prior in the case of the Mixed Linear Model
As in Kleinman and Ibrahim (1998) we will present a mixed linear model for which the random
effects have a non-parametric distribution. The non-parametric Bayesian approach for the random
effects is to specify a prior distribution on the space of all possible distribution functions. This prior
for the mixed linear model is applied to the general prior of the distribution of the random effects.
This can be accomplished with a Dirichlet process prior distribution. This means that the usual
normal prior on the random effects is replaced with a non-parametric prior, followed by a Oirichlet
prior on the general distribution. The foundation of this technology is discussed in Ferguson (1973),
where the Dirichlet process and its usefulness as a prior distribution are discussed. The practical
application of such models, using Gibbs sampling, has been pioneered by several researchers, e.g.
Doss (1994), MacEachem (1994), Escobar (1996), Lui (1996) and West et al. (1994).
Assume that G is sampled from a Dirichlet process with parameter Go and M, where Go is a
probability measure and M is a positive real constant, i.e.
G-DP(M' G,) (3.2)
The parameter Go , often called the base prior, is a location parameter for the Oirichlet process
prior and it approximates the true non-parametric shape of G. Thus, it is the best guess at what G is
believed to be and E(G) = Go. The role of Go for the Oirichlet process prior is similar to the role that
the median and mean play in the typical prior distribution; it is the location parameter. It is thus our
best guess of where the true values are. Therefore, if there are prior subjective beliefs, prior expert
opinions, or theoretical considerations that G belongs to a small, finite set of possible distributions,
then the prior distribution of Go should have support on this set. If the set of distributions is not
-75-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
finite, then. a finite subset of "typical" distributions that belong to this set could be chosen that
represents the larger set (Escobar, 1994). Because the algorithms developed in the present thesis will
average over the posterior distribution of Go, a natural smoothing occurs. Also, this Dirichlet process
prior is a third-stage prior in a hierarchical Bayesian structure. When estimating normal means, it is
common to assume that G is a normal distribution with unknown mean and variance. We will let Go
be a normal distribution and use the data to estimate the model parameters.
The parameter M, a type of dispersion parameter for the Dirichlet process prior, is a measure of
the strength in the belief that G is Go. Although it may be hard to quantify, M is a positive scalar that
is related to how "clumpy" the data are (often called a precision parameter). Clumpy data occur
when the different sires are concentrated into a few clusters. In practice it is difficult to select
appropriate values for this parameter. Instead, it is suggested to place a prior distribution on this
parameter, and simulate it given the data. West (1992) assumed that M - Gaïa.b) a gamma prior
with a > 0 and scale b > O. We may extend this idea to include a reference prior (uniform for
log(M) by letting a~ 0 and b~ O. In the final section of the chapter we use the latter, which means
that p(M) cx M-I and M> O.
To simplify the use of the Dirichlet process prior, note that when G is integrated over its prior
distribution, the sequence of Yi follows a general Polya urn scheme; that is
(i = }, ...,q), (3.3)
= r , with probability
r, M +q-lI YI""'Yq-1 (3.4)
-G Mwith probability0 M+q-l
-76-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
From (3.4) it is easy to sample a sequence of Yl, Y2, ... , Yq given Go and M. There are two special
cases in which the mixture ofDirichlet process models lead to the fully parametric case. As M ~ co,
G ~ Go so that the base prior is the prior distribution of the random effects. Also, if Yi == Yj for
all i, the same is true. When G is fully parametric, the joint posterior can easily be found. For the
implementation of the Dirichlet process prior, the different Yi 's must be considered separately. We
find that the conditional posterior of Yi is given by
q
P(Yi 1j3,a;,a:'Y_i,M)oc I~(~IiXij3+ziYj,a;In)·5Yj
+ {M ljl(L IXJ3 +z,y"a;I")jl(y, IQ,a;)dY,} (3.5)
x ~(Yi I O,a: )p(L I Yi,j3,a; ,~))
with mean J..i and variance d. Also, Y -i denotes the vector of random effects for the sires excluding
sire i and 5.\. is a degenerated distribution with point mass at s.
Consider the integral
00
Ai =M J ~(~i IXij3+ziYi,a;Ini~(Yi IO,a:)dYi
-00
[ --21y-" J~
2
exp{---Y2i }' dYi'
2a z«;
-77-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
2
The exponent of the integral is -21 ~a Y - X,/3 - ZiYi
)~Y,) - X,/3 - ZiYi + -r,
1 1 a2 ' This can be written
c r
as
2
= -,1 _ ,- 2 1 -, 1 2, r ,Y Y - -2 Y ZiYi + -2 r, Zi z, +-2
.0'.; _I _I a" _I ac ar
where
y =Y -XJ3.
-I _I
Following usual algebra routes, i.e. completing the square with respect to Yi' we find
=M 2;r -~(2I+II ) ( 2 )-~ ( 2 )_!l {I( ) i ar 2 ac 2 exp -2 (y -X,/J)'(/>i(Y -XJ3) }[-21 (Zi'Z;)+-, 1 ]-~ (2;r)2~
. 20' _I -, a a-
eer
(3.7)
where
-78-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Thus, after implementing the results in (3.7) into (3;6), we get
1 1 n
M nl (a:)-ï (a; t"t x r!J(y, I O,a: )p(~; I r.. ,13, a;, L)·
(3.8)
In the above specification, each summand in the conditional posterior of y; is separated into two
elements. The first element is a mixing probability, and the second is a distribution to be mixed. So
with probability proportional to
(3.9)
q
A; + Lr!J(L I X,p + z;y};a;In)
}=I;}'1';
we select y; from distribution bYl' which means that we set y; = y}. Also, with probability
proportional to
(3.10)
q
A; + Lr!J(~; IX;p + z;y};a;In)
}=I;}'1';
we select y; from
(3.11 )
-79-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Thus, we sample Y, from its full conditional posterior,
This results in a mixture distribution where one piece is a normal distribution and all of the others are
point masses. There is some plausible intuition behind this above mixture scheme. If the breeding
value, Y i of sire i has a relatively large residual using sire j's breeding value, then Y j is relatively
less likely to be chosen as the breeding value of sire i. Conversely, if the breeding value of sire i has
a relatively small residual using sire j's breeding value, then the random effect Y j is relatively more
likely to be chosen as the breeding value of sire i. On the other hand, the greater the residual for sire
i, the greater the probability that sire iwill get a new value from pc· , .) in (3.12).
This scheme results in what MacEachem (1994) calls a cluster structure among the different sires.
This cluster structure partitions the q different sires into k groups, where 0 < k ~ q. Thus, all the sires
in .a specific cluster will have identical breeding values and sires in different clusters will have
different breeding values. This may sound farfetched, but since the Gibbs sampler is repeated several
times, the algorithm leads to reduced variation and hence faster convergence of the estimated random
effects- to ·their true values. The average of the simulated values for each breeding value is then
computed, thus every sire will have its own unique breeding value.
The fully conditional posterior density for each of the other unknowns is obtained by regarding all
other parameters in the joint posterior as known.
-80-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
3.4.4 The Uniform Prior for pand a/
The full conditionals for jJ and CY,} in the non-parametric model (3.1) are the same as in the
paramétric model. Thus, an uniform prior distribution is assigned to jJ and CY/ as to represent lack of
prior knowledge about the vector of fixed effects and error variance. Therefore
p(jJ ,CY,}) = p(j3) p (cy,}) cc constant. (3.13)
The required full conditional for the fixed effects, is multivariate normal:
(3.14)
For the variance component, CY/ the conditional posterior is
n,
pea; I p,y,"f_ ) = KcI)(;2J2 exp{- 2~2 ("f_;-X;P-Z;Y)'("f_; -X;/J-Z;Y)}
. I-I c c
(3.15)
an Inverse Gamma density where
-81-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
n-2
Iq(~, 2-X;/3-Z,Y)'(L -X;/3-z;y)
1
K e = 2
q
Also, y=(YI,Y2, ···,Yq), Y = (y ',y ', ...,y-q ')' and In; = n, the sample size._ _I _2
;=1
3.4.5 Prior for a/
Typically, the variance a/ in the base measure of the Dirichlet process in (3.2) is unknown and
therefore a suitable prior distribution must be specified for it. Note that, once this has been
accomplished the base measure is no longer marginally normal. For convenience, suppose
p (a/ ) cx. constant
to present lack of prior knowledge about a/. After choosing random effects for each of the sires, the
sires will be grouped into clusters (groups) in which the sires have equal y; 'so That is, after
selecting a new y; for each sire i in the sample, there will be some number k, 0 < k ~ q, of unique
values among the random Y; 'So Denote these unique values by A/ , 1=1, ... ,k. Additionally let 1
represent the set of sires with common random effect ,.1,/.
Note that knowing the random effects is equivalent to knowing k, all of the A/ 's and the cluster
membership I. Then for the purposes of calculating the full conditional of a/ , the A/ are k
independent 'observations from N(O, a/Jo
-82-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Thus
(3.16)
an Inverse Gamma density where
k-2
K, ~{ ;[~;[}' r(T J
and
Bush and MacEachern (1996) and Kleinman and Ibrahim (1998) recommended one additional
piece of the model as an aid to convergence for the Gibbs sampler. To speed mixing over the entire
parameter space, they suggest moving around the A's after determining how the Yi 's are grouped.
Thus, in addition, a conditional posterior density is derived for the A's, i.e.
p(A, I /],O';,o':,~) o: rjJ(A,IO,O':)Ilp(~i 1/],0';) (3.17)
id
which implies that
-83-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
(3.18)
where'
(3.19)
This additional piece is now incorporated into the final Gibbs sampler. Before the Gibbs sampler
is presented, we first address the simulation of the precision parameter, M. In the preceding section
we assumed that this parameter for the Oirichlet process prior was fixed. In practice it is difficult to
select appropriate values for M. Instead, a prior distribution is placed on M and a posterior
distribution is derived. Because this parameter has an important influence on the estimation, special
care is going into the selection of a broad range of values for M and into the simulation thereof.
3.4.6 Simulation of the Precision Parameter M
When defining a Oirichlet process prior, recall that M represents the weight of our believe that G
is the distribution of Go. This parameter thus determines. the prior distribution of k, the number of
additional normal components in the mixture, and is a critical smoothing parameter of the mixed
linear model. M is also, as mentioned before, related to how "clumpy" the data are. When there are
only a few clusters among the sires in the model, the estimate of the normal means from the Oirichlet
process prior will be similar to the non-parametric Bayes estimator. When there are almost q
(random effects) different clusters, the estimator from the Oirichlet process prior will be similar to
the parametric Bayes estimator. Thus, the parameter M adjusts this estimator to behave like either a
parametric estimator, which uses the data in a global manner, or a non-parametric estimator, which
-84-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
uses the data in a local manner. In Antoniak (1974) it is shown that the prior distribution of k, the
number of clusters, may be written as
pek I M,q) = cq(k)q!Mk reM) k = 1,2,...,q (3.20)
reM +q)
and clk) = p{k I M = l.q), not involving M. West (1992) mentioned that if required, the factors cq{k)
can easily be computed using recurrence formulae for Stirling numbers. It is also shown that the
conditional posterior distribution of M is given by
p(M I k,fJ,r,a; ,a: ,y) = p(M I k) a: p(M)p(k IM) (3.21 )
where p(M) is the prior and the likelihood function is defined in (3.20). West (1992) also assumed
M - Gaïa.b), a Gamma prior with shape a > 0 and scale b > 0 (which we may extend to include a
reference prior, Uniform for 10g(M), by letting a 4 0 and b 4 O. In this section we will use the
latter, which means that
p(M) cc M-I M>O. (3.22)
Equation (3.21) can be expressed as a mixture of two gamma posteriors, and the conditional
distribution of the mixing parameter, x given M and k is a simple beta. This can be illustrated as
follows. For M> 0, the gamma functions in (3.20)can·be written as
_ r_e_M;_) --'--= -(,M--_+...q:.).B:.e..(._M:.._+__l,.q:.)....:....:.. (3.23)
reM +q) Mf(q)
-85-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
where Bee,.) is the usual beta function. Then in (3.21) and for any k= 1,2, ... ,q the posterior of M for
k, is
p(M I k) cx: p(M)Mk-I(M +q)Be(M +I,q)
I (3.24)
cx: Mk-2(M + q) JXM(1- X)q-Idx,
o
using the definition of the beta function.
From (3.24) it also follows that the joint posterior density of M and x is
p(M,x I k) cx: Mk-2(M + qix" (1- xtl, 0< M, 0 < x < 1
and the conditional posteriors p(M I x,k) and p(x I M,k) can be determined as follows.
Firstly
p(M I x,k) cx: Mk-2 (M + q)exp{- M(log(x))},
cx: Mk-I exp{-M(log(x))}+qMk-2 exp{-M(-log(x)} M> 0 (3.25)
which reduces easily to a mixture of two gamma densities, viz.
'.
:M I x, k - 1r.rGa(k,-log(x)) + (I-1rJGa(k -I,-log(x)) (3.26)
with weights 1[x defined by ~ = k -1 . Also note that log(x) = logÁx) = In(x).
I-1r... q(-log(x))
-86-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
Secondly
p(x I M,k) o: XM (1- X)q-I 0< x < 1 (3.27)
M+1
so that x] M,k - Be(M+ I.q), a beta distribution with mean ----
M+q+1
It should now be clear how M could be sampled at each stage of the simulation. Hence, at each
Gibbs iteration, the currently sampled values of M and k allow us to draw a new value of M by first
sampling an x value from the simple beta distribution in (3.27), conditional on M and k, both fixed at
the most recent values; then M is sampled from the mixture of gammas in (3.26) based on the same k
and the x value just generated. On completion of the simulation, we will have a series of sampled
values of M, k, x and all the other parameters. Note that only the sampled values k and x are needed
in estimating the posterior p(M ly) via the usual Monte Carlo average of conditional posteriors,
VIZ.
N
p(M I y) == N-1 LP(M I X(i) ,k(i)) (3.28)
i=1
where the summands are simply the conditional gamma mixtures in (3.26).
To calculate the posterior distribution of all the model parameters, we developed an important
Gibbs sampler algorithm for simulation. On completion of the simulation, we will have a series of
sampled values for all the model parameters. The next section illustrates an application of the most
recent developed Gibbs sampler for an animal breeding experiment.
-87-
The Dirichlet Process and Non-parametric Modeling in Animal Breeding Experiments
3.4.7 The Gibbs Sampler
Markov chain Monte Carlo methods, particularly Gibbs sampling, are now very often used in the
thesis and once again the model described in paragraph 3.4.2 can be implemented through this
sampling technique. As usual in Gibbs sampling, we identify collections of complete conditional
posterior distributions that determine the marginal posteriors for all the parameters. Hence, the Gibbs
sampler for pep, a; ,y, a: ,M ! ~) can be described as follows.
(0) Select starting values for r'" and a;(i) . Set i = 0
(1) Sample,8 (i./) from p(P I r'" ,a;(i) ,~) according to (3.14)
(2) Sample a;(i+I) from p( a; ! p(i+ll, r'" ,~) according to (3.15)
(3 .1) Sample Yl {i+l] tirom P (!Yl /3(i+I).o;2(i+l),ar'2Y-I' (i) M),~ accor d'mg to (3.9) or
(3.10)
(3.2)
(3.q) SampeI Yq {i+l] tirompY (I /3(i+l).a;2(i+l),ar,2y_ , (i) M)q q ,y accormdigto(3.)or 9
(3.10)
(4.2)
(4.k) Sample "A)H) from peAk ! /3(i+I),a;(i+I), a:(iO ,y) according to (3.18)
(5) Sample a: fromp(a: !A(i+I),y) accordingto(3.16)
(6.1) Sample X(i+l) from p(x Iu», kei») according to (3.27)
(6.2) Sample M(i+I) from p(M I X(i+I), kei») according to (3.26)
(7) Seti=i+l andretumto(l)
-88-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
3.5 An Example
3.5.1 The Data
The example used for illustrative purposes are based on an experiment undertaken at the
International Livestock Research Institute (!LR!) at the University of Nairobi, Kenya in the early
90's (Duchateau, et al., 1998). The data are shown in APPENDIX C. The goal of the research was
to select for improved Helminth resistance in sheep .
.The female sheep used in the experiment are from three different breeds, whereas the males are
from two breeds. In each of the six crosses, there are at least 25 and at most 42 different sires, and
each sire within a crossbreed has on average offspring of 6.4 lambs. The weaning weight is
measured for each lamb. The age at which lambs are weaned may differ from animal to animal, and
therefore, a variable expressing the age of the animal at weaning is included as a fixed effect as well
as the sex of the lambs. Finally, the sires are included as random effects.
Although the same sire is mated to ewes from different breeds, the sire nested in breed is taken as
a single random effect Yi and it is assumed that these random effects are independent. A total of
n = 1277 weaning weight records, from the progeny of q = 200 sires are available after editing, and
breed, sex and age are included as fixed effects in the final model.
-89-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
The mixed linear model used for this data structure, is the SIre model of section (1.2),
Y = XfJ + Zy + e , where y (1277 x l ) vector of weaning weights. jJ (8 xl) is the vector of fixed
effects, and the design matrix Z, a (1277 x 200) matrix identifying the (200 x I) vector of random
effects consisting of the breeding values for the 200 sires for which the data is observed, X a
(1277 x 8) matrix of full rank.
MATLAB software has once again been developed to generate the samples that enable us to
obtain the finite sample post-data parameter densities, using the Gibbs sampler. The full conditional
posteriors are updated after every iteration. The first 1 000 draws of each chain are discarded, and
then every 10th draw is saved. By saving every to" draw, the chain yielded a posterior sample of
1 000 approximately uncorrelated draws. All posterior analyses are based on these m = I 000 draws.
3.5.2 Analysis of Variance Components
Posterior modes of the Traditional Bayesian analysis, 95% credibility intervals, and the REML
estimates are summarized in Table 3.1. Note that the REML point estimates of a/ and a/ and the
posterior modes obtained from the Gibbs sampler (Traditional Bayes) do not differ much. This was
because we assigned uniform or "flat" priors to the vector of fixed effects and variance components,
and a normal prior to the vector of random effects (Harville, 1974; Searle, CaselIa & McCulloch,
1992).
-90-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Table 3.1 REML and Traditional Bayesian Estimates (posterior modes) of the Variance
Components, along with 95% Credibility Intervals.
95 % Credibility
Parameters REML Trad. Bayes Interval
a/ 4.8639 4.8885 4.4617 ; 5.3059
a2r 0.6802 0.7211 0.4312; 1.0496
In practice it is difficult to select appropriate values for the parameter, M Recall that M is a
positive scalar that is related to how "clumpy" the data are (often called a precision parameter), and
clumpy data occur when the different sires are concentrated into a few clusters. Because the
parameter value has an important influence on the estimation, a broad range of possible fixed values
for M is chosen, i.e. M = 5, 50, 100 and 1000. Furthermore, a prior distribution is placed on M and
values from the estimated posterior distribution of M are used in the simulations. The results are
summarized in Tables 3.2 and 3.3 below.
Table 3.2 Posterior Estimates of the Error Variance Component, a/ (different values of M),
along with 95% Credibility Intervals.
M Posterior 95 % Credibility
Mode Interval
5 4.9183 4.4788; 5.3612
50 4.8907 4.4649; 5.3286
100 4.8615 4.4192 ; 5.2731
1000 4.8743 4.4480; 5.3107
SimM 4.8667 4.4527; 5.2810
-91-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Table 3.2 Posterior Estimates of the Model Variance Component, er} (different values of M),
along with 95% Credibility Intervals.
M Posterior Posterior 95 % Credibility.
Mode Mean Interval
5 0.6310 0.8835 0.4213 ; 1.7733
50 0.6090 0.8659 0.4407 ; 1.0276
100 0.6290 0.8859 0.4407 ; 1.0276
1000 0.7275 0.7320 0.4213; 1.0476
SimM 0.6340 0.6532 0.3790; 1.0026
For large values of M, the "Oirichiet" estimates coincide with the Traditional Bayes and REML
estimates. In these cases there are almost 200 different clusters/groups among the different sires,
resulting in a similar behaviour of the estimates from the Oirichlet process prior and the Traditional
Bayes procedure.
Using the posterior densities for er/ and er/, the marginal posterior densities are estimated as the
average of the posterior densities and are displayed in Figures 3.1 and 3.2. Also, the distributions in
Figure 3.2 are quite skew, resulting in a difference between the posterior means and posterior modes.
The density for M = 100 is omitted from figure 3.1 since the estimated marginal density for this value
of M and the density obtained from the Traditional Bayes analysis are basically the same.
-92-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
- - Trad. Bayes
- SimM
Figure 3.1· Estimated Marginal Posterior Densities of the Variance Component, a/ for different
cases of M, (Sim M); Traditional Bayes and when Mis fixed at M= 5.
-. - Trad. Bayes
- SimM
Figure 3.2 Estimated Marginal Posterior Densities of the Variance Component (7/ .
-93-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
The similarity of the a/ results in Tables 3.1 and 3.2 is an indication, for this data set, that the
results are not sensitive to the choice of the precision parameter, M From the marginal posterior
densities of a/ in Figure 3.1 the effect of changing M on the distribution of the random effects is also
minimal. These densities are virtually identical, and it is clear that there is no uncertainty about the
exact location and height in the different densities. However, the posterior density for a/ when M is
fixed at 5 (Figure 3.2) shows some uncertainty in the shape of the density. This can be expected
because M is part of the prior for the random effects and will therefore influence a/ much more than
a/. This may also be related to the large variation in the importance sampling weights for smaller
values of M The other densities have similar shapes with only noticeable shifts in the posterior
modes.
The posterior means and modes of the Traditional Bayesian analysis; different values for M, and
95% credibility intervals of functions of variance components like the intraclass correlation
a2 a2
coefficient, p = " 2' and the variance ratio, v = _, are summarized in Tables 3.3 and 3.4.
a; +aó a;
Once again it is evident from this table that the credibility interval for the intraclass correlation
coefficient does not contain 0.5. This result corresponds well with the statement made by Wang et
al. (1992) namely that from a genetic point of view, an intraclass correlation coefficient of 0.5 is not
possible in a sire model.
Using again the conditional posterior densities for these functions of the variance components, the
marginal posterior densities are estimated as the average of the conditional posterior densities and are
displayed in Figures 3.3 and 3.4.
-94-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
(J2
Table 3.3 Estimates of p = 2 y 2' the Intraclass Correlation Coefficient for different cases
(Jy + (J&
of M, Simulated M and Traditional Bayes Results, along with 95% Credibility
Intervals.
M Posterior Posterior 95% Credibility
Mode Mean Interval
5 0.114 0.1497 0.0765 ; 0.2677
100 0.115 0.1189 0.0696 ; 0.1706
SimM 0.120 0.1179 0.0710; 0.1723
Trad. Bayes 0.130 0.1282 0.0779 ; 0.1807
;.::", ..'->,
jo:': '.
-- Trad Bayes
- SimM
Figure 3.3 The Estimated Marginal Posterior Density of the Intraclass Correlation
(J2
Coefficient, p = 2 Y 2.
(Jy + (J&
-95-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
= -a
2
Table 3.4 Estimates of v --T' a Function of the Variance Components, along with 95%
aE
Credibility Intervals.
M Posterior Posterior 95% Credibility
Mode Mean Interval
5 0.130 0.1803 0.0829 ; 0.3655
100 0,135 0.l359 0.0748; 0.2057
SimM 0.135 0.l347 0.0765 ; 0.2082
Trad. Bayes 0.l45 0.1481 0.0844 ; 0.2205
-- Trad. Bayes
- SimM
a2
Figure 3.4 The Estimated Marginal Posterior Density of the Variance Ratio, v = ---T .
aE
-96-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
The same conclusion can be drawn for the posterior distributions of the intraclass correlation
coefficient, p and variance ratio, v (which are functions of the variance components) as for the
variance components, i.e. more or less identical marginal posterior densities except for small values
of M.
Another commonly derived statistic, the heritability (h2) of the trait, which is also function of the
two variance components is calculated and reported in Table 3.5. This statistic describes the
proportion of the total variation in the environment of the study attributable to genetics. In this
formula h 2 = 4a: 22 2 ' ay is multiplied by 4 in the numerator to account for the fact that lambs
ay +ac
from the same sire are half siblings and the sire accounts for half of the inherited genetic component,
and a: +a; is the phenotypic variance. The higher the heritability, which lies between 0 and 1,
the hig.her .the proportion of the total variation that can be assumed .to be genetic in origin.
4a2
Table 3.4 REML and Bayesian Estimates of h2 = 2 Y 2 ,the heritability of the trait.
ay +ac
REML Trad Bayes M=5 M=100 SimM
0.49 0.5127 0.4756 0.5179 0.5097
Some caution is needed in variance component problems in genetics. For example, some genetic
models dictate bounds for a particular variable. If one employs the Sire model, as in the case of the
present thesis, the intraclass correlation coefficient must lie inside the [0, .!.. ] interval, because
4
-97-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
. a2 . 1
heritability is between 0 and I. This implies that the variance ratio v = -T IS between [0, -], and
aE 3
a2
that 0 ~ a: ~ _E • These profound concerns are evident in the above example, except for the results
3
of the variance components and functions thereof when M is set to small values (M = 5), but this is
due to some uncertainty about the real distribution of the posterior densities.
3.5~3 Analysis of Random Effects
As mentioned before (see section 3.4) the non-parametric Bayesian approach for the random
effects is to specify a prior distribution on the space of all possible distribution functions. This prior
for the mixed linear model is applied for the general prior of the distribution of the random effects.
This can be accomplished with a Dirichlet process prior distribution. This means that the usual
normal prior on the random effects is replaced with a non-parametric prior, followed by a Dirichlet
prior with precision parameter M, on the general distribution. Because this precision parameter value
has an. important influence on the estimation, a broad range of possible fixed values for Mis chosen
(M = 5, 50, 100 and 1000) to reflect small, moderate and large departures from normality in the case
of the random effects. Furthermore, a prior distribution is placed on M and the conditional posterior
distribution of M becomes part of the Gibbs sampler.
Rather than report results for all 200 sires, we focus our discussion on the first 10 sires in the
analysis, The tables referred to in this section contain only these 10 animals. The rest of the results
are summarized in APPENDIX E.' Since animal breeders might be interested in the breeding value of
specific sires in order to determine which sires should be retained for future selection, we used these
breeding values to find the REML-, Traditional Bayes-, and Dirichlet process ranks of the different
-98-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
sires. The results are summarized in Table 3.4 and 3.5, along with 95% credibility intervals.
Moreover, the REML estimates represent the mode of the marginal likelihood and thus might be
better compared to the modes of the posterior distributions of the random effects. These modes
(breeding values) are provided in the aforementioned tables.
Table 3.4 REML Estimates and their Standard Errors (SE's), Traditional Bayes Estimates along
with 95% Credibility Intervals and Posterior Rankings of the first 100f the 200 Sires.
REML Posterior Trad. Bayes 95% Credibility Posterior
SIRE_ID Estimate SE Rank Estimate Interval Rank
1971 -0.1061 0.6654 9 -0.1024 -1.4569 , l.l898 9
1972 0.5241 0.5349 5 0.5689 -0.4500 , 1.5455 5
1973 0.3888 0.6399 ·6 0.4464 -0.9447 , 1.6482 6
1974 1.9339 0.5486 1 1.9627 0.7862 ; 3.1311 1
1980 0.9299 0.5975 3 0.9635 -0.2615 , 2.1269 4
1991 0.3611 0.6396 7 0.3741 -1.0761 , 1.6095 7
1999 0.9266 0.6654 4 0.9939 -0.2873 , 2.2231 3
4907 0.2289 0.5790 8 0.2676 -0.8033 , 1.4348 8
4908 1.6614 0.4921 2 1.6662 0.5431 ; 2.8049 2
4909 -0.7628 0.6653 10 -0.7645 -2.1227 ; 0.4956 10
Note that the REML estimates and the posterior means (see Pretorius and Van der Merwe, 2000;
APPENDIX E) obtained from the Gibbs sampler for M = 5 are quite different. There are two factors
to keep in mind when examining this difference. Firstly, there is considerable uncertainty about the
distribution and central values of the breeding values when M = 5. Secondly, as mentioned earlier,
the REML estimate of the breeding value is more similar to the mode of the posterior distribution
than the mean. Figures 3.5 - 3.16 show the posterior distributions of the first 3 sires in the data set,
where it is evident that the posterior modes obtained from the Gibbs sampler are in fact similar to the
REML estimates of the breeding values.
-99-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Table 3.5 Oirichlet Process Estimates for different values of M, along with 95% Credibility
Intervals and Posterior Rankings of the first lOof the 200 Sires.
M=5 95% Credibility Posterior SimM 95% Credibility Posterior
SIRE ID Modes interval Rank Modes Interval Rank
1971 -0.0125 -1.3874 , 1.2378 9 -0.1097 -1.4280 , 1.160 I 9
1972 0.2510 -1.3396 , 1.4029 7 0.5081 -0.5712 ; 1.5426 5
1973 0.3580 -1.3874 , 1.5452 4 0.3934 -0.9254 ; 1.7685 6
1974 1.4265 0.3051 ; 2.6513 I 1.9026 0.8062 ; 3.1085 I
1980 1.0350 -1.2272 ; 2.1293 3 0.9193 -0.2371 , 2.1947 3
1991 0.3420 -1.3396; 1.4029 5 0.3438 -0.8887 , 1.6423 7
1999 0.3224 -1.2685 ; 2.4430 6 0.8936 -0.4606 , 2.2582 4
4907 0.1910 -1.3874 ; 1.2378 8 0.2293 -0.8966 , 1.3925 8
490:8 1.1420 0.2435 ; 2.6513 2 1.6490 0.6079 ; 2.7037 2
4909 -1.2825 -1.8916 ; 0.3827 10 -0.7711 -1.9938 ; 0.4418 10
Uncertainty in the values of the breeding values when M equals 5 is indicated in the large
posterior credibility intervals. Thus a wide range of values for the breeding values is quite possible.
However, as might be expected, the best sires from the REML, Traditional Bayes, and Oirichlet
process analyses (Sim M) overlap, with a minor disagreement between rankings of SireID 1980 and
Sirell) 1999 (ranked visa versa in the Traditional Bayes analysis).
Beyond the top three sires (according to the Oirichlet process when M was fixed at 5), there are
major differences in the order of the best to the worst sire. This is because of the great deal of
uncertainty about the values of the breeding values. This uncertainty is also reflected in the wide
credibility intervals. Note that the estimated breeding values take account of the variability in the sire
variance, a/ depicted in Figure 3.2.
-100-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Taking account of this variability can be important in evaluating the breeding potential of the
animals. This is a clear example of why it is important to correctly model the distribution of the
random effects; very different results may be obtained as a result of changing the precision
parameter, M.
To -further illustrate the uncertainty in some of the posterior densities, we have calculated the
marginal posterior densities of the breeding values for the first 3 sires in Figures 3.5 - 3.16.
(SireIDI972, SireIDl973 and SireIDI974). In these figures we plotted: the densities when M is
fixed at 5 and 50, the densities obtained from the Traditional Bayes analysis, and densities when M is
...,. simulated from a mixture of distributions in the Oirichlet process, given the data. We noted that
j these posterior densities have many features of interest.
,,~
The' next section is devoted to the overriding influence of the precision parameter M on the
~.
Jj\f posterior densities of the 200 breeding values of the different sires. M is, as mentioned before,
~
related to how "clurnpy" the data are. When there are only a few Clusters among the sires in the
model, the estimate of the normal means from the Oirichlet process prior will be similar to the non-
parametric Bayes estimator. When there are almost 200 different clusters, the estimator from the
Oirichlet process prior will be similar to the parametric Bayes estimator.
-101-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
0.6
0.5
0.4
0.3
0.2
0.1
O.OL-----~~--------~--------~------------~----------~
-3 -2 -1 o 2 3
Figure 3.5 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID 1971·
(Yl) when M= 5 in the Dirichlet Process.
0.6
0.5
0.4
0.3
0.2
0.1
0.0L-__ =~ ~ --=:::::" ~
-3 -2 -1 0 . . 2 3
Figure ·3.6 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1971
(Yl) when M= 50 in the Dirichlet Process.
-102-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
0.6
0.5
0.4
0.3
0.2
0.1
o.o~----~----~~------------o~------------~~----~------~-4 -3 -2 -1 2 3 4
Figur~ 3.7 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1971
(Yl) from the Traditional Bayes Analysis.
0.6
0.5
0.4
0.3
0.2
0.1
o.o~----~----~~------~-----o -------------~~------------~-4 -3 -2 -1 2 3 4
Figure 3.8 The Estimated Marginal Posterior Density of the Breeding Value for Sire lD1971
(Yl) when M is Simulated in the Dirichlet Process.
-103-
The Dirichlet Process and Non-parametric Model/ing in Animal Breeding Experiments
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-1.5 -0.5 0.5 1.5 2.5
Figure 3.9 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID 1972
(Y2) when M = 5 in the Dirichlet Process.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
o.o~------~~-------------o-----------------------~--------~-2 -1 2 3
Figure 3.10 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1972
(Y2) when M = 50 in the Dirichlet Process.
-104-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
0.8
0.6
0.4
0.2
o.o~---=~----------~------~----~----------~~--------~
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.11 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID 1972
(Y2) from the Traditional Bayes Analysis.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-1.0 -O.~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.12 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1972
(Y2) when M is simulated in the Dirichlet Process.
-105-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
0.6
0.5
0.4
0.3
0.2
0.1
O.o·~----~----------~------------------~------~~------~
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
Figure 3.13 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1973
(Y3) when M = 5 in the Dirichlet Process.
0.6
0.5
0.4
0.3
0.2
0.1
o.o~-----=~--------------------------~----~--~------~
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
Figure 3.14 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1973
(13) when M = 50 in the Dirichlet Process.
-106-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
0.7r---------------~------~------~------------------------~
0.6
0.5
0.4
0.3
0.2
0.1
o.o~---------=~--~------~o------~----------~=-----------~-3 -2 -1 2 3 4
Figure 3.15 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1973
(r3) from the Traditional Bayes Analysis.
0.7r---------------~------~------~------------------------~
Figure 3.16 The Estimated Marginal Posterior Density of the Breeding Value for Sire ID1973
(r3) when M is Simulated in the Dirichlet Process.
-107-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Unlike the densities of the fixed effects, the values of Mhave a large effect on the posterior
densities of the random effects, since the random effects are directly affected by the relaxation of the
normal assumption when M is set equal to 5 and 50. While for M = 50 it is clear that there are some
uncertainty about the shape and boundaries of the densities. This is even more pronounced with
M = 5, and may be related to the larger variation in the importance sampling weights for smaller M.
We also note from these first two densities that as M increases, the shapes of the densities tend to
become more bell-shaped and symmetrical. Thus, a value of M = 5 reflects a large departure from
normality in the posterior density of the breeding value.
For smaller values of M the sires are grouped into less clusters, with the average number ·of
clusters, k = 124 when M = 5. Furthermore, a value of M = 50 reflects a moderate to small
departure from normality, with k = 145. Thus, for small values of M, the estimates of the normal
means. from the Dirichlet process prior are similar to the values of the non-parametric Bayes
estimates.
However, when M = 1000 the estimator from the Dirichlet process prior is similar to the
parametric Bayes estimator and the density reveals no departure from normality (Figures 3.7, 3.11
and 3.15), with k = 196. The estimated marginal posterior density for M given k = 196 is
presented in Figure 3.1 7.
-108-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.17 Estimated Marginal Posterior Density of M with k = 196, and Posterior Mode
M; = 820.
When M is simulated, given the data, the average number of clusters, k = 140 with M = 220.
The estimated marginal posterior density and the unconditional marginal posterior density for the
simulated Mvalues are displayed in Figures 3.18 and 3.19. Finally, the observed histograms for the
number of clusters, k for different values of M are presented in Figures 3.20 - 3.22.
-109-
The Dirich/et Process and Non-parametric Modelling in Anima/ Breeding Experiments
Figure 3.18 Estimated Marginal Posterior Density of M with k = 140, M= 220 and Posterior
Mode M; = 200.
Figure 3.19 Estimated Unconditional Marginal Posterior Density of M with k = 140,
M = 220 and Posterior Mode M; = 320.
-110-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.20 Observed Histogram for the Number of Clusters, k when M= 5;
k = 124
Figure 3.21 Observed Histogram for the Number of Clusters, k when M = 1000;
k =196
-111-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.22 Observed Histogram for the Number of Clusters, k when M is Simulated from a
Mixture of Distributions, given the Data; k = 140 .
A substantial statistical issue that remains to be tackled is the great discrepancy between pictures
of the posterior densities of the random effects as the value of M changes. Indeed, if data are very
sparse and not very clumpy, then non-parametric maximum likelihood methods may not work very
well. But if the data are very clumpy, with modes that are spread out, then standard parametric
Bayes methods do not work very well, and non-parametric Bayes methods work quite well. The
question can be asked "why non-parametrics?" According to Walker et al., (1999), the answer
depends on the particular problem and the procedure under consideration, but many, if not most
statisticians appear to feel that it is desirable in many contexts to make fewer assumptions about the
underlying population from whichthe data are obtained than are required from a parametric analysis.
Also, the mixture of Dirichlet process priors will cover departures from symmetry and cases where
the assumptions of unimodality do not hold for the random effects in the mixed linear model.
-112-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
3.5.4 Analysis of Fixed Effeets
The estimates for the different fixed effects are summarized in Tables 3.6 and 3.7. The results
obtained when M is fixed at 5 and 500, along with 95% credibility intervals are given in Table 3.6.
The REML estimates, estimates from the Traditional Bayes analysis, and estimates when M is
simulated from the data, are presented in Table 3.7. From these tables it is clear that the changing of
the va.ues of M have a minor effect on the posterior estimates of the fixed effects. These minor
differences are attributable to the little mass of the Dirichlet process prior on the fixed effects.
To put these estimates into perspective, we focus our discussion only on the results of the analy.sis
when M is simulated given the data (Sim. M column). As might be expected, there is a significant
effect of sex with female lambs weighing 'on average 0.7057 kg (C10.95 = [0.4575 ; 0.9523]) less at
weaning than males. Furthermore, there is a significant effect of age at weaning with weaning
weights increasing by 0.0464 kg per daily increase in age (C10.95 = [0.0400 ; 0.0532]). There are also
significant differences among breeds, with breeds 1 - 4 having significant higher weaning weights
than breeds 5 - 6.
-113-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Table 3.6 Estimated Values of the Fixed Effects for M = 5 and 500, as well as the 95%
Credibility Intervals.
M=5 95% Credibility M=500 95% Credibility
Interval Interval
f30 (Intercept) 4.7406 5.3418 ; 6.1000 4.8115 3.5054; 6.0282
/3/ (Breed I) 1.5575 0.7430; 2.3669 1.5898 0.6819; 2.2147
/32 (Breed 2) 1.4068 0.5691 ; 2.1645 1.4941 0.7420; 2.2386
/33 (Breed 3) 1.3466 0.6641 ; 2.0364 1.3960 0.6406 ; 2.1626
/3. (Breed 4) 0.9581 0.2810; 1.5885 0.9337 0.1968 ; 1.6439
/3j (Breed 5) -0.4066 -1.2002 ; 0.3532 -0.3256 -1.2729 ; 0.4002
/36 (Sex) 0.699 0.4375; 0.9576 0.7092 0.4755; 0.9681
/37 (Age) 0.0461 0.0399 ; 0.0529 0.0457 0.0398 ; 0.0526
Table 3.7 REML estimates and SE's, Traditional Bayes Estimates with 95% Credibility
Intervals, and Dirichlet Process Prior Estimates when M is simulated given the data.
REML SE for Trad. Bayes 95% Credibility SimM 95% Credibility
REML Interval Interval
f30 (Intercept) 5.0030 0.537 4.3439 3.2871 ;,5.3149 4.3267 3.2806; 5.4281
/3/ (Breed I) 1.6820 0.376 1.6706 0.9035 ; 2.3790 1.6760 0.8870 ; 2.4189
/32 (Breed 2) . ·1.5150 0.383 1.5017 0.8259 ; 2.2473 1.5043 0.7582 ; 2.2397
/33 (Breed 3) 1.4360 0.357 1.4326 0.7821 ; 2.0731 1.4267 0.7189; 2.1200
/3. (Breed 4) 0.9480 0.357 0.9278 0.2524 ; 1.5857 0.9381 0.2249 ; 1.6617
/3j (Breed 5) -0.4060 0.393 -0.4258 -1.2394 ; 0.3043 -0.4079 -1.1983; 0.3194
/36 (Sex) 0.7040 0.128 0.7046 0.4648 ; 0.9461 0.7057 0.4575 ; 0.9523
/37 (Age) 0.0465 0.003· 0.0463 0.OJ98 ; 0.0529 0.0464 0.0400 ; 0.0532
-114-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
The estimated marginal posterior densities are now been calculated and depicted in Figures
3.23 - 3.30. Note once again that the posterior densities are only calculated for the fixed effects
when M is simulated, given the data. These figures also show the minor effect of changing M on the
posterior distributions of the fixed effects. Unlike the densities for the random effects, the plots are,
as expected, bell-shaped and symmetrical since the fixed effects are not directly affected by the
relaxation of the normal assumptions (when M is small).
Figure 3.23 Estimated Marginal Posterior Density of Po , the Intercept.
-115-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.24 Estimated Marginal Posterior Density of p" the Expected Difference in
Average Weaning Weight between lambs of Breed 1 and Breed 6.
Figure 3.25 Estimated Marginal Posterior Density of P2, the Expected Difference in Average
Weaning Weight between lambs of Breed 2 and Breed 6.
-116-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.26 Estimated Marginal Posterior Density of PJ, the Expected Difference in Average
Weaning Weight between lambs of Breed 3 and Breed 6.
Figure 3.27 Estimated Marginal Posterior Density of P., the Expected Difference in Average
Weaning Weight between lambs of Breed 4 and Breed 6.
-117-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.28 Estimated Marginal Posterior Density of /3j, the Expected Difference in Average
Weaning Weight between lambs of Breed 5 and Breed 6.
Figure 3.29 Estimated Marginal Posterior Density of /36, the Expected Difference in Average
Weaning Weight between Male and Female lambs.
-118-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Figure 3.30 Estimated Marginal Posterior Density of /37, the Average Increase m Weaning
Weight per Daily Increase in Age.
3.6 An Experimental Design - Model Validation
Much of the current research focused on the distributional properties of Bayesian models
compared to classical models. At present one would suggest the results of these models should be
compared by means of partial F-tests, residual analysis, cross validation (data-splitting) or tests of
overal! model adequacy. However, model validation involves an assessment of how the fitted
models will perform in practice, i.e. how successful it will be when applied to new or future data. An
experiment will thus be conducted where all the genetic parameters in the mixed linear model are
known, and the dependent variable will be built. These parameters, using REML and Bayesian
methods (Dirichiet process) will then be estimated.
-119-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Again, consider the following mixed linear model
where 'Si - N(O; d ó1n) and for which the random effects have a non-parametric Dirichlet process prior
distribution, i.e. Yi - G where G - DP(M . Go). The parameters of the Dirichlet process are
Gu= N(O, dy) , a probability measure, and M = 100. The values for the variance components are
dó= 4.88 and dy= 0.7211. The only fixed effect in the model is sex (/31 = 0.705). Thus, the male
lambs weigh on average 0.705 kg more than female lambs. A total of 200 sires are added as random
sire effects to the model. For each sire, 10male and 10 female weaning weights are generated using
the Dirichlet process.
Table 3.8 reports the estimated variance components used for the experimental data set. A small
difference between the estimates and the actual values of these variance components are observed,
indicating that the Bayesian approach using the Gibbs sampler is certainly valuable and worthwhile
in the context of animal breeding and selection.
Table 3.9 contains the estimated breeding values of the first 10 sires in the data set along with
their rankings. The second column (EXP) in the table is the actual breeding values of the sires. The
fourth and sixth columns contain the results when a Dirichlet process is implemented in the Gibbs
sampler. For the fourth column, the precision parameter M is set equal to the true value, i.e. 100,
whereas the six column contains results when this parameter is simulated given the data
-120-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
Table 3.8 Estimated Variance Components for the Experimental Data using the Dirichlet
Process Prior and REML Analysis.
MDP,M=100 MDP, Sim M REML
1
(fr 0.765 0.768 0.772
cic 4.932 4.935 4.929
Note that the true values for the variance components are, for cle = 4.88 and clr = 0.7211.
Table 3.9 Estimated Breeding Values of 100fthe 200 Sires for the Experimental Data along
with their Posterior Rankings.
Rank EXP Sire ID MDP,M= 100 Sire ID MDP,SimM Sire ID REML Sire ID
1 1.4702 9 1.413 9 1.4188 9 1.44597 9
2 .1.3975 2 1.3183 2 1.3239 2 1.32166 2
3 0.381 5 0.913 10 0.8727 10 0.87709 10
4 0.2997 I 0.2711 5 0.256 5 0.26953 6
5 0.146 10 0.2381 6 0.2142 6 0.2679 5
6 -0.082 6 -0.0503 7 -0.1292 7 -0.02923 7
7 -0.2163 8 -0.1544 3 -0.2231 3 -0.15932 3
8 -0.2835 II -0.2535 1 -0.2281 I -0.16895 I
9 -0.3977 4 -0.2812 II -0.3324 II -0.26275 II
10 -0.4936 7 -0.4713 8 -0.4889 8 -0.44905 8
-121-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
From the table we can show that the Oirichlet process in Bayesian inference regarding breeding
experiments is a very promising method. According to the experimental data, it is known that sires
9,2,5,1 and 10 are ranked as the five best sires in the model. The Oirichlet process ranked sires
9,2,10,5 and 6 as the best animals. This is an 80% success rate in the ranking procedure.
Sires 9,2,10,6 and 5 are also ranked as the best sires by the REML analysis. In the next section
dealing with the model adequacy, the SSE (sum square errors) is calculated and reported in Table
3.10 below:
Table 3.10 The Calculated Sum Square Errors for the different Analysis.
REML MDP,M= 100 MDP,SimM
I SSE 46.145 46.44 "44.833
From the results, it is believed that the Bayesian non-parametries, using the Gibbs sampler, have
as much to 'offer as the REML analysis. Since the posterior densities resulting from the Gibbs
sampler can easily be used to construct confidence intervals for the model parameters, the potential
mathematical consequences of the tooikit that is explored here in the world of the animal breeder is
evident.
-122-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
3.7 Chapter Summary
The important contribution of this chapter revolves around the non-parametric modelling of the
random effects. We have applied a general technique for Bayesian non-parametries to this important
class of models, the mixed linear model for animal breeding experiments.
Our technique involved specifying a non-parametric prior for thé distribution of the random
effects and a Oirichlet process prior on the space of prior distributions for that non-parametric prior.
The mixed linear model was then fitted with a Gibbs sampler, which turned an analytical intractable
multidimensional integration problem into a feasible numerical one, overcoming most of the
computational difficulties usually experience with the Dirichlet process. This proposed procedure
also represents a new application of the mixture of Oirichlet process model to problems arising from
animal breeding experiments. The application to and discussion of the breeding experiment from
Kenya is helpful for understanding the importance and utility of the Oirichlet process, and inference
for all the mixed linear model parameters.
As far as non-parametric versus parametric analysis are concerned, in relatively 'well-behaved'
cases, where a parametric analysis would have coped, we typically obtain similar forms of posterior
inference, particularly posterior modes, but with appropriately greater range of uncertainty in
posterior means, as indicated in the case when the precision parameter is relatively small. When the
appropriate form of the posterior should be 'badly behaved', the non-parametric analyses will reflect
this, whereas most parametric analyses would not reveal. this fact.
-123-
The Dirichlet Process and Non-parametric Modelling in Animal Breeding Experiments
However, as mentioned before, a substantial statistical issue that still remains to be tackled is the
great discrepancy between resulting posterior densities of the random effects as the value of the
precision parameter, M changes. The work in this area is ongoing and needs a careful understanding,
especially where inferences may be sensitive to the distributional assumption on the random effects.
We believe that Bayesian non-parametries have much to offer, and can be applied to a wide range
of statistical procedures. As far as Bayesian versus Classical approaches are concerned, we note the
very real advantage of being able to input broad prior ideas of characteristics such as location, scale
and shape. Moreover, the much richer and more tractable forms of inference that are presented as a
consequence of the Gibbs simulation-based approach to computation are quite profound.
© Parts of this chapter (simulation study) have been published in the South African Journal of Animal Science.
(See Pretorius & Van der Merwe, 2000)
© Parts of this chapter (Dirichlet process results) are submitted and in press in Genetics, Selection and
Evolution.
(See Van der Merwe and Pretorius, 2000 (in press))
© Parts of this chapter (Dirichiet process results) have been accepted for publication in 'Collection of Refereed
Articles' -ISBA2000'.
(See Pretorius and Van der Merwe, 2001)
, International Society for Bayesian Analysis
-124-
The Dirichlet Process with an Application in Veterinary Medicine Research
CHAPTER4
«The Dirichlet Process in Veterinary Medicine Research»
Introductory words: In the present thesis, parallel developments of mixed linear models have also taken place
in veterinary medicine research. The aim of the present chapter is to expand the Dirichlet process prior to a
veterinary medicine problem in which the observations are correlated with each other.
4.1 Prologue
.As mentioned in the previous chapter, Mixture priors, especially Dirichlet Mixtures have opened
the way to serious Bayesian developments in Non-parametric Modeling and Density Estimation.
Moreover, mixtures of Dirichlet process models (MDP) have become increasingly popular for
mode ling when conventional parametric models would impose unreasonably stiff constraints on the
distributional assumptions. There has been some work towards this end in the classical setting,
however in the Bayesian paradigm, non-parametric mode ling is still very scant and introductory.
As mentioned in Chapter 3, from the Bayesian perspective, inferential interest focuses on the
posterior distribution of the random and fixed effects .. Allowing distributions other than the normal
for the random effects' may more accurately model our prior beliefs, or it may allow us to better
express out uncertainty about the true distribution of the random effects. However, one major
question arising in Bayesian analysis concerns the sensitivity of the results to the chosen prior
-125-
The Dirichlet Process with an Application in Veterinary Medicine Research
In the next sections, we describe how the MOP model can be applied to the mixed linear model.
We show the full conditional distributions and how the Gibbs sampling can be implemented for both
the conjugate (section 4.3) and non-conjugate case (section 4.4). Indeed, if we assume the sampling
distribution for y. to be normal, then a normal base measure for the random effects completes a
-I
conjugate MOP model. If on the other hand, the base measure is specified as a multivariate t -
distribution (i.e. a scale mixture of the multivariate normal distribution), then the base measure for
the random effects completes a non-conjugate MOP model.
We .provide a detailed exploration of a veterinary medicine application of interest in an
experiment where the mixed linear model is appropriate. Finally, we present an extension of the
work to a non-conjugate mixture of Dirichlet process model, and compare the results of this analysis
to that of the conjugate MDP model.
4.2 The Experiment and Model Structure
.An important assumption In the use of mixed linear models is that the observations are
independent from each other. In many practical situations this assumption does not hold. The most
common situation is where different measurements are taken on the same individual, leading to what
is known as a repeated measures design. Often these measurements are taken periodically over time.
Alternatively, observations may be spatially collected in which case those closest together may be
most alike. To demonstrate how the mixed linear model can be used for repeated measures design,
the following experiment is analyzed.
-126-
The Dirichlet Process with an Application in Veterinary Medicine Research
The aim of the study was to see whether there are differences in the change in pev between the
two breeds of cattle, N'Dama and Boran, following a trypanosome infection'. A variable often
measured to evaluate the severity of the diseases is packed cell volume (peV), which is the
percentage of the volume of the blood serum taken up by the red blood cells. Low pev corresponds
to anaemia and can indicate infection with the disease.
Depending on the design of the experiment, different models could be fitted to the data, but it
will be shown that the mixed linear model framework provides a unified way to investigate the
changes over time of pev in the case of the two different breeds. Moreover, the Gibbs sampler
developed in the previous chapter can also be useful here to show how posterior computations via
Gibbs sampling simulations can be routinely applied to the experiment. The data are shown in
APPENDIX D.
The appropriate mixed linear model for the experiment is given by
(4.1 )
y. is a n, x 1 vector of pev measurements for the lh animal; fJ(P x I) is a vector of uniquely defined
_I
fixed effects and that the corresponding design matrix Xi is tn, x p). For the present example, we fit
the average slopes and intercepts for the two breeds as the fixed effects.
I Parasitic disease transmitted by Tse-tse flies
-127-
The Dirichlet Process with an Application in Veterinary Medicine Research
Also, Zi (n, x v) is a matrix of covariates for the v x 1 vector of random effects. Further, Yi is the
unobservable random effect, and for the unobservable vector of random errors, e, (n, x 1), it is
common to assume independent nonnal distributions.
For 'r, (v x 1), the vector of unobservable random effects which is usually taken to be normally
distributed, the normal prior is replaced with a non-parametric prior, followed by a Oirichlet process
prior on the general distribution. For the present example, it is also assumed that each animal has a
random slope and intercept, and that this random slope and intercept are normally distributed. Thus,
the intercept and slope parameters describing the linear relationship between PCV and breed contain
both fixed and random effects.
Considering the above model structure, the covariates for the fixed effects are
Xo: Intercept
. XI: Time in days
I if N' Darna breed
{o if Boran breed
X3 : XI X2 - the time by treatment interaction,
and the covariates for the random effects are (with v = 2)
Zo: Intercept
Z, : Time in days
-128-
The Dirichlet Process with an Application in Veterinary Medicine Research
Further
Y Ol -- Intercept for the /h individual
YII = Slope for the /h individual,
and
/30 = Intercept for pev measurements on Boran breed,
/31= Slope for pev measurements over time on Boranbreed,
/32 = Difference between the intercepts of the N'Dama and Boran breeds, and
/33::; Difference between the slopes of the N'Dama and Boran breeds.
Therefore we have
Xo XI x2 x3 21 22
0 0 0 1 0
2 0 0 1 2
1 4 0 0 1 4
1 7 0 0 1 7
9 0 0 1 9
1 14 0 0 /30 J 14
I/3= 1 17 0 0 /31X ZiYi = 1 17 [ro;]
1 18 0 0 /32 1 18 YIi
21 0 0 /33 21
1 23 0 0 23
1 25 0 0 1 25
1 29 0 0 1 29
31 0 0 31
1 35 0 0 35
with XI = X2 = ... = ~, and
-129-
The Dirichlet Process with an Application in Veterinary Medicine Research
xo x, x2 x3
0 0
2 2
4 4
7 7
9 9
/30
14 14
7/3 = /3,X 17 17 /3
18 18 2
21 21 /33
23 23
25 25
1 29 29
1 31 1 31
1 35 1 35
with X7 = Xg = ... = X'2'
Further
36.2 30.4
35.9 33.0
~, = and y_7 =
21.3 24.5
17.8 22.6
-130-
The Dirichlet Process with an Application in Veterinary Medicine Research
As in the previous chapter we will present a mixed linear model for which the vector of random
effects have a non-parametric distribution. The non-parametric Bayesian approach for the random
effects is to specify a prior distribution on the space of all possible distribution functions. This prior
for the mixed linear model is applied to the general prior of the distribution of the random effects.
Indeed, as shown before, this can be accomplished with a Dirichlet process prior distribution.
This means that the usual normal prior on the random effects is replaced with a non-parametric prior,
followed by a Dirichlet prior on the general distribution .
.In the next section the required conditional posterior distributions for the model parameters are
given .. Note that although the detailed derivations of these distributions can be seen in Chapter 3,
some extra derivations will be given in the present chapter. Only minor changes are made to the
Gibbs sampler because of the correlated observations (repeated design).
4.3 Priors and Conditional Posterior Distributions for the Conjugate MDP Model
4.3.1 The Uniform Prior for jJ and a/
The full conditionals for fJ and a/ for the conjugate MDP model in the case of repeated measures,
are the same as in Chapter 3, i.e. an uniform prior distribution for both fJ and a/ as to represent lack
of prior knowledge about the vector of fixed effects and error variance. Therefore
p( fJ .a/) =p(/3 ) p (a/ ) cx: constant (4.2)
-131-
The Dirichlet Process with an Application in Veterinary Medicine Research
The required full conditional for the fixed effects, is multivariate normal:
(4.3)
For the variance component, (7/ the conditional is
n,
pea; If3,Y,y_ ) = x,I1(~)2 exp{- ~(L - X;f3 - Z;y;)'(y_; - X;f3 - Z;YJ}
1=1 ac. ac
(4.4)
an Inverse Gamma density where
n-2
q 2
I(y_ - X;f3 - Z;y;)'(y_ - X;f3 - Z;y;)
K e = ;=1 2
Also, Y =(n,YJ, ... , Yq), Y = (y ',y ', ...,y ')', in; = n the sample size, and Y; = [YO;]
- _I _2 -q ;=1 Yl;
where Y Oi = Intercept for the jth individual and h= Slope for the ;th individual, as defined before.
-132-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.3.2 Prior for D
The variance covariance matrix D in the base measure of the Dirichlet process is unknown and
therefore a suitable prior distribution must be specified for it, i.e.
P (D) oc constant
to present lack of prior knowledge about D. After choosing random effects for each subject, the
different subjects will be grouped into clusters (groups) in which the subjects have equal Yi 's (equal
intercepts and equal slopes). That is, after selecting a new Yi for each subject i in the sample, there
will be some number r;, 0 < r; ~ q, of uriique values among the random Yi's. Denote these unique
values by ,.1,/, 1= 1 ... r;. Additionally let 1 represent the set of subjects with common random effect ,.1,/.
Note that knowing the random effects is equivalent to knowing r; , all of the Yi 's and the cluster
membership I. Then for the purpose of calculating the full conditional of D, the A/ are r; independent
observations from N(O,D).
Thus
(4.5)
an Inverse Wishart distribution where
-133-
The Dirichlet Process with an Application in Veterinary Medicine Research
See Chapter 3 for more details on the additional piece of the model to be added as an aid to
convergence for the Gibbs sampler (equations (3.17) with a: = D). This additional piece is
incorporated into the final Gibbs sampler. The simulation procedure of the precision parameter, M
remains the same as in Chapter 3, and an algorithm (1.3) for simulating from a Wishart distribution is
given in APPENDIX A. Simulation from the Wishart distribution can also easily be done by using
the algorithm of Odell and Feiveson (1966).
4.3.3 Dirichlet Process Prior for n
As mentioned in the previous chapter the mixture of Dirichlet Process Prior is simplified in
practice. by the Polya urn representation, u.sing the fact that marginally, the Yi are distributed as the
base measure along with the added property that p(y i = Y j' i '* j) > 0 .
Therefore
(i = 1, ... ,q), (4.6)
= Y j with probability
M+q-l
M (4.7)- Go with probability
M +q-l
We find that the conditional posterior of Yi is given by ..
-134-
The Dirich/et Process with an Application in Veterinary Medicine Research
q
P(Yi I fJ,CJ; ,D'Y_i,M) o: L~(~i I XifJ + Z,yJ,CJ;ln,)' bYl
J"'"
+ {M _[~(LIXJ3 + Z,y,,(Y;I •.)~(y, IO,D)dY,} (4.8)
x~(y, IO,D)p(y I Yi'fJ,CJ;,y)
-, -J
.'? 2 2 .
where p(y IY i' fJ, CJ; ,Y ) = ~(y I XfJ + Z iY i' CJJ n ) and ~(·I JL, CJ ) denotes the normal density
-I -J -I I
with mean f.1. and variance cl . Also, Y -i denotes the vector of random effects for the subjects
excluding subject i and b" is a degenerate distribution with point mass at s.
Consider again the integral
s, = 'J"M ~(LIXifJ + ZiYi,a;IIl; )t/J(Yi IO,D)dYi
-'"
n;
~ M12;:"r ex+ 2~(;l', - X,fI- Z'Y')'(l', - X,fI- Z,y,+
v
(2~)2 IDI-~ exp{ - ~Yi' D-1 r. }dYi.
(4.9)
Following the usual algebraic routes, i.e. completing the square with respect to Yi' it follows from
(4.9) that
(4.10)
where
o, ~[ ;: (Z,' Z,) +D-'T' and '1', ~ ( ;: z,n,z,'- I.,)
-135-
The Dirichlet Process with an Application in Veterinary Medicine Research
Proof:
The exponent of the integral is ~~ -XJ3-ZiYi)'~ -XJ3-Ziy,)+y,'D-IYi' This can bea ' ,
e
written as
= -21 -a Y
,-Y - 2 1 - 'z-J y. .r, + 1 2Z 'z· 'D-l-J r, i i + r, r,
-, -, a; -, a;
e
where
y =y -XJ].
-I _I
Following the usual algebraic routes, i.e. completing the square with respect to Yi where Yi is a
v x I vector, it follows from (4.9) that
-136-
The Dirichlet Process with an Application in Veterinary Medicine Research
From the above expression it follows,
and we find
Therefore
where
-137-
The Dirichlet Process with an Application in Veterinary Medicine Research
Thus, as explained in Chapter 3, each summand in the conditional posterior distribution of y,
(equation (4.8)) is separated into two elements. The first element is a mixing probability, and the
second is a distribution to be mixed. So with probability
(4.11 )
q
Ai + I~(L I XJ3 + Ziyj;a;In)
i=i.) ..i
we select from distribution (\ ' which means that we set Yi = Yj. Also, with probability
(4.12)
q
A, + I~(L IX,/3+Ziy};a;In,)
.i=I;j .. i
we select from
P(Yi I [J,a; ,D,y)-, oc ~(Yi IO,D)p(y -,I-Y}i,[J,a; ,y .i, (4.13)
meaning we sample Yi from its full conditional,
(4.14)
Before applying the conjugate MDP model to a veterinary medicine experiment, we first turn to
the non-conjugate MDP model, and show how to apply this model structure to our mixed linear
model.
-138-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.4 Priors and Conditional Posterior Distributions for the Non-Conjugate MDP
Model (Modified MDP Model)
As mentioned before, samples of the Yi from a modified distribution can be obtained by
specifying the base measure as a multivariate normal scale mixture. The multivariate I - distribution
can be obtained as a scale mixture of the multivariate normal distribution as follows. If
(4.15)
then the marginal distribution of x is St p (s, jl, "f.) , where St p (s, jl, "f.) is ap - dimensional Student
I - distribution with s degrees of freedom, mean jl, and dispersion matrix "f.. Also ga = Gamma
distribution. From a sampling perspective, sampling from
and then from
is equivalent to sampling from St p (s, jl, "f.). Thus, marginally, the distribution of x is multivariate I,
with 'I being integrated out. This representation is often used in the Gibbs sampling literature
(Kleinman & Ibrahim, 1998; also see Wakefield et al., 1995 and the references therein). Note that
with a common 'I for all i, we generate dependent samples for the Yi'S. To obtain independent
samples for the Yi 's, we must specify a separate 'Ii for each Yi and take the 'li'S to be i.i.d. gamma
variates. The specifications for the modified MDP model are as follows.
-139-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.4.1 The Uniform Prior for jJ and a/
The full conditionals for jJ and (7,/ for the multivariate t model (non-conjugate MDP model) are
the same as in the conjugate MDP model, i.e. a uniform prior distribution for both jJ and (7,/ as to
represent lack of prior knowledge about the vector of fixed effects and error variance (see equations
(3.13) and (4.2)). Thus the required full conditional for the fixed effects, jJ is again multivariate
norma! (see equations (3.14) and (4.3)), and for the variance component, (7,/ an Inverse Gamma
density (see equations (3.15) and (4.4)).
4.4.2 Prior and Conditional Posterior for TJ
From the discussion in section (4.4) we will specify for TJi the prior
. TJ - ga (p-, -p) , I..e.
2 2
Cp) ~ TJ~-I exp{ - -T}
p(TJ) = e ( ) (4.16)
22[ P
2
In the example that follows, we will use p = 4, which means that instead of using a normal prior as
base measure, we will use a t - distribution with 4 degrees of freedom for Go.
The conditional posterior distribution for TJ is given by
-140-
The Dirichlet Process with an Application in Veterinary Medicine Research
(4.17)
which leads to
(4.18)
a Gamma distribution. Hence, from a sampling perspective, 'if - X:k+P' i.e. " is sampled from a
Chi-square distribution with vk+p degrees offreedom and" can be calculated as
(4.19)
Equations (4.17), (4.18) and (4.19) also follow from the fact that after selecting a new Y i for each
subject i in the sample, there will be some nurnber Z, 0 < ; ~ q, of unique values A" I = 1,... ,;
among the random Yi 's (see also section (4.3.2) for more details on ;). Thus, for the modified
MDP model an additional piece is added to the Gibbs sampler. Also, note that we use a somewhat
simpler generation of the multivariate t than is usually found in applications of the Gibbs sampler.
-141-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.4.3 Prior for D
As in the normal case, section (4.3 .2) we specify for D the prior
p(D) cc constant
to present lack of prior knowledge for D. For given 17 the posterior is therefore given by
(4.20)
Further, as shown in Chapter 3, to speed mixing over the entire parameter space, it is suggested to
move around the A's after determining how the Yi 's are grouped. Thus, in addition, a posterior
density is derived for the A's, i.e,
p().;, I j3,a;, 17,D,~) cc </JUt, 10,17-1 D)Il P(~i 113,0";) (4.21)
id
which implies that
(4.22)
-142-
The Dirichlet Process with an Application in Veterinary Medicine Research
where
(4.23)
4.4.4 Dirichlet Process Prior for n
From the discussion of the conjugate MDP model in section (4.3), we find that he conditional
posterior of y; is given by
q
p(y; I (J,er;, 17, D,y -i' M) cx. I~I(X;~(J + Z;;Yj ,er; 1/1)' 5y}
.+ {M_1~(L IX,fJ + Z,y"a;I")~(y, IO,~-' D)dY,} (4.24)
x~(y; IO,17-'D)P(y ly;,(J,er;,y )
. . _I _J
where p(y ly; ,f3,er;,y ) =~(y IX(J + Z;Yi'er; In) and ~(-I ,u,er2) denotes the normal density
-I -J _I I .
with mean J.i and variance cl . Also, Y _; denotes the vector of random effects for the subjects
excluding subject i and 5s is a degenerate distribution with point mass at s. Consider again the
integral
0f0Ai = M ~(~; I X;(J + Z;y;,er;In, ~(y; IO,17-'D)dy;
-00
.=M R -2-1 J72er :r exp{--2 12er (y .-X;(J-Z'.;y;)'(y -X;(J-Z;y;) } x (4.25)& & _I _I-00
v
(2~)'11(' DI-iex+~y,'~"D'"'y, }dY,
-143-
The Dirichlet Process with an Application in Veterinary Medicine Research
Completing the square with respect to Y; where Y; is a v x 1 vector, it follows from (4.25) that
A"1 =M(21r t~ITJ-J DI-~1Q.71(~an-~exp{_2l-aJ- (y - X;/3)''t';'' (y - X;/3)}-I -I
C
(4.26)
where
Thus, as explained in Chapter 3, each summand in the conditional posterior distribution of Y;
(equation 4.24) is separated into two elements. The first element is a mixing probability, and the
second is a distribution to be mixed. So with probability
q (4.27)
A; + IqJ(~; I X;/3+Z;Yj;a;In,)
i=ï.!»!
we select from distribution 5r} , which means that we set Y; = Yj' Also, with probability
N'I
q (4.28)
A; + IqJ(~; I X;/3+Z;Yj;a;In,)
l=ï.!»!
we select from
(4.29)
-144-
The Dirichlet Process with an Application in Veterinary Medicine Research
meaning we sample Yi from its full conditional,
(4.30)
Since all the necessary conditional posterior densities are derived and given, we illustrate our
methodology with an experiment from veterinary medicine research.
4.5. AVeterinary Medicine Example
As mentioned in the introductory sections of the chapter, the aim of the study was to see whether
there are differences in the change in pev between the two breeds following a trypanosomal
infection. The estimates are obtained from the Gibbs sampler, in which the full conditional
posteriors are updated after every iteration. All posterior analyses are based on I 000 draws, giving
us a full non-parametric Bayesian solution to all the mixed linear model parameters.
4.5.1 REML Solution
The following REML results are taken from Duchateau, et al. (1998). The variance components
of the random effects can be obtained from Table 1. The first line in this table, named 'd(l, I)',
corresponds to the variability of the intercepts within breed, the second line, named' d(2, 1)', to the
covariance between the intercept and the slope, and the third line, 'd(2,2)', to the variability of the
slopes within breed, and 'Residual' to a/. The variance of the intercepts is the largest component
-145-
The Dirichlet Process with an Application in Veterinary Medicine Research
and much larger than the variance of the slopes. There is a negative covariance of -0.2551 between
the slope and the intercepts, implying that the higher the initial PCV the greater it's subsequent fall.
The intercept and slope parameters describing the linear relationship between PCV and breed
contain both fixed and random effects. Average fixed intercepts and slopes are assumed for each
breed, and each animal has a random slope and intercept.
Table 4.1 Covariance Parameter Estimates obtained from the REML Analysis.
CovParm Estimate
d(1,I) 12.4646
d(2,I) -0.2551
d(2,i) 0.0058
a2 4.3531
E:
The average linear relationship for each breed can be obtained from Table 4.2 below.
Table 4.2 REML Solution for the Estimates of the Fixed Effects.
Effect Breed Estimate SE DF t Pr> ItI
Intercept 35.06 1.501 10 23.35 0.0001
Breed BO -0.842 2.123 144 -0.4 0.692
Breed NO 0
Time*Breed BO -0.413 0.0374 144 -11.04 0.0001
Time*Breed NO -0.276 ·0.0375 144 -7.37 0.0001
From this table, the linear regression equation for the two breeds can be obtained as
-146-
The Dirichlet Process with an Application in Veterinary Medicine Research
BO: PCV = 34.00':'" 0.4131
ND: PCV= 35.06 - 0.2761
Both the intercept and slope of the fitted relationship is thus different from animal to animal. .
IndividuaIregression lines for each animal can be obtained from Table 4.3.
Table 4.3 Individual Intercepts and Slopes for the Different Animals (random effects) obtained
from the REML Analysis.
Effect Anim ID Estimate SE DF t Pr> ItI
Intercept 80241 -1.836 1.638 144 -1.12 0.264
Time 80241 0.0563 0.0417 144 1.35 0.179
. Intercept 80322 -4.014 1.638 144 -2.45 0.0154
Time 80322 0.084 0.0417 144 2.03 0.044
Intercept B0326 -3.53 1.638 144 -2.16 0.032
Time 80326 0.0676 0.0417 144 1.62 0.107
Intercept B0209 4.698 1.638 144 2.87 0.0047
Time B0209 -0.095 0.0417 144 -2.28 0.024
Intercept B037 0.8398 1.638 144 0.51 0.608
Time B037 -0.0359 0.0417 144 -0.86 0.39
Intercept BOl 3.841 1.638 144 2.35 0.02
Time BOl -0.0777 0.0417 144 -1.86 0.064
Intercept ND60 -2.1658 1.638 144 -1.32 0.188
Time ND60 0.0338 0.0417 144 0.81 0.418
Intercept ND66 3.3628 1.638 144 2.05 0.041
Time ND66 -0.0512 0.0417 144 -1.23 0.222
Intercept ND72 -2.688 1.638 144 -1.64 0.102
Time ND72 0.057 0.0417 144 1.37 0.174
Intercept ND73 0.2487 1.638 144 0.15 0.879
Time ND73 -0.0248 0.0417 144 -0.60 0.552
Intercept ND74 -2.8312 1.638 144 -1.73 0.086
Time ND74 0.06 0.0417 144 1.44 0.152
Intercept ND75 4.073 1.638 144 2.49 0.014
Time ND75 -0.0749 0.0417 144 -1.80 0.0747
-147-
The Dirichlet Process with an Application in Veterinary Medicine Research
For instance, for the first animal, B0241, the fitted relationship is
B0241: PCV = (34.22 - 1.836)+( -0.413 + 0.0563) t
= 32.38 - 0.357 t
These .regression lines are determined for each animal and are presented in Figure 4.1. This figure
demonstrates that PCV tend to decrease more rapidly the higher the initial PCV. This illustrates the
negative covariance between slope and intercept observed in the fitting of the model and Table 4.1.
37
<,
" ~
....... 32 :::::::. :::::::.-ti. ::::::. -::::...'~ ,,~> -...:::..",.,_-...:::. .0
Il.. 27
'-..
22 1-
17
0 10 20 30 40
1ime(days)
Figure 4.1 Change of PCV in Time for Individual Animals based on the REML Analysis. (This
figure is taken from Duchateau et al. (1998».
-148-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.5,2 Non-Parametric Bayesian Solution
The covariance parameter estimates (modes of the posterior distributions) for the conjugate MDP
model and the modified MDP model are displayed in Table 4.4 below. The reason for determining
the posterior modes and not the posterior means is because the REML estimate is more similar to the
mode of the posterior distribution than the mean. Also, it is well known that the REML estimate is
the mode of the marginal likelihood.
Table 4.4 Covariance Parameter Estimates (modes of the posterior distributions) from the,
Conjugate MDP and Non-conjugate MDP Model as obtained by the Non-parametric
Bayesian Analysis.
Cov Parm Conjugate MDP 95% Credibility Non-Conjugate MDP 95% Credibility
Estimate Interval Estimate Interval
D(l,l) 13.50 7.4122; 209.5653 15.75 5.2152; 225.2366
D(2,l) -0.28 -4.6765 ; -0.0643 -0.31 -5.4513 ;-0.0788
D(2,2) 0.0076 0.0024 ; 0.1198 0.01 0.0008; 0.1357
a2 4.610 3.8070;7.172 4.690 3.7915; 8.007
E
The main difference between the results of the conjugate MDP model and the non-conjugate MDP
model is that the 95% credibility intervals for the variance components are wider under the modified
base measure (non-conjugate MDP model). This is to be expected, as the variance covariance matrix
D is directly affected by the relaxation of the normal assumption. Moreover, the similarity of the
results for the a; indicates that this variance parameter is not sensitive to the choice of one ofthese
-149-
The Dirichlet Process with an Application in Veterinary Medicine Research
two base measures, Furthermore, as evident from the above tables it is also clear that the Bayesian
estimates coincide well with the REML estimates. The observed histogram for a/ from the
conjugate MDP model is given in Figure 4.2. Using the conditional posterior densities for d(l,l) and
d(2,2) the marginal posterior densities are estimated as the average of the posterior densities and are
displayed in Figures 4.3 - 4.4. The attenuation of the width of the 95% credibility intervals is also
evident on Figures 4.3 and 4.4.
Figure 4.2 Histogram of the Posterior Distribution of a/ (Error Variance) with Mean = 6.032
and Mode = 4.610 (Conjugate MOP Model).
-150-
The Dirichlet Process with an Application in Veterinary Medicine Research
Figure 4.3 Posterior Density of the Intercept Variance: p(dJ/I D) for the Conjugate MOP Model,
Mode = 13.50; and for the Non-conjugate MOP Model, Mode = 15.75.
- Non-conjuqate MOP model
- - - Conjugate MOP model
Figure 4.4 Posterior Density of the Slope Variance: p(dnl D) for the Conjugate MOP Model,
Mode = 0.0076; and for the Non-conjugate MDP Model, Mode = 0.01.
-151-
The Dirichlet Process with an Application in Veterinary Medicine Research
Only the results of the conjugate MDP model will be reported in the next sections. The average
linear relationship for each breed can be obtained from Table 4.5 along with 95% credibility
intervals. According to the proposed model, /30 is the intercept of the Boran breed and /3, the slope of
the Boran breed. /32measures the difference between intercepts for N'Dama and Boran breeds, and
/33 the difference between slopes for these two breeds.
Table 4.5 Bayesian Solution and 95% Credibility Intervals for the Estimates of the Fixed
Effects from the Conjugate MDP Model.
Effect Breed Estimate 95% Credibility Interval
/30 BO 35.397 30.9278 ; 39.2172
/3, BO -0.43 -0.5492 ; -0.3325
/32 ND-BO 0.9851 -4.0721 ; 5.9692
/33 ND-BO 0.1345 -0.0019; 0.2641
If one considers /33, the difference in the rate of decrease of PCV measurement between N'Dama and
Boran breeds it follows from the 95% credibility interval (-0.0019 ; 0.2641) that there is no
difference in the rate since this interval includes zero. However, a 90% credibility interval does not
include: zero meaning that there is a significant difference in the rate of decrease of PCV
measurements between the two breeds at a 0.1 level of significance. This conclusion can also be
drawn 'from Figure 4.8.
Further, also from this table, the linear regression equation for the two breeds can be obtained as
BO: pcv= 35.39 - 0.43 I {PCV = /30+/3, I}
ND: PCV = 36.38 - 0.30 I {PCV =(/30 + /32) + (/3, +/33) I}
-152-
The Dirichlet Process with an Application in Veterinary Medicine Research
Both the intercept and slope of the fitted relationship are again different from animal to animal.
Using the 'conditional posterior densities for the above parameters and Gibbs sampling, the marginal
posterior densities are then estimated and displayed in Figures 4.5 - 4.8.
Figure 4.5 Posterior Density of the Intercept for Boran: p(f30 ID), Mean = 35.3971.
-153-
The Dirichlet Process with an Application in Veterinary Medicine Research
Figure 4.6 Posterior Density of the Slope for Boran: p(/3, I D), Mean = -0.4373.
-154-
The Dirichlet Process with an Application in Veterinary Medicine Research
Figure 4.7 Posterior Density of the Difference between Intercepts for N'Dama and Boran
Breeds: pUhI D), Mean = 0.9851.
-155-
The Dirichlet Process with an Application in Veterinary Medicine Research
Figure 4.8 Posterior Density of the Difference in the Rate of Decrease of pev Measurement
between N'Dama and Boran Breeds: p(/3J1 D), Mean = 0.1345.
-156-
The Dirichlet Process with an Application in Veterinary. Medicine Research
Individual regression lines for each animal can be obtained from Table 6.
Table 6 Individual Intercepts and Slopes for the Different Animals (random effects) obtained
from the Dirichlet Process.
Effect Anim ID Estimate
Intercept 80241 -2.6158
Time 80241 0.0759
Intercept 80322 -4.5198
Time 80322 0.0982
Intercept 80326 -4.2742
Time 80326 0.0857
Intercept 80209 3.3598
Time 80209 -0.0685
Intercept 8037 -0.1336
Time 8037 -0.0246
Intercept 801 1.2138
Time 801 -0.0235
Intercept ND60 -3.1529
Time ND60 0.0512
Intercept ND66 1.8883
Time ND66 -0.0194
Intercept ND72 -3.5944
Time ND72 0.0748
Iritercept ND73 -0.9109
Time ND73 -0.0127
Intercept ND74 -3.7435
Time ND74 0.0793
Intercept ND75 1.694
Time ND75 -0.0223
-157-
The Dirichlet Process with an Application in Veterinary Medicine Research
As in the REML analysis, these regression lines can be determined for each animal and are
presented in Figure 4.9. This figure also complements the REML results, i.e. that PCY tends to
decrease more rapidly the higher the initial PCY, and once again illustrates the negative covariance
between slope and intercept observed in the fitting of the model. Moreover, there seems to be a
difference between the N'Dama and Boran breeds.
Figure 4.9 Change of PCY in Time for Individual Animals based on the Bayesian Non-
Parametric Mixed Model. (Dashed = N'Dama; Solid = Boran). The lines are
numbered from top to bottom.
-158-
The Dirichlet Process with an Application in Veterinary Medicine Research
Let us now turn to the important parameter of the Oirichlet process, M. Recall that the parameter
M, a type of dispersion parameter for the Oirichlet process prior, is a measure of the strength in the
belief that G is Go. Although it may be hard to quantify, M is a positive scalar that is related to how
"clumpy" the data are (often called a precision parameter). Clumpy data occur when the different
subjects are concentrated into a few clusters. In practice it is difficult to select appropriate values for
this parameter. Instead, it is suggested to place a prior distribution on this parameter, and simulate it
given the data. West (1992), assumed that M - Gaïa.b) a gamma prior with a > 0 and scale b > O.
We may extend this idea to include a reference prior (uniform for log(M)) for the repeated measure
design by letting a-? 0 and b-? O.
.Moreover, when defining a Oirichlet process prior, recall that M determines the prior distribution
of ; , the number of additional normal components in the mixture, and it is a critical smoothing
parameter of the mixed linear model. When there are only a few clusters among the animals in the
model, the estimate of the normal means from the Oirichlet process prior will be similar to the non-
parametric Bayes estimator, and when there are almost q = 12 (total number of random effects)
different clusters, the estimator from the Oirichlet process prior will be similar to the parametric
Bayes estimator.
-
.When M is simulated, given the data, the average number of clusters, ; = 9 with mode
Mo = 16.1 0 . The estimated marginal posterior density and the unconditional marginal posterior
density for the simulated Mvaluesare displayed in Figures 4.10 and 4.11.
-159-
The Dirichlet Process with an Application in Veterinary Medicine Research
-
Figure 4.10 Estimated Marginal Posterior Density of M with ~ = 9 , and Posterior Mode
M; = 10.0.
Figure 4.11 Estimated Unconditional Marginal Posterior Density of M with Posterior Mode
Mo=16.10.
-160-
The Dirichlet Process with an Application in Veterinary Medicine Research
Finally, the observed histogram for the number of clusters, ~ for the simulated values of M is
presented in Figure 4.12.
Figure 4.12 Observed Histogram for the Number of Clusters, ~ when M is simulated;
According to the above figure, the different animals for the two breeds are grouped into 9 different
clusters. Figure 4.9 also supports this conclusion. In this figure it is clear that the lines for animals
4,5 and 6 of the N'Dama breed are very similar. Hence these three animals are grouped into one
cluster, having the same slope and intercept. Moreover, the same conclusion can also be drawn for
animals 5 and 6 from the Boran breed. Thus, with these different animals forming two groups, we
have an average of 9 groups/clusters among the twelve different animals.
-161-
The Dirichlet Process with an Application in Veterinary Medicine Research
4.6 Chapter Summary
In this chapter, we have applied a general technique for Bayesian non-parametries to the mixed
linear model. Our technique involved specifying a non-parametric prior for the distribution of the
random effects and a Oirichlet process prior on the space of prior distributions for the non-parametric
prior. Moreover, we also present the use of a modified MOP model. The resulting model was fitted
with a Gibbs sampler. The modified MOP model is a new generalization, as is the computational
imputation of this model.
The application to and discussion of an interesting data set from veterinary medicine research was
helpful for understanding the importance and utility of these two MDP models, and the Bayesian
mixed linear model framework provided a unified way to investigate the changes over time of pev
in the two different breeds. Future work suggested by researchers (Kleinman & Ibrahim, 1998),
includes allowing a different 17;for each i in the modified base measure and perhaps more
complicated base measures. The MOP model for the random effects would be particularly useful in
these kinds of models since they depend heavily on the random effects, which are greatly affected by
the MOP model.
-162-
Reference and Probability-Matching Priors for the Mixed Linear Model
CHAPTERS
«Reference and Probability-Matching Priors»
Introductory words: Besides the intrinsic interest of developing good non-informative priors for the variance
components problem (mixed /inear model), a number of theoretically interesting issues arise in application of
the proposed procedures. For example, in animal breeding experiments, interest may be in making inferences
about ratios of variance components or functions thereof, rather than about individual variance components
themselves. This important aspect is explored in the present chapter.
5.1 . Prologue
Box and Tiao (1973) wrote: "The sampling theory approach to the variance component problem
encounters a number of snags. These have bothered statisticians for many years, as is evident by the
great variety of attempts which have been made to resolve the problems". Determination of
reasonable non-informative priors in multiparameter problems is not easy; common non-informative
priors, such as Jeffrey's prior, can have features that have an unexpectedly dramatic effect on the
posterior. In recognition of this problem, Bemardo (1979), proposed the Reference Prior approach
to the development of non-informative priors, the key feature of which was a possible dependence of
the reference prior on specification of parameters of interest and nuisance parameters.
-163-
Reference and Probability-Matching Priors for the Mixed Linear Model
In this chapter the reference prior of Berger and Bemardo (1992) is derived for the mixed linear
model and the solution depends on the ordering of the parameters and how the parameter vector is
divided into sub-vectors. In spite of these difficulties iere is growing evidence, mainly through
examples that reference priors provide "sensible" answers from a Bayesian point of view and some
more limited evidence that frequentist properties of inference from reference posteriors are
asymptotically "reasonable". We will also examine whether the reference priors satisfy the
probability-matching criterion.
5.2 The Mixed Linear Model
From section 1.2, we again have the mixed linear model, which postulates that the observable
random vector Y is a linear combination of the fixed effects and random effects plus a random error
term. In its simplest form the univariate mixed linear model can be written in matrix notation as
y = xp +Zy + e . (5.1)
As before, Y (n xl) is a vector of observed values for the trait on which selection is desired,
jJ (p xl) is a vector of fixed effects uniquely defined so that the corresponding design matrix X
(n x p) has full column rank, p. Also, y (q x 1) is a vector of unobservable random effects with
y - N(Q,Aa}) and design matrix Z (n x q). a/ is an unknown scalar and A (q x q) is called a
relationship (genetic covariance) matrix.
-164-
Reference and Probability-Matching Priors for the Mixed Linear Model
for the unobservable vector of random errors terms, e (n xl), it is common to assume
independent normal distributions with mean vector Q and variance-covariance matrix cr/ In where In
represents a n x n identity matrix and G/ an unknown scalar.
As before, cr/ and cr/ are the variance components.
Lemma 1
In model (5.1) y is a random parameter vector whilst /3, cr/ and a/ are population parameters.
Therefore the likelihood function depends only on /3, cr/ and a/,
Since Yl p,y,a; - N(Xp + Zy,a;IJ and y - N(Q,Aa:) it follows that the marginal
distribution of Y is
(5.2)
Proof:
Since
E(Y I p,y,a;) = Xp + Zy and
E(Y 1,0,0';) = Xp because (5.3)
y - N(Q,Iq). (5.4)
-165-
Reference and Probability-Matching Priors for the Mixed Linear Model
Further,
Var(Y I (J,y,a.)
2 = aJ2n
E(YY'I (J,y,a;) = Var(Y I (J,y,a;) + E(Y I (J,y,a; )E(Y'I (J,y,a;)
=a;Jn +(X(J+Zy)(X(J+Zy)' and
= a;Jn + X(J(J'X'+Zy(J'X'+X(Jy'Z'+Zyy'Z'. (5.5)
Therefore,
E(YY'I (J,a;) = a;Jn + X(J(J'X'+a:ZAZ' which follows from the fact that
E(y) = 0,
and
E(yy') = E(yy') - £(y)£(y') = Var(y) = a:A.
Also
= a;Jn + X(J(J'X'+a:ZAZ'-X(J(J'X'
(5.7)
Thus, we have from (5.3) and (5.7) that
Therefore, the integrated likelihood function ignoring the constant (see also Chen (1994» is given by
-166-
Reference and Probability-Matching Priors for the Mixed Linear Model
(5.8)
5.3 . Reference and Probability-Matching Priors
5.3.1 Background
Prior distributions are needed to complete the Bayesian specification of the mixed linear model.
In the following section the reference prior algorithm of Berger and Bemardo will be used to obtain
the reference prior. The prior is motivated by an asymptotic argument i.e. of maximizing asymptotic
missing information. In the case of a scalar parameter the reference prior is the Jeffreys prior, which
does h.ave.the feature of providing accurate frequentist inference. In the multiparameter setting the
situation is much less clear. The reference prior algorithm is relatively complicated and as mentioned
the solution depends on the ordering of the parameters and how the parameter vector is divided into
sub-vectors.
5.3.2 The Fisher Information Matrix
In the case of the mixed linear model, we are concerned with inferences of jJ, a/ and a/. This is
a typical situation where reference priors had been shown to be very promising (Yang & Chen,
a2
1995). It is sometimes reasonable to consider the parameters (jJ, a/ , v), where v = -T rather than
ae
(jJ, a/ ,a/) (Box & Tiao, 1973). The reason for this is that the between group variance parameter,
-167-
Reference and Probability-Matching Priors for the Mixed Linear Model
(J/ can always be written as a function of v in Henderson's mixed model equations (Ye, 1994).
Using v instead of (J/ will facilitate the calculations considerable.
From (5.8) the log-likelihood is obtained:
(5.9)
As in the case of the Jeffreys prior, the reference prior method is derived from the Fisher information
matrix. To obtain the Fisher information matrix the expected values of the second derivatives must
be calculated, i.e.
l' ( 2 1 2 2E (" a21 a 1 (aa(a 2 2 ' E aaJ2)v ' E -a)1] (a 1]v- and E af3(af3)' for example, must be obtained.E)
Thus,
(5.10)
and
-168-
Reference and Probability-Matching Priors for the Mixed Linear Model
Taking the expectation of 5.11) with respect to Y, it follows that
= -n ( ,)-,0'; - -lr (, )-3 I ( , \r0'; (vZAZ'+Inf 0'; }yZAZ'+J )
2 n
= -n (0'&2 )-2 - (20'&)-2 trl ;
2
-_ -- n2 (0' 2 )-2& '
and
-E( a21 ) =!!2_(0'2)-2 =.!2.I{0'2). . a( (5.12)0'; ) 2 e e r :
Next, differentiate I with respect to a/ and v
(5.13)
and
a::~= -key - XP)'(O'; t\vZAZ'+Infl(ZAZ')(vZAZ'+Infl(y -XP)·
e
(5.14)
(See Searle, CaselIa and McCulloch, (1992) P 456)
-169-
Reference and Probability-Matching Priors for the Mixed Linear Model
If we now take the expectation of (5.14), we have
-E( af J = !tr(O"; r (vZAZ'+Inrl (ZAZ')(vZAZ'+IJ-1 (0"; )(vZAZ'+I )
aO";8v 2 n
= !tr(O"; t (vZAZ'+In r'(ZAZ')2
1 ,
=-I(O";,v).
2
(5.15)
Differentiate twice with respect to vand consider I in two parts, first
then
~ = ++ZAZ'+IY a(VZ~'+IJ}
= -±tr{(VZAZ'+In r' (ZAZ')}
and
2
a /21 = !tr~vZAZ'+IJ-1 (ZAZ')(vZAZ'+IJ-1 (ZAZ')}
av 2
= _!_tr~vZAZ '+ In ti (ZAZ ')Y.
2
(5.16)
Therefore, from (5. I6) it follows that
(5.17)
-170-
Reference and Probability-Matching Priors for the Mixed Linear Model
Consider now the second part of /, i.e.
and
~ = ~«();r(lX - Xf3)'(vZAZ'+lnrl (ZAZ')(vZAZ'+lnrl (Y - Xf3),
a;:; = -~«()r;l (Y - Xf3)'(vZAZ'+ln rl (ZAZ')(vZAZ'+ln rl
x-a{cvZAZ'+ln )(ZAZ'r
l (vZAZ'+IIl)}
av
x (vZAZ'+ln r'(ZAZ')(vZAZ'+ln rl (X - Xf3) (5.18)
= -~«();rl(Y - Xf3)'(vZAZ'+lnrl (ZAZ')(vZAZ'+lnrl
x 2(vZAZ'+lnrl (vZAZ'+ln rl (ZAZ')(vZAZ'+lnrl (Y - Xf3).
a2[
Thus, taking the expectation of the above -+ we have
av
- E( a;:i ) = trKvZAZ'+I"f'(ZAZ'l}'. (5.19)
If we now combine the results in (5.17) and (5.19), we have
_E(~) = -E(~ a2+ [2)
av2 av2 av2
= .!.tr{(vZAZ'+lnrl(ZAZ')Y
2
= 1-/(v). (5.20)
2
-171-
Reference and Probability-Matching Priors for the Mixed Linear Model
Finally, if we differentiate I with respect to jJ, we have
[«(J) ~ [a~~)]ta;t'(VZAZ'+U'[a~)J
(5.21 )
= (0"; t X'(vZAZ'+lnrl X.
The expected values of the other second order derivates are equal to zero. The Fisher information
matrix. is therefore given by
lUl) o. 0
l(jJ,O";, v) = 0' I.I(O";) -1 I(O"c2, v) (5.22)
2 2
0' 1-11(v,O"c) 2 -lev)
2 2
Although the Jeffreys prior is invariant under reparameterization and as mentioned, has been
proven to be a successful non-informative prior for one-dimensional parameter problems, Jeffreys
himself had noticed difficulties in multi-dimensional parameter problems, especially when nuisance
parameters are present. In the following section the reference priors for jJ, 0/ and v are derived.
Note that the reference priors depend on the "group ordering" of the parameters.
5.3.3 Reference Prior for jJ, 0"/ and v
Berger and Bemardo suggested that one allows multiple groups "ordered" in terms of inferential
importance, with the reference prior being determined through a succession of analyses for the
implied conditional problems. They particularly recommended the reference prior based on having
-172-
Reference and Probability-Matching Priors for the Mixed Linear Model
each parameter in its own group, i.e. having each conditional reference prior be only one-
dimen~ion~l. Notations such as {jJ, er/, v} will be used to specify the groups and the importance of
the parameters; {jJ, er/, v} means that there are three groups, with jJ being the most important, and v
the least.
Lemma2
For .the mixed linear model X = Xj3 + Zy +8 the reference prior for the group ordering
{jJ, er/, v} is given by
(5.23)
Note that only the reference prior for the ordering {jJ, er/, v} will be derived, since the reference
priors for the orderings, {er/, jJ, v} and {er/. v. jJ} can be computed in a similar fashion. These priors
are the same as the one for {jJ, er/, v } given in (5.23).
Proof: .
Following the notations in Berger and Bernardo (1992), (see also Yang and Chen, (1995) and Ye,
(1994)) the functions hj, which are needed to calculate the reference prior for the group ordering
{jJ, er/, v} are obtained from I(j3,a;, v), see equation (5.22):
-173-
Reference and Probability-Matching Priors for the Mixed Linear Model
Thus,
hl = IJ(P)I = 1(0-; t X'(vZAZ'+Jnrl Xl· (5.24)
To obtain hl. calculate the following matrix
and consider the (2,2) element, then
h = !J(a2)_! J2(a;,v)
2 2 E 2 J(v)
'=~{'__n_ ¥r(vZAZ'+Jnt(ZAZ')Y}
2 (5.25)(0";)2 (0";)2
= ~(a;r2 ~_ ¥r (vZAZ'+ In t (ZAZ')Y }.
Also
(5.26)
The conditional prior of v given,8 and a/ is
(5.27)
-174-
Reference and Probability-Matching Priors for the Mixed Linear Model
Hence, E[loge Ih211 jJ, 0';] = -loge 0'; + C. SO the conditional prior of vand 0'; given jJ is
It implies that E[loge Ihlll jJ] = c and therefore, for the mixed linear model Y = XjJ + Zy + e the
reference prior for the group ordering {jJ, a/, v} is given by
Lemma3
For ·the mixed linear model Y = XjJ + Zy + e the reference prior for the group ordering
{jJ, v,a/} is given by
I
1rR (P,v,u;) acu;' {tr[(vZAZ'+ Jy (ZAZ')J - ~ HvZAZ'+ I.r'(ZAZ')l } '.
(5.28)
As will be proved later, this prior is also a probability-matching prior for v.
Proof:
Once again, the functions hj (j= I ,2,3), which are needed to calculate the reference prior for the
group ordering {jJ, v,a/}, are obtained. The Fisher information matrix for this ordering is given by
-175-
Reference and Probability-Matching Priors for the Mixed Linear Model
Q. o
1 1 .?
-lev) -l(v,a;) (5.29)
2 2
1 ?
-l(a;, v) .!_ l(a;)
2 2
Also,
(5.30)
From the Fisher information matrix in (5.29), h2 follows by calculating the following matrix
1(jJ) Q] {
[ Q' ~1(V) - "
121 }-I [ Q ][ 1 ](a;) ~1(a;,V) Q: "21(a;,v)
and the (2,2) element is
. h =.!_ lev) _ 1 l\a;, v)
2 2 2 l(a;)
= .!_{tr{(vZAZ'+lnt (ZAZ')Y _ (a;)2 ~r(vZA~'~Int (ZAZ')Y}
2 (ac) n
~~HVZAZ'+ I" t' (ZAZ'))' -; Vr(vzAi '+ I" r' (ZAZ'))' }
(5.31 )
Also
(5.32)
Now·
because it does not contain f3
-176-
Reference and Probability-Matching Priors for the Mixed Linear Model
I
?l( V I jJ) <XC (h,) : {trKvZAZ'+ ly (zAz')l' - ~ HvZAZ'+ IJ-I(zAz')l' } ï ,
and
I
;r(a; I v,,B) cx: (hJ)2 = (a;rl.
Therefore we have
I
<XC a;'{tr[(VZAZ'+IJ-'(ZAZ')j -~~(VZAZ'+I"t'(ZAZ,)j V.
(5.33)
Interesting, the priors in Lemmas 2 and 3 are not dependent on any Iimits of the compact subsets,
nor do they yield improper posterior distributions. Therefore, it is more convenient to deal for
example with the reparameterization {jJ, a/, v}, providing of course that a/ is not of independent
interest. Note also that (5.33) is also the reference prior for the group orderings {v, jJ, a/} and
{v ,a/, jJ}. Additionally, one can also consider the following priors for the different group orderings,
namely
• ;r(a; ,v) = constant, and
• ;r(a;, v) cx: a;
In the second case we have an uniform or "flat" prior on a; and a: ' in other words
• ;r(a;,a:) = constant.
-177-
Reference and Probability-Matching Priors/or the Mixed Linear Model
5.3.4 Reference Prior for the Intraclass Correlation Coefficient, p
There are two main reasons for considering intraclass correlation coefficients instead of variance
components. First of all, the interest might center on them due to the nature of the study. Animal
breeding provides an excellent example, where making inferences about heritability requires
modeling the intraclass correlation coefficients. However, even if that is not the case, it may be
easier to elicit priors using intraclass correlation coefficients since they are defined on the unit
hypercube, as apposed to variance components which take values on the product span of the positive
half of the real line. For more details see Gënen (2000).
Frequentists analysis of data often results in a negative estimate of the variance component as well
as a confidence interval for the intraclass correlation coefficient that covers the entire parameter span.
In contrast the Bayesian method produces sensible answers with different prior densities. Hence
these methods and likelihood based methods leave a lot to be desired, a void which Bayesian analysis
can fill and as mentioned before a Bayesian approach is intuitively appealing as well, since the
parameter space is naturally restricted.
(J'2
The reference prior for the intraclass correlation coefficient p = 2 Y 2 can also be derived for
(J'y + a;
different group orderings of p, f3 and a/. Only the reference prior for the group ordering {p, IJ. a/}
will be derived since, as mentioned before, the reference priors for the other group orderings can be
computed in a similar way.
-178-
Reference and Probability-Matching Priors for the Mixed Linear Model
Lemma4
For the mixed linear model Y = XjJ + Zy + e the reference pnor for the group ordering
{p, jJ, a/} is given by
I
TrIl.(p,jJ;a;)oc a;2 2 {tr[(_P_ZAZ'+In)-I(ZAz,)]2 _![tr(_P_ZAZ'+In)-I(ZAZ,)]2}ï
(l-p) 1-p n 1-p .
(5.34)
Proof'
According to Ye (1994), the Fisher information matrix I(p, jJ, a/) can be derived from
I(v, jJ, a/) by making the transformation
I(p,jJ,a;) = P'I(v,jJ,u;)P (5.35)
a2
where 'v = ----T = _p_ and
a. 1-p
1
0'
(1- p)2 0
P = B(v,jJ,a;) = 0 I 0
B(p,jJ,a;)
0 0 I
From equation (5.35) it follows that
-179-
Reference and Probability-Matching Priors for the Mixed Linear Model
1
-l(p) Ol -1l(p,a;) J
2 2
l(p,j3,a;) = 0 1(j3) 0 (5.36)
-1 1(0';J, p) Ol _!_ 1(0';)
2 2
where
•_!_l(p) = 1 4 tr[(ZAZI)(_f!_ZAZI+lnJ-I]2,
2 2(1-p) I-p,
_!_I(p,a;) = _!_l(a;,p) = 2 1 2 tr[(ZAZI)(_f!_ZAZI+lnJ-I],
2 2 20's (1- p) 1- P
and
.1(j3) = (a; rXI(_f!_ ZAZI+ln J-IX.1- p
From (5.36) it follows that
-180-
Reference and Probability-Matching Priors for the Mixed Linear Model
Calculate the following matrix
and consider the (2,2) element of this matrix, then
Now
;rep) cx: (hl)~ = 1 2 [tr{(ZAZ')(_P_ZAZ'+In)-I}~
(l-p) I-p
I
- Mtr(ZAZ')C ~ p ZAZ'+/" rrr
and
I
;r(/) lp) cx: (h2) 2 = 1 because it does not contain /), and
I
;rea; I p,/) cx: (h)2 = a;2 .
Therefore we have
-181-
Reference and Probability-Matching Priors for the Mixed Linear Model
and thus the reference prior for the group ordering {p, fJ, a/} is given by
1
JrR (p, jJ, cr;) o: 0';2 2 {tr[(__p_ ZAZ'+J n )-1 (ZAZ,)]2 _ .!.[tr(__P_ ZAZ'+J )-1 (ZAZ,)]2}ï
. (l-p) I-p n I-p n
Corollary 1
Equation (5.34) is also the reference prior for the group ordering {fJ, p , o-/} and {p, 0-/, jJ}. Also,
in a similarly way, it follows that the reference prior for the group orderings {fJ, 0-/, p},
{o-/, fJ, :p} and {o-/, p, m is given by
1
7r1l (jJ, a;, p) cx. a;2 2 {tr[(__f!_ ZAZ'+/n J-I (ZAZ,)]2}ï (5.37)(l-p) I-p
5.3.5 Probability-Matching Priors
In this section we will derive the probability-matching priors for the parameters of the mixed
linear model and see whether they include some of the reference priors developed in the previous
sections.
Recently Datta and Ghosh (1995) derived the differential equation that a prior must satisfy if the
posterior probability of an one-sided credibility interval for a parametric function and its frequentist
-182-
Reference and Probability-Matching Priors for the Mixed Linear Model
probability agree up to O(n-') where n is the sample size. They proved that the agreement between
the posterior probability and the frequentist probability holds if and only if
In a
I-{1](B)n(B)} = 0 (5.38)
a=1 aBa
where 'n(e) is the probability-matching prior for Bthe vector of unknown parameters.
Also
11( = [~atB(B), ...,~eto(B; )ll (5.39)I
and
(5.40)
It is clear that 1]'(B)1(B)1](B) = 1 for all B where /"1(6) is the inverse of 1(8), the Fisher information
matrix of Band t( 8) the parameter of interest. The following lemma can now be stated.
-183-
Reference and Probability-Matching Priors for the Mixed Linear Model
Lemma 5
eY2
For the mixed linear model Y = Xp + Zy + e the probability-matching prior for v = -T is
eYG
given by
,
. "M (v,/3,rT;) = rT;'[trkvZAZ'+JJ-'(ZAZ'l/ - ~ Vr(vZAZ'+J"t'(ZAZ'l/ J'.
(5.41)
Proof:'
From the equation (5.22) it follows that the inverse if the Fisher information matrix is given by
r' (/]) 0 0
1
-/(v) --/(12eY
2 v)
r' G'(/],eY;, v) = 0' 2
H H
--/(12veY
2 ) !/(eY;)
'G
0' 2
H H
1 1
whereH = -4 l(eY;)/(v) - -41
2 (eY;, v).
Now
t(B) = v , ataepB) = -0' aateeYB;) = 0 and ate B) = 1acv)
St (B) = [Q' Q 1]
-184-
Reference and Probability-Matching Priors for the Mixed Linear Model
and
1 )
- -2l(er v)E'
H
.ll(a;)
Also 11', (B)rl (B)I1(B) = 2 Hand
: 'l( B) = rl = ~~ J(a;l' ['II (B)](B)I1, (B) 0
--'----1- = '12 (B) .
~ 11', (B)rl (B)I1, (B) H2 rh (B)
The prior JrM (B) = JrM ([J, a;,v) must be chosen in such a manner that the differential equation
(5.38) is satisfied, i.e. that
(5.42)
I
Hence, if we choose Jr( B) = JrM (B) = JrM ([J, a; ,v) = H ï then (5.42) is true and
I
JrM([J,a;,v) =[I(a;)I(v)-12(a;,v)]ï
I
=[ n2 2 .ltr{(vZAZ+IJ-I(ZAz)}2 _(.l)2 (a;r2~r(vZAZ+lnt(ZAZ)}2]ï.2(a&) 2 2
-185-
Reference and Probability-Matching Priors for the Mixed Linear Model
Therefore
I
."M (V,fl,u;) = u;'[ trKvZAZ'+IJ'(ZAZ')1' - ~~r(VZAZ'+IY (ZAZ'))' r
(5.43)
Corollary 2
Since equation (5.43) is exactly the same as (5.28) it follows that the probability-matching prior
JrM (V, j3,a;) is also a reference prior for the group orderings {jJ, v, a/ }, {v, jJ, a/} and {v, a/, jJ}.
A very important property of probability-matching priors is that, unlike uniform priors, they
always remain invariant under any one-to-one transformation of parameters. Therefore, since
= -- jJ' av 1v and - = the probability-matching prior for the intraclass correlation
1- p ap (1- py
coefficient, p,i.e. JrM(p,j3,a;) is given by (5.37).
5.4 Reference Posterior Distributions
5.4.1 Background
Applications of reference priors III the case of the mixed linear model with two variance
components complicate the Gibbs sampling procedure somewhat. However, for more than two
variance components the algorithm becomes quite difficult to apply. The reason for this is that the
-186-
Reference and Probability-Matching Priors for the Mixed Linear Model
a2
conditional posterior distribution of a: or v = -T 10 our case IS not the well-known Inverse
aE:
Gamma density anymore, but an unknown distribution from which it is difficult to simulate.
One: way to overcome this problem is to replace the Gibbs sampling procedure as discussed in
Chapter I with ordinary Monte Carlo simulations. Therefore, instead of using the well-known
conditional posterior densities p(jJ I r,a; ,a:, Y), p(r I jJ,a; ,a:, V), pea; I r,a: ,jJ, Y) and
pea: IjJ,a;,r,Y), posterior densities of the form p(jJlv,a;,Y)p(a; Iv,Y)p(vIY) and
p(r I v,a;, Y)p(a; I v, Y)p(v I Y) will be used to simulate joint and marginal posterior densities.
From Searle et al. (1992) (p 359) it follows after some algebraic manipulations that
(5.44)
where
E(jJ I v,a;, Y)= [X'(vZAZ'+IJ-1 x]" X'(vZAZ'+IJ-1 Y, (5.45)
and
(5.46)
..
-187-
Reference and Probability-Matching Priors for the Mixed Linear Model
Also from Searle et al., (1992) it can be shown that
rl v,a;, V - N{E(r Iv,a;, Y);Var(y Iv,a;, v)} (5.47)
where
: E(y I v,O';, V)= vAZ'(vZAZ'+Int'{V -E(fJ I v,O';, V)} (5.48)
and
Var(y I v,a;, V)= 0'; {vA- vAZ'[(vZAZ'+I,,)-1 -
(vZAZ'+IlItl X(X'(vZAZ'+Int' x]" X'(vZAZ'+Int ]ZAV}
(5.49)
E(fJ I v,O';, V) is defined in equation (5.45).
Further, to facilitate the calculations we will make use of the fact that
where G = v-lA -I + Z'Z.
Therefore, instead of calculating the inverse of a n - dimensional matrix, the results can be
obtained from the inverse of a q - dimensional matrix. Equations (5.44 - 5.49) are true whether the
uniform ("tlat"), or reference priors are used in the simulation procedure because the parameters fJ
or r do not occur in the priors.
-188-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.4.2 Calculation of the Posterior Density p(cr;, v IX)
From equation (5.8) (see also equation (5.9)) it follows that the likelihood function
L(P, o-~,v) oe(o-:t~ IvZAZ'+ I" I-i x ex~ - 2~; (Y - xp)' (vZAZ'+ IJ' (Y - X,B)}
and the marginal likelihood
L(cr;, v)= fL(,a,cr;, v}i,a.
f3
By completing the square with respect to ,a it follows that
(vZAZ'+In t X{X'(vZAZ'+In t x]" X'(vZAZ'+In t ]y .
(5.50)
By multiplying L(cr; , v) with the prior :r(cr; , v), the joint posterior
follows.
-189-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.4.3 Calculation of the Posterior Density pea:; I v, Y)
In the case of the reference priors
p(O'; J I v, Y) = KlO('; )-~{n-p+ J) exp {Y2 - X - '20'; lr(vZAZ'+In rI -
(vZAZ'+In r'X{X'(vZAZ'+In t x]" X'(vZAZ'+In t~},
(5.51)
an Inverse Gamma density where
n-p
K, = k[(VZAZ'+IS' -(vZAZ'+/J-'X(X'(vZAZ'+IS'X)' X'(vZAz'+IYkr {~-(-') }
f-(n-p)
2
(5.52)
The posterior distribution p(O'; I v,Y) using "(0';, v) = constant (uniform prior on 0'; and v)
can easily be derived by substituting p - 2 for p in (5.51). On the other hand, if a uniform prior is
used o~ 0'; and 0':, then p(O'; I v,Y) can also be obtained from (5.51) by substituting p - 2 for
p + 2.
-190-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.4.4 Calculation of the Posterior Density p(v IY)
(I) For the reference prior (see also equation (5.23»
the reference posterior density of v is given by
r' r'xt I r'~tl-(n-p)(vZAZ'+In X(X'(vZAZ'+In X'(vZAZ'+In 2 X
(5.53)
(I~ For-the reference prior (see also equation (5.28»
I
2
Jr R (v,p,(j;) cx: (j;2{tr[(VZAZ'+In t (ZAZ,)]2_:~ ~r(vZAZ'+Int (ZAZ')j }
the reference posterior density is given by
-191-
Reference and Probability-Matching Priors for the Mixed Linear Model
I
(vZAZ'+In t I --(n-p)X(X'(vZAZ'+In ti xt X'(vZAZ'+In t ~} 2 X
I
{tr[(VZAZ'+In t (ZAZ')Y - ~~r(VZAZ'+In t (ZAZ')J}"2.
(5.54)
(III) For the reference prior Tre (v, jJ, a; ) cc constant, the posterior density is given by
I I
Pc (v,1 Y) o: lX' (vZAZ'+In yl xl-"2lvZAZ'+InP x {X [(vZAZ'+In t -
t ti I tiLJXft-.!.(n-p-2)(vZAZ'+In X (X'(vZAZ'+In X )- X'(vZAZ'+In 2 .
(5.55)
(IV): For the prior Tr K (jJ,a: ,a;)oc constant, the posterior density is given by
. I I
PK (v.] Y) = 1X'(vZAZ'+In r'XI-"2lvZAZ'+InP X {X'[(vZAZ'+In t -
(5.56)
-192-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.4.5 Posterior Distribution of e = Xp + Zy and Predictive Distribution of
Yf =x~P+z~y+&f
In this section the posterior density of e = Xp + Zy as well as the predictive density of a future
observation Yf =x~f3+z~y+&f will be derived. Here jy is(lxl),xf is (pxl), zf is (qxl) and
ef - N( 0, a; ). These densities will enable us to estimate or predict the weaning weights of lambs
for the different sires. Also the year, age of dam, sex of the lamb and birth status effects will be
taken into account. However, the following lemma must first be proved.
Lemma 6.
The conditional posterior mean and variance of e = Xp + Zy are given by
E(e 1 v,a;, Y)= XE(f31 a~. v,xl- vZAZ'(vZAZ'+Int {Y- XE(f31 v,a;, V)},
(5.57)
where
E(PI a;, v, Y)= (X'(vZAZ'+Int x}' X'(vZAZ'+IJ-Iy,
and
Var(e 1 v,a;, Y)= a;Z'(v-1A -I +zz}' Z'+[In - vZAZ'(vZAZ'+Int]x x
Var(f31 a;, v, Y)X.'[In - vZAZ'(vZAZ'+IJ-1],
(5.58)
-193-
Reference and Probability-Matching Priors for the Mixed Linear Model
where
Var(jJl 0';, v,Y)= 0'; (X'{vZAZ'+In tix]" .
Proof'
Equation (5.57) follows from equations (5.45) and (5.48).
Further,
Var(B I jJ, Y,O';, v)= Var{(XjJ + Zy I jJ, Y,O'; , v)
= Var{Zy I jJl Y,O'; '}= O';Z(v-IA -I +zz)" Z'
his follows from Searle, et al., (1992), P 357, equation (40).
Also, the unconditional variance of B is given by
(5.58)
-194-
Reference and Probability-Matching Priors for the Mixed Linear Model
where
Corollary 3
Since e = Xf3 + Zy is a linear combination of normally distributed random variables, the
posterior distribution of e is also normal for given v and er; . Therefore
elY:, v, er; - N{E(e I v, er; ,V}Var(e I v, er; .x) . (5.59)
Further, the conditional predictive density óf a future observation
is normal with mean
E(y J I er;, v,V)= x~E(f31 er;, v,V)+ vz~AZ'(vZAZ'+IJ-1 {y - XE(f31 v, er; ,V)},
(5.60)
and variance
:Var(y J Ier;, v,X) = er; ~ + z , (V-IA -I + Z' Z t' Z J + ~~ - vz~AZ'(vZAZ'+In t' X}x
{X'(vZAZ'+Int'Xr ~~ -vz~AZ'(vZAZ'+Int' X}}.
-195-
Reference and Probability-Matching Priors for the Mixed Linear Model
The unconditional posterior predictive distribution of an unobserved observation p(y f IY) , has
therefore two components of uncertainty: (1) the fundamental variability of the model, represented
by the variance a; in Yr not accounted for by x~j3 and z~y; and (2) the posterior uncertainty in
j3,y,a; and a: (or v) due to the finite sample size of Y. According to Geisser (1975) the
prediction of observables or potential observables is of much greater relevance than the estimation of
what are often artificial constructed parameters.
5.5 An Example
5.5.1 The Data
Consider again the Donner sheep stud of Elsenburg (see section (1.8.1)). A total of n = 879
weaning weight records, from the progeny of q = 17 sires were available after editing, and p = 17
fixed effects were included in the final model.
As before the mixed linear model used for this data structure, is the sire model
X = Xj3 + Zy + e , where Y (879 x I) vector of weaning weights. jJ (17 x I) is the vector of fixed
effects, X a (879 x 17) incident matrix, and the design matrix Z, a (879 x 17) matrix identifying the
(17 xl) vector of random effects including the breeding values for the 17 sires for which the data are
observed.
In this section estimation of fixed and random effects as well as prediction of future weaning
weights will be obtained for the Donner sheep stud using reference and uniform priors.
-196-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.5.2 Estimation and Prediction using Uniform and Reference Priors
In the example that follows, the following reference priors will be considered for the model
parameters. For the group ordering {jJ, a/, v}:
1
Ref2: R(P, v,O";) cc {tr[(VZAZ'+IJ-11Z" 0";2 (ZAZ')Y - ~ ~r(vZAZ'+In t (ZAZ·)j y,
and uniform prior in section (IV) page 192.
5.5:3 Analysis of Variance Components
Posterior modes for the uniform prior, 95% credibility intervals, and the estimates obtained when
the different reference priors (Reference estimates) are used, are summarized in Table 5.1.
Table 5.1 Point - and Reference Estimates (posterior modes) and 95% Credibility Intervals
for the Variance Components.
Analysis Estimate 95% Credibility
Interval
Uniform 21.2620 19.2579; 23.3612
Refl 21.2028 19.2887; 23.3191
Ref2 21.2015 19.2847; 23.2233
-197-
Reference and Probability-Matching Priors for the Mixed Linear Model
Using the conditional posterior densities for a/ , and Monte Carlo simulations the marginal
posterior densities are estimated as the average of the posterior densities and are displayed in
Figure 5.1. Considering the results in the above table and the figure below, we conclude that for our
practical problem the posterior densities using the first reference prior (Refl), those derived from the
second reference prior (Ref2) and when the uniform prior (Uniform) is used, are for all purposes the
same. This comes as no surprise since the posterior density of the error variance component a/ is
not directly influenced by the different reference priors.
Figure 5.1 Estimated Marginal Posterior Densities of the Variance Component, a/ .
-198-
Reference and Probability-Matching Priors for the Mixed Linear Model
However, as mentioned in paragraph (5.3.4) researchers are often more interested in functions of
the variance components. This is due to the nature of their study, e.g. in breeding experiments,
animal breeders are making inferences about heritability, which requires modeling the intraclass
correlation coefficients.
(]'2
Table 5.2 reports the estimates for the intraclass correlation coefficient, P = r and the
(]'2r + (]'2&
-(]'2variance ratio v = T' The 95% credibility intervals are also reported in this table. Note that the
(]'&
reference priors for the group ordering, {po [J. o/} and {[J. a/, p} are used in the analysis and will be
called Refl and Rej2. The results yield that the estimates from the two reference prior analysis are
very much the same. Hence, the main result of the different reference priors on the sire variance is
attenuation of the width of the 95% credibility intervals. Note that these intervals are wider under the
uniform prior than the different reference priors.
Also, the 95% credibility intervals for the intraclass correlation coefficients do not contain 0.5.
As mentioned before, this result corresponds well to the statement made by Wang et al., (1992)
namely that from a genetic point of view, an intraclass correlation of 0.5 is not possible in a sire
modei. The marginal posterior densities are also estimated and displayed in Figures 5.2 and 5.3. The
wider intervals are quite evident in the shape of the marginal posterior densities displayed in the
figures.
-199-
Reference and Probability-Matching Priors for the Mixed Linear Model
Table 5.2 Estimates (posterior modes) using the Uniform and Reference Priors of Functions of
the Variance Components, along with 95% Credibility Intervals
95 % Credibility 95 % Credibility
Parameters Uniform Refl Interval Ref2 Interval
p 0.133 0.120 0.0338; 0.3031 0.115 0.0340 ; 0.3030
v 0.140 0.120 0.0350; 0.4350 0.110 0.0355 ; 0.4350
Uniform
(J2
Figure 5.2 The Estimated Marginal Posterior Density of the Variance Ratio, v = --T .
(Jc
-200-
Reference and Probability-Matching Priors for the Mixed Linear Model
- Uniform
Figure 5.3 The Estimated Marginal Posterior Density of the Intraclass Correlation Coefficient,
-201-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.5.4 Analysis of Random Effects
In Table 5.3 the estimated breeding values (posterior modes) of the 17 sires and the 95%
credibility intervals are given. The marginal posterior densities for the breeding values of sire 3 and
10 are displayed in Figures 5.4 and 5.5 respectively. In all three analyses, the results of the estimates
are very much the same, with minor differences in the width of the 95% credibility intervals.
Table 5.3 Estimated Breeding Values for 17 Sires from the Elsenburg Dormer Stud, and 95%
Credibility Intervals.
Sire1D Refl 95% Credibility Interval Ref2 95% Credibility Interval
41037 0.6538 -1.3640; 3.0539 0.6575 -1.4208; 2.9119
41004 0.1900 -1.6150; 2.3719 0.1906 -1.6480; 2.3881
41019 3.3282 1.3131; 5.7251 3.3314 1.3208 ; 5.8563
43002 -1.1093 -3.5516; 1.3940 -1.1046 -3.5189; 1.2828
44170 -0.0857 -2.5505 ; 2.4474 -0.0926 -2.5478; 2.3214
44174 -0.5709 -3.2919; 2.0173 -0.5661 -3.3822 ; 2.1433
44042 -1.1895 -3.2556; 0.7401 -1.1858 -3.1829; 0.6891
45070 -1.0288 -3.2473 ; 0.7452 -1.0189 -3.1385; 0.8123
451J5 -0.5068 -2.8742 ; 1.8211 -0.499 -2.8643 ; 1.7676
46015 -1.6490 -3.9451 ; 0.2502 -1.6401 -3.7808; 0.2536
46037 -0.7021 -2.8284; 1.0831 -0.6912 -2.7557; 1.1339
48014 -0.8675 -3.0533 ; 1.l59~ -0.8557 -3.0533 ; 1.0817
48052 -0.3462 -2.6278 ; 1.7172 -0.3328 -2.5345; 1.7411
48148 -1.2589 -3.7277 ; 1.0333 -1.2445 -3.5498 ; 0.9884
49053 0.4203 -2.5061 ; 3.5098 0.4302 -2.4457; 3.4350
49134 0.7776 -1.9565; 3.6495 0.8125 -1.9465; 3.8679
49046 0.4152 -2.6140; 3.2126 0.4230 -2.6800 ; 3.6002
-202-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.4 The Estimated Marginal Posterior Density of the Breeding Value for Sire 3
(ID41019) (Y3).
J
. :~Qt~,~;
:~"-, ,
0:1'
Figure 5.5 The Estimated Marginal Posterior Density of the Breeding Value for Sire 10
(ID46015) (YIO).
-203-
Reference and Probability-Matching Priors for the Mixed Linear Model
The similarity of the results indicates that, for this example, the results are not that sensitive to the
choice of either the two reference priors or the uniform or "flat" prior. However, the application to
and discussion of the example is helpful for understanding the implementation of the different priors
via the Gibbs sampler. For smaller sample sizes the differences can be quite substantial (large).
5.5.5 Analysis of Fixed Effects
Table 5.4 summarizes the estimated fixed effects. Selected parameters in the case of the mixed
linear model with corresponding joint marginal posterior densities are presented in Figures 5.6 -
5.10.
Table 5.4 Estimated Values of Selected Fixed Effects, 95% Credibility Intervals, and
Reference Estimates.
Parameter Refl 95% Credibility Ref2 95% Credibility
Interval Interval
/30 22.8796 18.9515 ; 26.5691 22.8927 18.9515; 26.5691
/37 5.2692 4.1121; 6.3856 5.2680 4.1003 ; 6.3796
/3J. 3.6501 3.0255 ; 4.2836 3.6494 3.0256; 4.2851
/3J5 9.5337 7.3747; 11.6265 9.5340 7.4161 ; 11.6158
/3J6 3.0345 0.8671 ; 5.1672 3.0339 0.8988; 5.1642
As expected, the estimates of the fixed effects using the different priors are for all practical
purposes thesame, and in particular the results from the two reference priors. There is one factors to
keep in mind when examining this similarity in the results, i.e. that the different priors do not directly
influence the posterior densities of the fixed effects to the same extend as in the case of the random
effects.
-204-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.6 Estimated Marginal Posterior Density of {Jo, the average weaning weight of female
lambs born in 1950 if the age of the dam is 8 years or older, and the birth status
"triplets" .
- Uniform
~,;,~
Figure 5.7 Estimated Marginal Posterior Density of {J7, the expected difference III average
weaning weight between lambs born in 1949 and in 1950.
-205-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.8 Estimated Marginal Posterior Density of fJN , the expected difference in average
weaning weight between male and female lambs .
.. ;1;1'105'
Figure 5.9 Estimated Marginal Posterior Density of Pl5 , the expected difference in average
weaning weight between single births and triplets.
-206-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.10 Estimated Marginal Posterior Density of /3/6 , the expected difference in average
weaning weight between a pare of twins at birth and triplets.
5.5.6 Predictive Density of a Future Observation
The densities derived in Lemma 6 and Corollary 3 will now enable us to estimate or predict the
weaning wei~hts of lambs for the different sires. Also the year, age of dam, sex of the lamb and birth
status effects will be taken into account. Using the notation in Chapter 1, the weaning weight of a
female lamb born in 1950 if the age of the dam is 8 years or older, and the birth status "triplets", will
be predicted. Figure 5.11 displays such a predictive density if Sire 15 (1049053) is used to
impregnate the dam.
-207-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.11 Predictive Density of the Expected Weaning Weight of a Female Lamb born in 1950
if the Age of the Dam is 8 years or older, and the Birth Status "Triplets".
The Posterior Modes are Uniform = 25.10; Refl = 23.90; Ref2 = 23.85.
The respective 95% Bayesian predictive intervals for the three analyses under the different prior
specification are Uniform = [12.56 ; 37.01], Refl = [15.98 ; 34.17] and Ref2 = [16.01 ; 34.12]
respectively. Thus, the analyses for our Dormer data result in predictive densities that are close to
each other which can be observed both in the predictive density and in the posterior modes and
credibility intervals. In the derivations of the reference priors it was mentioned that the second
reference prior (Ref2) is also a probability-matching prior, which means that its credibility intervals
will have the correct coverage probability from a frequentist point of view.
-208-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.6 Priors for the Mixed Linear Model in the Case of Three Variance Components
5.6.1 Reference Prior for the Three Variance Components
In this section, the reference priors for the mixed linear model in the case of three variance
comporients will be derived. The application to and discussion of an example conforming to a mixed
linear model with unbalanced data concludes the chapter. A uniform prior ("flat") and a proper prior
(also see Theobaid, Fivat and Thompson, (1997) for more details) are considered for the example.
The Gibbs sampler is once again used to implement the different priors and to obtain marginal
posterior densities for the different variance components.
Consider the following mixed linear model:
(5.61 )
where Y(nxl), X(nxp),j3(pxl),Z;(nxq;) andy;(q;xl) wherei=1,2.
Also,
Therefore as before, the likelihood function is given by
1 .
L(p,a; ,a:, ,a:,) cc ta:.Z,A,Z,' +Cf;I, -, ex+~(Y - XfJ!( ta:,Z,AZ,'+Cf;I,f (Y- XP)}
(5.62)
-209-
Reference and Probability-Matching Priors for the Mixed Linear Model
(Y
As in (5.) we wi Il use the transformation Vi = ---4- (i = 1,2), then
• (Ye
(5.63)
Here as in the case of the previous sections, the reference priors will be derived from the Fisher
information matrix. To obtain the Fisher information matrix, the expected values of the second order
derivatives must be calculated. These can be found as follows:
Differentiate equation (5.63) twice with respect to jJ, then
(5.64)
Differentiate equation (5.63) twice with respect to a/:
and
-210-
Reference and Probability-Matching Priors for the Mixed Linear Model
Therefore
(5.65)
Hence, differentiate equation (5.63) with respect to .(7/ and Vi:
Thus, taking the expected value, we have that
i=1 i=1
(5.66)
If we differentiate equation (5.63) twice with respect to Vi:
-211-
Reference and Probability-Matching Priors for the Mixed Linear Model
all . _I •
E ( -22/ J = +tr{2(Lv,Z,A,Z, + In) (Z,A,Z,) } 2av, 2 ,=1
Now consider
where
and
-212-
Reference and Probability-Matching Priors for the Mixed Linear Model
2
2
-0O/vj2 = -21 cx - XP) ,((l'e2)-1C~f:( VjZjAjZj '-I+( In) ZjAjZj '\",~f:( vjZjAjZj '-I+ IJ X
a{Ct. v,z, A,Z; + I" )(Z,A,ZJ'Ct. v,Z,A, z, + I") }
----'---- --- X
êv,
ct vjZjAjZj' + IJ-I (ZjAjZj' xt vjZjAjZj' + IIIr CX - XP)
j=1 1=1
2
CLVjZjAjZj' +IlIrlCx -XP),
j=1
Thus
Combining the above results, we have
1
:=--ICvj).
2
(5,67)
-213-
Reference and Probability-Matching Priors for the Mixed Linear Model
Finally,
(5.68)
As before the expected values of the other second order derivatives are equal to zero. The Fisher
information matrix is therefore given by
1(/3) 0 0 0
0 _!_ 1(0';) -1/(ae,v,2) -1/(a 2e,v2)2 2 2
1(/3,0';, v" v2) = 1 . 20 1 1-/(v"ae) - I(v,) (5.69)2 - I(v" v )2 2 2
0 -1/(v2,a
2 1 1
e) -/(v2,v,) - l(v )2 2 2 2
The following lemma can now be stated:
Lemma 7
For the mixed linear model X = Xp + Z,y, + Z2Y 2 + e the reference prior for the group
. ,
"II {B,a;, (v, v2)} = (0'; r{/(v, )/(v 122) - (v" v2 )}ï
-214-
Reference and Probability-Matching Priors for the Mixed Linear Model
where l(vl)' l(v2) andl(vl,v2) aredefinedineq~ations(5.67)and(5.68).
Proof: .
The reference prior for the group ordering {B,CY;, (VI' V2)} is obtained from the Fisher
information matrix l(f3,cy;, VI' v2) (equation 5.69) by calculating the functions h/j = 1,2,3). Now
.!.l(a;) 1 2 -1-1(ac'v ) l(a 2., V?)
2 2 l 2 6 -
hl = Iep) - [Q 0 Q]
-1 11(vl,CY 2) 1c -1(vl)2 -1(v"v )2 2 2 [~]
1
-1 l(v2 ,ac
2) -2l(v?,v )
1
-l -1(v2)2 2
= 1(f3)
;rep) = constant.
To calculate hl. consider the matrix
then
-215-
Reference and Probability-Matching Priors for the Mixed Linear Model
multiply with terms that do not contain a;2 ,and therefore
Further
and
, ,
7Z"( v" v2 IjJ, a;) och} = {I (v,)1 (v2) - 12 (v" v2)}ï 0
The reference prior for the group ordering {f3,a; , (v, , V 2 )} is thus given by
. ,
. 7Z"~ {p,a;, (v, v2)} = (a;t' {levi )/(v2) - 12 (v" v2 )}ï.
5.6.2 Proper Prior for the Variance Components (Theobald et al. (1997)
Consider the following proper priors for the different variance components, i.e.
(I) i = 1,2. (5.70)
where k;2 can be interpreted as the prior expectation of a;2 .
-216-
Reference and Probability-Matching Priors for the Mixed Linear Model
(In p (a-) I V , k ) a: (2a )-.!_2('v, , +2) J Vy k; }v exp - _' _, . ,
p r, Y, y,,, 20'2 i = 1,2.
L r,
(5.71)
where, as before, k;,2 can be interpreted as the prior expectation of 0';,2 .
5.7 Joint and Conditional Posterior Densities for the Mixed Linear Model in the
:Case of Three Variance Components'
(a: rT exp{- _210' 2 Yt' Yt} x (a 2y tT exp{- _210_' 2 Y 2' Y 2}' (5.72)I 1
: y, Yl
(I) Uniform Prior
The conditional posterior distributions for the variance components when the uniform or "flat"
prior is used, i.e.
are given by
-217-
Reference and Probability-Matching Priors for the Mixed Linear Model
n
K e ( __12_J2 exp{- _221-_ (Y - Xp - Z r - Z r )'(Y - Xp - Z r - Z r )}.a a 1122_ 112)e e
a;? > 0,
(5.73)
an Inverse Gamma density where
and
P.( cr;, lP,r, cr; ,Y)~K r, [ crlJ~ex+ 2~;,r:r,}
2
ar, > °
(5.74)
also an Inverse Gamma where i= 1,2 and
Further, for the proper priors in equations (5.70) and (5.71) we have
-218-
Reference and Probability-Matching Priors for the Mixed Linear Model
(10 Proper Prior
The conditional posterior distributions for the variance components a; and a:, where i = 1,2,
are given by
pp(a; I p'YI 'Y2 ,a:, ,a:" Y) =
(n+vc+2)
x, (--;-)-2 - exp{- _1_2[CY- Xp - ZIYI - Z2Y2)'CY - Xp - ZIYI - Z2Y2) + V£k }
ac 2a , Jc
a; > 0,
(5.75)
an Inverse Gamma density where
and firially
q;+vr;+2
P,(CT;, I P,y"CT;, Y) ~ K" ( ;;,]-'-2 - ex+ 2~;, b, +v"k" l}
a2r. >0
(5.76)
also an Inverse Gamma where
The conditional posterior densities for the random and fixed effects are give in Chapter 1 where Zy
-219-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.8 An Example
5.8.1 The Data
As an example conforming to a model of random effects only, with unbalanced data, consider
age-adjusted milk production records (305 days) obtained in the same year and herd from cows
whose' sires and dams were considered randomly representative of a large population. The example
consists of 44 production records (in kg) of full-sib daughters and is shown in APPENDIX F. The
model for this example is (see also equation (5.61)),
where ,Y(44xl), X(44xl),,8(1xl),Z;(44xq;) andy;(q; x l) where ï e l.Z.
Further,
r, - N(Q,Ip:i),
e - N(Q,Ina;),
ql = 4 and q2 = 20.
Thus, the study includes 4 sires, 20 dams and 44 milk production records from the daughters. Also,
there is .only one fixed effect included in the final model. For further details see Gill (1978).
-220-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.8.2 Analysis of Variance Components
Bayesian analyses of the data set are given using uniform priors and proper priors from Theobald
et al., (1997) (see equations (5.70) and (5.71)) with
v. =vY =1, k =151380, k =126735 and k , =859997Y1 Y1
the variance components where ky1' k and kc are the ANOVA estimates for a: , a: and a;Y1 1 1
respectively.
The prior distribution for the variance components is proper and leads to a proper joint posterior
distribution, but the small values chosen for Vc and vY correspond to a very dispersed distribution;
this is intended to reveal any problems with convergence of the Gibbs sampler algorithm. The
sample data provide little information about the variance components, as the number of observations
is only 44, the number of sires 4, and the number of dames 20.
Ta.ble 5.5 contains the posterior modes of the estimates obtained under the uniform or "flat" prior
specification, and the posterior modes of the estimates obtained using the proper priors for the
variance components. The respective 95% credibility intervals are also displayed in the table, as well
as the ANOV A estimates and ANOV A-based 95% confidence intervals based on a Safterthwaite
(1946) approximation to the distribution of a linear combination of X 2 random variables (see Gill
(1978)). This table reveals little difference between the implementations in the marginal
-221-
Reference and Probability-Matching Priors for the Mixed Linear Model
posterior modes for the model variance, (J"; , except for a tendency of the 95% credibility interval to
be wider under the uniform prior specification. This is observed both in the table and the marginal
posterior density of (J";, displayed in Figure 5.12. The estimated marginal posterior densities of
(J";?I a;?and , under the different prior specifications are also calculated and shown in Figures 5.13
and5.14.
Table 5.5 Marginal Posterior Modes of the Variance Component under Uniform and Proper
specifications and 95% Credibility Intervals. Also, ANOV A Estimates and 95%
Approximated ConfidenceIntervals. Note that a: (sires) = a: , and a: (Dams) =
l
Parameter ANOVA 95% Confidence Uniform 95% Credibility Proper 95% Credibility
Estimates Intervals Estimates Intervals Estimates Intervals
a/ 859997 524335 ; I 664376 830 OIO 533060; I 481 9000 801200 551240; 1432500
a/(sires) 151380 28812; 144751070 174126 60207; 104020011 140250 30 145 ; 13793000
a/(dams) 126735 16683; 83814925 195 148 9 878 ; 11 100 200 87850 12910; 728 040
-222-
Reference and Probability-Matching Priorsfor the Mixed Linear Model
Figure 5.12 Estimated Marginal Posterior Densities of CT; using Uniform and Proper Priors for
this Variance Component.
Uniform Prior
- Proper Prior
3
2
. 1.5
0.5 '"- - - ...- ....
2
Figure 5.13 Estimated Marginal Posterior Densities of CT:, using Uniform and Proper Priors for
this Variance Component.
-223-
Reference and Probability-Matching Priors for the Mixed Linear Model
Figure 5.14. Estimated Marginal Posterior Densities of 0-: using Uniform and Proper Priors for
2
this Variance Component.
These figures reveal quite a difference in the marginal posterior densities of O-y2 and 0-2 under
I Y2
the uniform prior and proper prior specifications. Not only are there differences between the
posterior modes, but also the length and values covered in the 95% credibility intervals differ quite
substantially. The proper prior also results in posterior densities with more mass close to zero,
whereas the posterior densities using the uniform prior yield much more uncertainty about the true
posterior distribution of the variance components.
Fro~ Table 5.5 it is also clear that the 95% confidence intervals obtained from the X 2
approximations are in general wider than the corresponding Bayesian intervals. According to
Hamada and Weerahandi (2000) the coverage of the ANOVA-based confidence intervals could in
certain cases be drastically different than the nominal values.
-224-
Reference and Probability-Matching Priors for the Mixed Linear Model
5.9 Chapter Summary
For many Bayesians finding an appropriate prior distribution, when faced with a specific decision
problem, can be quite difficult. It is often recommended to choose conjugate priors whenever
possible because they are so computationally convenient. However, this is quite a limited family and
there are many instances where they should not be used.
Another possibility is to approximate our prior beliefs by developing non-informative priors
especi~lIy for the parameters of the mixed linear model via the Reference Prior algorithm. At the
very least, this algorithm can be thought of as a method for generating interesting candidate non-
informative priors, either for sensitivity studies of for investigation of their performances. As stated
in Bernardo (1979) the motivation and idea for the reference prior is basically to choose the prior,
which in a certain asymptotic sense maximizes the information in the posterior that is provided by the
data. From the different lemmas it is evident that the group orderings of the model parameters are
very important since different orderings wi II frequently result in different reference priors. This
dependence of the reference prior on the group chosen and their ordering is unavoidable. Berger and
Bernardo (1992) stated that many examples exist which illustrate that no single non-informative prior
will work well for all functions of a given high dimensional parameter. As mentioned and more fully
discussed in Berger and Bernardo (1989b) it is suggested to use the reference prior corresponding to
single element groups, with the group ordered according to the inferential importance of the
parameters. That different orderings of the nuisance parameters can yield different answers even has
positive aspects; one can then conduct a sensitivity study over the choice if the non-informative prior.
Whilst our feeling is that study of performance of reference priors is certainly to be encouraged, we
have found it to be generally high satisfactory. Indeed, we would feel reasonably confident in using
them in situations in which further study is impossible.
-225-
Conclusion and Summary
CHAPTER6
«Conclusion and Summary»
6.1 Conclusion
Arguing from a Bayesian viewpoint, the Gibbs Sampling Algorithms presented in the thesis
turned analytically intractable multidimensional integration problems arising from animal breeding
theory, into' feasible and appealing numerical problems. It is clear from the analyses that the
Bayesian practitioner does not need to commit him to a point estimate of the variance components in
order to obtain a point predictor for the variables of interest. Also, all the available information about
the random variables to be predicted is contained in the posterior distributions of the random
variables. Therefore, the practitioner can base all of his inferences on these distributions.
We believe that BMOM and Bayesian Non-parametries have much to offer. In the case of
BMOM, if not enough information is available to specify a form for the likelihood function, then
clearly there will be problems in both the traditional likelihood and Bayesian approaches. In
situations like this, some resort to non-likelihood based methods is proposed, e.g. the Bayesian
Method of Moments (BMOM), first introduced by Arnold Zellner. Given the data, BMOM then
enables researchers to compute post data densities for parameters and future observations if the form
of the likelihood function is unknown, and provides a solution to the famous inverse problem
proposed by Bayes (1763).
-226-
Conclusion and Summary
As far as Bayesian parametric versus nonparametrie analyses are concerned, in relatively 'welI-
behaved' cases, where a parametric analysis would have coped, we typically obtain similar forms of
posterior inference, particularly posterior densities. When the appropriate forms of posterior
inference should be 'badly behaved' the nonparametrie analysis will reflect this, whereas most·
parametric analyses would not reveal this fact.
A further question that arises is the ever known" ... Bayesian or Classical. .. ?" We now note the
very real advantage of being able to input broad prior ideas of different characteristics such as
location, scale and shape. The much richer and more tractable forms of inference that are presented
as a consequence of the Gibbs simulation-based approach to computation are quite profound and
significant.
Finally, it is evident that a full Bayesian solution to the problem of inference about variance
components, functions thereof, and random effects in the Mixed Linear Model is possible, and
contribute significantly to the theory of animal breeding.
With that in mind, we would like to conclude this thesis with the inspiring words of Daniel Gianola
(1986):
.....fn navigating through the waters of prediction of breeding values,
estimation of genetic parameters and of inferences about populations
undergoing selection or assortative mating, we found that the Bayesian
inference brought us to familiar harbors or to new exiting lands.
However, a great deal of exploration remains ahead .....
-227-
Conclusion and Summary
6.2 Summary
Chapter 1 illustrated an extension of the Gibbs sampler to solve problems arising in animal
breeding theory. Formulae were derived and presented to implement the Gibbs sampler where-after
marginal densities, posterior means, modes and credibility intervals were obtained from the Gibbs
sampler.
In the Bayesian Method of Moment chapter we have illustrated how this approach, based on a few
relatively weak assumptions, is used to obtain maximum entropy densities, realized error terms and
future values of the parameters for the mixed linear model. Given the data, it enables researchers to
compute post data densities for parameters and future observations if the form of the likelihood
function is unknown. On introducing and proving simple assumptions relating to the moments of the
realized error terms and the future, as yet unobserved error terms, we derived post-data moments of
parameters and future values of the dependent variable. Using these moments as side conditions,
proper maxent densities for the model parameters were derived and could easily be computed. It was
also shown that in the computed example, where use was made of the Gibbs sampler to compute
finite sample post-data parameter densities, some BMOM maxent densities were very similar to the
traditional Bayesian densities, whilst others were not.
It should be appreciated that the BMOM approach yielded useful inverse inferences without using
assumed likelihood functions, prior densities for their parameters and Bayes' theorem, also it was the
case that the BMOM techniques extended in the present thesis to the mixed linear model provided
valuable and significant solutions in applying traditional likelihood or Bayesian analysis in animal
breeding problems.
-228-
Conclusion and Summary
The important contribution of Charter 3 and 4 revolved around the nonparametrie modeling of the
random effects. We have applied a general technique for Bayesian nonparametries to this important
class of models, the mixed linear model for animal breeding experiments. Our technique involved
specifying a nonparametric prior for the distribution of the random effects and a Dirichlet process
prior on the space of prior distributions for that nonparametric prior. The mixed linear model was
then fitted with a Gibbs sampler, which turned an analytical intractable multidimensional integration
problem into a feasible numerical one, overcoming most of the computational difficulties usually
experience with the Dirichlet process.
This proposed procedure also represented a new application of the mixture of Dirichlet process
model to problems arising from animal breeding experiment. The application to and discussion of
the breeding experiment from Kenya was helpful for understanding the importance and utility of the
Dirichlet process, and inference for all the mixed linear model parameters. However, as mentioned
before, a substantial statistical issue that still remains to be tackled is the great discrepancy between
resulting posterior densities of the random effects as the value of the precision parameter, M changes.
We believe that Bayesian nonparametries have much to offer, and can be applied to a wide range of
statistical procedures. In addition to the Dirichlet Process Prior, we will look in the future at other
nonparametric priors like the Pólya tree priors and Bernoulli trips.
Whilst our feeling in the final chapter was that study of performance of non-informative was
certainly to be encouraged, we have found the group reference priors to generally be high
satisfactory, and felt reasonably confident in using them in situations in which further study was
impossible. Results from the different theorems yielded that the group orderings of the mixed model
-229-
Conclusion and Summary
parameters are very important since different orderings will frequently result in different reference
priors. This dependencél of the reference prior on the group chosen and their ordering was
unavoidable. Our motivation and idea for the reference prior was basically to choose the prior, which
in a certain asymptotic sense maximized the information in the posterior that was provided by the
data.
The thesis has surveyed a range of current research in the area of Bayesian parametric and
nonparametrie inference in animal science. The work is ongoing and several problems remain
unresolved. In particular, more work is required in the following areas: a full Bayesian
nonparametrie analysis involving covariate information; multivariate priors based on stochastic
processes; multivariate error models involving Pólya trees; developing exchangeable processes to
cover a larger class of problems and nonparametric sensitivity analysis.
.~
.',../~. (:
.. 1'.' \,'. ,
: . :~!' :.:',: 1,J. ' ..~
1, ' ."
r F~:~':(1d~./~~
-230-
References
REFERENCES
Antoniak, C.E. (1974). Mixtures of Dirichlet processes with Applications to Non-parametric
problems. Ann. Statist., 2, 1152 - 74.
Barron, A.R., Schervish, M., and Wasserman, L. (1996). The Consistency of Posterior Distributions
in Non-parametric problems. Reprint.
Bayes, Rev. T. (1973). An Essay Toward Solving a Problem in the Doctrine of Chances. Phi!. Trans.
Roy. Soc. (London) 53, 370-418; reprinted in Biometrika, 45, 293-315 (1958), and Facsimiles of
Two Papers Bayes (commentary by W. Edwards Deming). New York; Hafner, (1963).
Berger, J .0. (1985). Statistical Decision Theory and Bayesian Analysis, New York: Springer-Verlag.
Berry, O.A., and Christensen, P. (1997). Empirical Bayes Estimation of a Binomial Parameter via
Mixture of Dirichlet Processes. Ann. Statist., 7, 558 - 568.
Berzuini, C., Best, N.G., Gilks, W.R., and Larizza, C. (1997). Dynamic Graphical Models and
Markov Chain Monte Carlo Methods. J. Amer. Statist. Assoc., 92, 1403 - 1412.
BlackweIl, D., and MacQueen, J.B. (1973). Ferguson Distribution via Polya Urn Schemes. Ann.
Statist., 7, 558 - 568.
Boldman, K.G., Kriese, L.A., Van Vleck, L.O., Van TasselI, C.P., and Kachman, S.O. (1995). A
Manual for use of MTDFREML. A Set of Programs to obtain Estimates of Varianees and
Covariances. US Dept. of Agric., Agricultural Research Service.
Brunner, Lf., and La, A.Y. (1989). Bayes Methods for a Symmetric Unimodal Density and its Mode.
Ann. Statist. 4, 1550 - 1566.
-231-
References
Bush, C.A., and MacEachern, S.N. (1996). A Semi-parametric Bayesian model for Randomized
Black Designs. Biometrika, 83,275 - 285.
Carlin, B.P., Gelfand, A.E., and Smith, A.F.M. (1992). Hierarchical Bayes Analysis of Change Point
Problems. Ann. Statist. 41, 389-405.
Chaloner, K. (1994). Residual Analysis and Outliers in Bayesian Hierarchical Models. In Aspects of
Uncertainty edited by P.R. Freeman and A.F.M. Smith. John Wiley and Sons Ltd, 149-151.
Chaloner, K., and Brant, R. (1988). A Bayesian Approach to Outlier Detection and Residual
Analysis. Biometrika, 75,651-659.
Clarke, G.P.Y. (1998). Some Aspects of BLUP Estimation of Breeding Values for Individual Trees
in a Breeding Programme. Presented at the Statistics Department, University of the Free State
Seminar Series.
Cover, T., and Thomas, J. (1991). Elements of Information Theory. New York: John Wiley and
Sons.
Chung, Y., and Dey., (1998). Bayesian Approach to Estimation of Intraclass Correlation Using
Reference Prior. Commun. Statist. - Theory Meth., 27, 2241 - 2255.
De Gruttola, V., and Lagakos, S.W. (1989). Analysis of Doubly-censored Survival Data, with
Application to AIDS. Biometrics, 45, 1 - Il.
Denison, D.G.T., and Mallick, B.K. (1998a). A Non-parametric Bayesian approach to Modeling
Nonlinear Time Series. Technical Report. Imperial College of Science, Technology and Medicine
London.
Doss, H. (1994). Bayesian Nonparametrie Estimation for Incomplete Data via Successive
Substitution Sampling. Ann. Statist., 22, 1763 - 1786.
Draper, D. (1999a). Discussion on Decision Models in Screening for Breast Cancer, by G.
Parmigiani. In Bayesian Statisties 6 (eds J. M. Bemardo, J. Berger, A.P. Dawid and A.F.M. Smith).
Oxford: Oxford University Press. To be published.
-232-
References
Draper.Tï., Cheal, R., and Sinclair, J. (1998). Fixing the Broken Bootstrap: Bayesian Non-parametric
Inference with Highly Skewed and Heavy-tailed data. Technical Report. Statistics Group, University
of Bath, Bath.
Draper, D., Hodges, J., Mallows, c., and Pregibon, D. (1993). Exchangeability and Data Analysis
(with discussion). JR. Statist. Soc. A 156,9 - 37.
Duchateau, L., Janssen, P., and Rowlands, G.L. (1998). Linear Mixed Models. An Introduction with
Applications in Veterinary Research. ILRI (International Livestock Research Institute), Nairobi,
Kenya.
Escobar, M.D. (1991). Estimating Normal Means with a Dirichlet Process Prior. Technical Report
No.512; Department of Statistics, Carnegie Mellon University.
Escobar, M.D. (1994). Estimating Normal Means with a Dirichlet Process Prior. JASA, 89(425),
268 - 275.
Escobar, M.D., and West, M. (1995). Bayesian Density Estimation and Inference using Mixtures. J
Amer. Statist. Assoc., 90, 578 - 588.
Ferguson, T.S. (1973). A Bayesian Analysis of Some Nonparametrie Problems. Ann. Statist., 1,
209 - 230.
Fernando, R.L., and Gianola, D. (1984). Optimal Properties of the Conditional Mean as a Selection
Criterion. J Anim. Sci., 59 (Suppl.) : 177.
Foulley, J.L., and Gianola, D. (1984). Estimation of Genetic Merit from Bivariate "all or none"
Responses. Genet. Sel. Evol. 16 : 401.
Foulley, J.L., Gianola, D., and Thompson, R. (1983). Prediction of Genetic Merit from Data and
Quantitative Variates with an Application to Calving Difficulty, Birth weight and Pelvic opening.
Genet. Sel. Evai., 15 : 401.
-233-
References
Gasparini, M. (1996). Bayesian Density Estimation via Mixtures of Dirichlet Processes. J
Nonparametrie Statist., 6,355 - 366 ..
Geisser, S. (1975). The Predictive Sample Re-use Method with Applications. Journal of American
Statistical Association, 70 (350),320 - 328.
Gelfand, A.E., and Kuo, L. (1991). Non-parametric Bayesian Bioassay including Ordered
Polytomous Response. Biometrika, 78, 657 - 666.
Gelfand, A.E., and Smith, A.F.M. (1990). Sampling-Based Approaches to Calculating Marginal
Densities. .LAmer. Statist. Assoc. 85,398-409.
Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990). Sampling-Based
Approaches to Calculating Marginal Densities. JAmer. Statist. Assoc. 85,972-985.
Gelfand, A.E., Smith, A.F.M., and Lee, T-M. (1992). Bayesian Analysis of Constrained Parameter
and Truncated Data Problems Using GibbsSarnpling, J Am. Statist. Ass., 87, 523 - 532.
Geman, S., and Geman, D. (1984). Stochastic Relaxation, Gibbs Distributions and the Bayesian
Restoration of Images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6,
721-741.
George, E.I., and McCulloch, R.E. (1991). Variable Selection via Gibbs Sampling. Technical Report,
University of Chicago, Graduate School of Business.
Geweke, J. (1992). Evaluating the Accuracy of Sampling-based Approaches to Calculating Posterior
Moments. In Bayesian Statistics, Volume 4, J.M. Bernardo, J.O., Berger, A.P. Dawid, and A.F.M.
Smith (eds), 169-193. Oxford: Clarendon Press.
Geyer, CJ. (1992). Practical Markov chain Monte Carlo. Statistical Sci., 7,467 - 511.
Geyer, CJ., and Thompson, E.A. (1992). Constrained Monte Carlo Maximum Likelihood for
Dependent data (with discussion). 1. Ray. Statist. Soc., B 54, 657 - 699.
-234-
References
Ghosh, J.K., and Mukerjee, R. (1992). Non-informative Priors. In Bayesian Statistics 4 (eds J.M.
Bernarde. J.O. Berger, A.P. Dawid ana A.F.M. Smith). Oxford: Oxford University Press.
Gianola, D. (1982). Theory and Analysis of Threshold Characters. J Anim. Sci., 54 : 1079.
Gianola, D., and Fernando, R.L. (1986). Bayesian Methods in Animal Breeding Theory. Journal of
Animal Science, 63, 217-244.
Gianola, D., Fernando, R.L., Im, S., and Foulley, J.L. (1989). Likelihood Estimation of Quantitative
Genetic Parameters when Selection occurs: Models and Problems. Genome, 31, 768 -777.
Gianola, D., and Foulley, J.L. (1982). Non-linear Prediction of Latent Genetic Liability with Binary
Expression: An Empirical Bayes Approach. Proc. 2nd World Congo Genet. Appl. Livestock Prod., 7 :
293.
Gianola, D., and Foulley, J.L. (1983). Sire Evaluation for Ordered Categorical data with a Threshold
Model. Genet. Sel. Evoi., 15: 201.
Gianola, D., and Foulley, J.L. (1990). Variance Estimation from Integrated Likelihoods (VEIL).
Genet. Sel. Evoi., 22,403-417.
Gianola, D., Foulley, J.L., and Fernando. R.L. (1986). Prediction of Breeding Values when Variances
are not known. Genet. Sel. Evol., 18,485 - 498.
Gianola, D., and Goffinet, B. (1982). Sire Evaluation with Best Linear Unbiased Predictors.
Biometrics,38, 1085 - 1088.
Gianola, D., Im, S., Fernando, R.L., and Foulley, J.L. (1990b). Mixed Model Methodology and the
Box-Cox Theory of Transformations: A Bayesian Approach. In: Advances in statistical methods for
genetic .improvement of livestock (Gianola, D. and Hammond, K. eds). New York, Heidelberg and
Berlin : Springer-Verlag. 15 - 20.
Gianola, D., Im, S., and Macedo, F.. (1990). A Framework for Prediction of Breeding Value, in
Advances in Statistical Methods for Genetic Improvement of Livestock, eds. D. Gianola and
K. Hammond, New York: Springer-Verlag, pp. 210 - 238.
-235-
References
Gianola, D., and Kachman, S.D. (1983). Prediction of Breeding Value in Situations with Nonlinear
Structure : Categorical responses, growth functions and lactation curves. 341h Annu. Meeting. Eur.
Assoc. Anim. Prod. Madrid. Summaries, p 172 (Abstr).
Gilks, W.R., and Roberts, G.O. (1996). Strategies for improving MCMC. In Markov Chain Monte.
Carlo in Practice, W.R. Gilks, S. Richardson, and DJ. Spiegelhalter (eds), 89 - 114. London:
Chapman and Hall.
Gill, J.L. (1975). Design and Analysis of Experiments in the Animal And Medical Sciences. The
lawa State University Press, lawa, U.S.A, Vol 1,204 - 207. .
Gënen; M: (2000). A Bayesian Analysis of the Intraclass Correlations in the Mixed Linear Model.
Commun. Statist. - Theory Meth., 29, 1451 - 1464.
Gopalan, R., and Berry, D.A. (1998). Bayesian Multiple Comparisons using Dirichlet Process Priors.
JAmer. Statist. Assoc., 93, 1130 - 1139.
Green, E., and Strawderman, W.E. (1996). A Bayesian Growth and Yield Model for Slash Pine
Plantations. Journal of Applied Statistics, 23, 285-299.
Green, :PJ., and Richardson, S. (1998). Modeling heterogeneity with and without the Dirichlet
process. Research Report S-98-02. Department of Mathematics, University of Bristol, Bristol.
Hamada, M. and Weerahandi, S. (2000). Measurement System Assessment via Generalized
Inference. Journal of Quality Technology, 32 (3), 241 - 253.
Harris, D.L. (1970). Breeding for Efficiency In Livestock Production: Defining the economic
objectives. J Anim. Sci., 30: 360.
Harville, D.A. (1974). Bayesian Inference for Variance Components Using Only Error Contrasts,
Biometrika, 61, 383 -385.
Harville, D.A. (1977). Maximum Likelihood Approaches to Variance Component Estimation and to
Related Problems. Journal of the American Statistical Association, 72, 320 - 339.
-236-
References
Harville, D.A. (1990). BLUP (Best. Linear Unbiased Prediction and Beyond. In : Advances of
Statistical Methods for Genetic Improvement of Livestock. (Gianola, D. Hammond, K. eds.)
Springer-Verlag, New York - Heidelberg - Berlin, 15-40.
Hastings, W.K. (1970). Monte Carlo Sample Methods using Markov Chain and their Applications ..
Biometrika, 57,97-109.
Henderson, C.R. (1953). Estimation of Variance and Covariance Components. Biometrics 9, 226 -
252.
Henderson, C.R. (1974). General Flexibility of Linear Model Techniques for Sire Evaluation.
Journal of Dairy Science, 57,963-972.
Henderson, C.R., Kempthorne, 0., Searle, S.R., and Van Krosig, C.N. (1959). Estimation of
Environmental and Genetic Trends from Records subject to Culling, Biometrics, 30, 583-588.
Henderson, C.R. (1984). Applications of Linear Models in Animal Breeding. University of Guelph,
Guelph, Ontario.
Hugo, J. (1998). A Bayesian Statistical Evaluation of the Elsenburg Dormer Stud using a Sire Model.
Unpublished Master thesis, University of the Free State.
Ibrahim, J.G., and Laud, P.W. (1994). A Predictive Approach to the Analysis of Designed
Experiments. J Am. Statist. Ass., 89,309 - 319.
Jaynes, E. (1982). On the Rationale of Maximum - Entropy Methods. Proceedings of the IEEE, 70,
939-952.
Jaynes, E. (1988). Comment on Optimal Information Processing and Bayes' Theorem. American
Statistician, 42, 280-281.
Kaplan, E.L., and Meier, P. (1958). Non-parametric Estimation from Incomplete Observations. 1.
Am. Statist. Ass. 53,457 - 481.
-237-
References
KendalI, M.G., and Buckland, W.R. (1971). A Dictionary of Statistical Terms. Hafner, New York.
Kleinman, K.P., and Ibrahim, J.G. (1998). A Semi-parametric Bayesian Approach to the Random
Effect Model. Biometrics, 54,921 - 938.
Kong, A., Liu, J.S., and Wong, W.H. (1994). Sequential Imputations and Bayesian missing data·
Problems. J. Amer. Statist. Assoc., 89,278 - 288.
Kuo, L. (1986). Computations of Mixtures of Dirichlet Process. SIAM J. Sci. Statist. Camp. 7,
60 - 71.
Kuo, L., and Smith, A.F.M. (1992). Bayesian Computations in Survival Models via the Gibbs
Sampler, in Survival Analysis: State of the Art (ed. Klein, J.P. and Goel, P.K.) Kluwer Academic
Publishers: Dordrecht, The Netherlands.
Laird, N.M. (1978). Non-parametric Maximum Likelihood Estimation of a Mixing Distribution, J.
Am. Statist. Ass., 73, 805 - 811.
Laird, N.M. (1981). Empirical Bayes Estimates Using Nonparametrie Maximum Likelihood Estimate
for the Prior. J. Statis. Camp. SimuI., 15, 211 - 220.
Laird, N.M. and Ware, J.H. (1982). Random-effects Models for Longitudinal Data. Biometrics, 38,
963 - 974.
Laud, P.W. and lbrahim, J.G. (1995). Predictive Model Selection. J. R. Statist. Soc., Series B, 57,
247 - 262.
Laud, P.W., and Ibrahim, J.G. (1996). Predictive Specification of Prior Model Probabilities In
Variable Selection. Biometrika, 83, 267 - 274.
Lindley, D.V. (1971). Bayesian Statistics, a Review. Regional Conf. Ser. In Appl. Math. SIAM.
Lindley, D.V., and Smith, A.F.M. (1972). Bayes Estimates for the Linear Model (with discussion).
Journal of the Royal Statistical Society, Ser. B, 34, 1 - 41.
-238-
References
Liu, J.S. (1996). Non-parametric Hierarchical Bayes via Sequential Imputation. Annals of Statistics,
24,911 - 930.
Liu, J.S, Wong, W., and Kong, A. (1991). Correlation Structure and Convergence Rate of the Gibbs
Sampler. Technical Report 304, Dept of Statistics, University of Chicago.
Liu, J.S. (1994). The Collapsed Gibbs Sampler in Bayesian Computations with Application to a Gene
Regulation Problem. J. Amer. Statist. Assoc., 98, 958 - 966.
Liu, J.S., and Chen, R. (1998). Monte Carlo Methods for Dynamic Systems. J. Amer. Statist. Assoc.,
93, 1032 - 1044.
MacEachern, S.N. (1998). Estimating Normal Means with a Conjugate Style Dirichlet Process Prior.
Comm. Statist. Simulation Comput., 23, 727 -741.
MacEachern, S.N. and Mueller, P. (1998). Estimating Mixtures of Dirichlet Process Models. J.
Comput. Graph. Statist., 7,223 - 238.
Mauldin, R.D., Sudderth, W.O., and Williams, S.c. (1992). Pólya Trees and Random Distributions.
Ann. Statist., 20, 1203 - 1221.
Meyer; K: (1983). Maximum Likelihood Procedures for Estimating Genetic Parameters for later
Lactations of Dairy Cattle. J. Dairy Sci., 66, 1988.
Meyer, K. (1989). Restricted Maximum Likelihood to Estimate Variance Components for Animal
Models with Several Random Effects using a Derivative-Free algorithm. Genet. Sel. Evol. 21,317.
Meyer, K. and Thompson, R. (1984). Bias in Variance and Covariance Component Estimators due to
Selection on a Correlated Trait. Z. Tierz. Zuchtung. sbiol., 101, 33..
Mori, M., Woodworth, G.G., and Woolson, R.F. (1992). Application of Empirical Bayes Inference to
Estimation of Rate of Change in the Presence of Informative Right Censoring. Statistics in Medicine,
11, 62 ~- 63 {.
-239-
References
Mueller, P. Erkanli, A., and West, M. (1996). Bayesian Curve Fitting using Multivariate Normal
Mixtures. Biometrika, 83, 67 - 79.
Mueller, P., and Walker, S.G. (1997). A Bayesian Non-parametric Approach to Survival Analysis
using Pólya trees. Scand. 1. Statist., 24, 331 - 340.
Newton, M., Czado, c., and Chappell, R. (1996). Semi-parametric Bayesian Inference for Binary
Regression.1. Am. Statist. Ass., 91,142 - 153.
Numrnelin, E. (1984). General Irreducible Markov Chains and Non-Negative Operators. Cambridge:
Cambridge University Press.
Odell, P.L., and Feiveson, A.H. (1966). A Numerical Procedure to generate a Sample Covariance
Matrix'. Journal of American Statistical Association, 61, 198 - 203.
Patterson, H.O., and Thompson, R. (1971). Recovery of Inter-block Information when Block Sizes
are Unequal. Biometrika, 58, 545-554.
Pretorius, A.L., and Van der Merwe, A.J. (1999). Semi-parametric Bayesian Inference for the Mixed
Linear Model. Technical Report No. 274. Department Mathematical Statistics, University of the Free
State, RSA
Pretorius, A.L., and Van der Merwe, A.J. (2000). A Non-parametric Bayesian Approach for Genetic
Evaluation in Animal Breeding. South African Journal of Animal Science, 30(2), 138 - 148.
Pretorius, A.L., and Van der Merwe, A.J. (2001). Nonparametrie Bayesian Estimation for the Mixed
Linear Model, # 167. Collection of Refereed Artiele from the 6th World Meeting for the International
Society of Bayesian Analysis - Crete.
Raftery, A.L., and Lewis, S. (1992). How many Iteration in the Gibbs Sampler? In Bayesian
Statisties, Volume 4, J.M. Bernardo, 1.0. Berger, A.P. Dawid, and A.F.M. Smith (eds), 763 - 773,
Oxford: Clarendon Press.
Rao, CR. (1971). Minimum Variance Quadratic Unbiased Estimation of Variance Components. 1.
Muit. Anal., 36 : 117.
-240-
References
Rao, C.R. (1973). Linear Statistical Inference and its Applications. P. 528. J. Wiley and Sons, New
York.
Richardson, S., and Green. P.J. (1997). On Bayesian Analysis of Mixtures with an Unknown Number
of Components (with discussion). JR. Statist. Soc. B, 59, 731 -792.
Roeder, K., and Wasserman, L. (1997). Practical Bayesian Density Estimation using Mixtures of
Normals. J Am. Statist. Ass. 92, 894 - 902.
Rothschild, M.F., Henderson, C.R., and Quaas, R.L. (1979): Effects of Selection on Variance and
Covariances of Simulated first and second lactations. J Dairy Sci., 62 : 996.
Sacks, B. (1996). Bayesian Method of Moments (BMOM) Analysis of the Multiple Regression
Model with Auto-correlated Errors. H.G.B. Alexander Research Foundation, University of
Chicago.
Safterthwaite, F.F. (1946). An Approximate Distribution of Estimates Variance Components. Biom.
Bull., 2, 110 - 114.
Searle, S.R. (1971). Linear Models. John Wiley and Sons, New York.
Searle, S.R. (1979). Notes on Variance Components Estimation. A detailed Account of Maximum
Likelihood and Kindred Methodology. Technical Report Bu - 673 - M, Biometrics Unit, CornelI
University, Ithaca, New York.
Searle, S.R. CaselIa, G., and McCulloch, C.E. (1992). Variance Components, New York: Wiley.
Shannon, C.E. (1948). The Mathematical Theory of Communication, Bell system Technical Journal,
July - October; reprinted in C.E. Shannon and W. Weaver, The mathematical Theory of
Communication (University of Illinois Press, Urbana, IL) 3-91.
Shore, J., and Johnson, R. (1980). Axiomatic Derivation of the Principle of Maximum Entropy and
the Principle of Minimum Cross-Entropy. IEEE Transactions IT-26, 26-37.
-241-
References
Sorensen, D., Wang, e.S. Jensen, J., and Gianola, D. (1994). Bayesian Analysis of Genetic Change
Due to Selection Using Gibbs Sampling. Genetics, Selection, Evolution, 26, 333 - 360.
Swart, J.e. (1967). The Origin and Development of the Dormer Sheep Breed. Meat Ind. 31, May-
June 1967.
Tao, H., Palta, M., Yandell, B.S., and Newton, M.A. (1999). An Estimation Method for the Semi-
parametric Mixed Effects Model. Biometrics, 55, 191 - 202.
Thompson, R. (1977). The Estimation of Heritability with Unbalanced data, II. Data available on
more than two Generations. Biometrics, 33 : 497. .
Thompson, R. (1982). Recent Developments in the Estimation of Variance Components and their
Application to the Estimation of Genetic Parameters. In : R.A. Barton and W.C. Smith (Ed) Proc. Of
the World Congo On Sheep and Beef Cattle Breeding. Vol. 1, pp 217 - 224. Dunmore Press,
Palmerston North, New Zealand.
Tiao, G.C., and Tan, W.Y. (1965). Bayesian Analysis of Random-effect Models in the Analysis of
Variance, I. Posterior distribution Of Variance Components. Biometrika, 52 : 37.
Tiao, G.C., and Tan, W.Y. (1966). Bayesian Analysis of Random-effect Models in the Analysis of
Variance, II. Effects of Auto-correlated Errors. Biometrika, 53 : 477.
Tierney, L. (1991). Markov Chains for Exploring Posterior Distributions. Technical Report No. 560,
School of Statistics, University of Minnesota.
Tierney, L. (1991). Markov Chains for Exploring Posterior Distributions. Technical Report 560,
University of Minnesota, School of Statistics.
Tierney, L. (1994). Markov Chains for Exploring Posterior Distributions (with discussion). Ann.
Statist; 4, .1701 - 1762.
Theobaid, e.M., Firat, M.Z., and Thompson, R. (1997). Gibbs Sampling, Adaptive Rejection
Sampling and Robustness to Prior Specification for the Mixed Linear Model. Genet. Sel Evol, 29,
57 -72.
-242-
References
Tobias, J., and Zellner, A. (1997). Further Results on the Bayesian Method of Moments Analysis of
the Multiple Regression Model. H.G.B. Alexander Research Foundation, University of Chicago.
Presented at Econometric Society Meeting, June 1997.
Van der Merwe, A.J. (2000). Reference and Probability Matching Priors in the Case of Mixed Linear
Models. Technical Report No.286, Department Mathematical Statistics, University of the Free State,
RSA
Van der Merwe, A.J., and Botha, T.J. (1993). Bayesian Estimation in Mixed Linear Models using the
Gibbs Sampler. South African Statistical Journal, 27, 149-180.
Van der Merwe, A.J., Pretorius, A.L., and Hugo, J. (1999). Bayesian Method of Moment Analysis
for the Mixed Linear Model. Technical Report No. 261, Department Mathematical Statistics,
University of the Free State, RSA
Van der Merwe, A.J., Pretorius, A.L., and Hugo, J. (2000). Traditional Bayes and the Bayesian
Method of Moment Analysis for the Mixed Linear Model with an Application to Animal Breeding.
South African Statistical Journal. (in press)
Van der Merwe, A.J., and Pretorius, A.L. (2000). Bayesian Estimation in Animal Breeding using
Dirichlet Process Prior. Technical report No. 277, Department Mathematical Statistics, University of
the Free State, RSA
Van der Merwe, A.J., and Pretorius, AL (2000). Bayesian Estimation in Animal Breeding using
Dirichlet Process Prior. Genet. Sel. Evol., (in press).
Van der Merwe, A.J., and Pretorius, A.L., (2001). Bayesian Method of Moments and the Mixed
Linear Model, #150. Collection of Refereed Artic/efrom the 6'h World Meetingfor the International
Society of Bayesian Analysis - Crete.
Van Vleck, D.L. (1979). Notes on the Theory and Application of Selection Principles for the Genetic
Improvement of Animals. Cornell Univ., Ithaca, NY.
-243-
References
Van Wyk, J.B. (1992). A Genetic Evaluation of the Elsenburg Dormer Stud. Dissertation submitted
to the 'Faculty of Agriculture in Partial Fulfillment of the Requirements for the Degree Philosophy
Doctor, The University of the Orange Free State, Department of Animal Science.
Walker, S.G., and Muliere, P. (1997a). A Characterization of Pólya Tree Distributions. Statist.
Probab. Lett., 31, 163 - 168.
Walker, S.G. (1998). A Non-parametric Approach to a Survival Study with Surrogate Endpoints.
Biometrics, 54, 662 - 672.
Walker, S.G., and Damien, P. (1998). A Full Bayesian Non-parametric Analysis involving a Neutral
to the Right Process. Scand. J Statist., 25, 669 - 680.
Walker, S.G., and Mueller, P. (1997). Beta-Stacy Processes and a Generalization of the Pólya-urn
Scheme. Ann. Statist., 25, 1762 - 1780.
Wang, C.S., Gianola, D., Sorensen, D.A:, Jensen, J., Christensen, A., and Rutledge, J. (1994a).
Response to Selection for Litter Size in Danish Landrace Pigs : A Bayesian Analysis. Theoretical
and Applied Genetics, 88,220 - 230.
Wang, C.S., Rutledge, J., and Gianola, D. (1994b). Bayesian Analysis of Mixed Linear Models via
Gibbs Sampling With an Application to Litter Size in Iberian Pigs. Genetics, Selection, Evolution. 26,
91-115.
Wang, C.S., Rutledge, JJ., and Gianola, D. (1993). Marginal Inferences about Variance Components
in a Mixed Linear Model using Gibbs Sampling. Genet. Sel. Evol., 25,41-62.
Wasserman, L. (1998). Asymptotic Properties of Non-parametric Bayesian Procedures. In Practical
Nonparametrie and Semi-parametric Bayesian Statisties (eds D. Dey, P. MUlier and D. Sinha). New
York: Springer.
West, M. (1990). Bayesian Kernel Density Estimation, ISDS Discussion Paper, 90 - A02, Duke
University.
-244-
References
West, M. (1992). Hyperparameter Estimation in Dirichlet Process Mixture Models. Technical Report
92 - A03, Duke University, (SOS.
West, M., and Cao, G. (1993). Assessing Mechanisms of Neural Synaptic Activity, in Bayesian
Statisties in Science and Technology: Case Studies, C. Gatsonis, J. Hodges, R. Kass and N.
Singpurwalla (eds), (to appear).
West, M., Mueller, P., and Escobar, M.D. (1994). Hierarchical Priors and Mixture Models, with
Applications in Regression and Density Estimation. In Aspects of Uncertainty: A tribute to D.V.
Lindley, A.F.M. Smith and P.R. Freeman (eds), London: Wiley.
Wilks, W.R., Wang, e.C., Yvonnet, B., and Coursaget, P. (1993). Random Effects Models for
Longitudinal Data using Gibbs Sampling. Biometrics, 49,441 - 453.
Zeger, S., and Rizaul Karim, M. (1991). Generalized Linear Models with Random Effects: A Gibbs
Sampling Approach. J Am. Statist. Ass., 86, 79 - 86.
Zellner, A: (1971). An Introduction to Bayesian Inference in Econometrics. J. Wiley and Sons, New
York.
Zellner, A. (1975). Bayesian Analysis of Regression Errors. Journal of the American Statistical
Association, 70, 138-144.
Zellner, A. (1996). Bayesian Methods of Moments/Instrumental Variable (BMOMlIV) Analysis of
Mean and Regression Models (in J. Lee, W. Johnson and A. Zellner (edsj). Modeling and Prediction:
Honoring Seymour Geyser. Springer Verlag, 61-74 (and Zellner (1997b».
Zellner, A. (1997). The Bayesian Method of Moments (BMOM) : Theory and Applications. (in T.
Fomby and R. Hill (eds.j). Advances in Econometrics, 12,85-105.
Zellner, A., and Highfield, R. (1988). Calculation of Maximum Entropy Distributions and
Approximation of Marginal Posterior Distribution. Journal of Econometrics, 37, 195-210.
-245-
References
Zellner, A., and Min, C. (1993). Bayesian Analysis, Mode Selection and Prediction, in: W.T. Grandy,
Jr. and P.W. Milonni, eds., Physics and probability: Essays in honor of Edwin T. Jaynes, Cambridge
University Press, Cambridge. 195 - 206.
Zellner, A., and Sacks, B. (1996). Bayesian Method of Moments (BMOM) Analysis of the Multiple
Regression Model with Auto-correlated Errors, H.G.B. Alexander Research Foundation ms., U. of
Chicago, presented to the Workshop in Economics and Econometrics, April 19.
Zellner, A., Min, C., Dallaire, D., and Currie, J. (1994). Bayesian Analysis of Simultaneous Equation,
Asset-Pricing and Related Models using Markov Chain Monte Carlo Techniques and Convergence
Checks. H.G.B. Alexander Research Foundation, University of Chicago.
Zellner, A., Tobias, J., and Ryu, H.K. (1999). Bayesian Method of Moments (BMOM) Analysis of
Parametric and Semi-parametric Regression Models. South African Statist. J. 3 1, 41-69.
-246-
Appendix A: Selective Algorithmsfor the Gibbs Sampler
APPENDIX A
«Selective Algorithms for the Gibbs Sampler»
1.1 Algorithm for the Traditional Bayes Analysis
%%Load data with X, Y and Z matrices
%%Initialize the different variables
N the sample size;
SimTot= the simulation total;
Gibbs save every ..th sampled value in the Gibbs sampler;
q the number of random e~fects;
p the number of fixed effects;
%%The Random Effects
UVec=[0.13957685
3.32999015
0.57818853
0.79500881); A qx1 ve~tor of starting values for the Gibbs
sampler (Random effects)
%%Calculate initial values to be used, e.g.
InvXtX=inv(X'*X);
Yster=Y-Z*UVec;
BetaHat=invXtX*(X'*Yster);
e=Yster-X*BetaHat;
%%The Gibbs Sampler
for i=l:SimTot
for j=l:Gibbs
%%Simtilate 0'; from an Inverse Gamma
%%Specify the degree of freedom. Note that a Gamma distribution is the
%%sum of df squared random numbers, i.e.
-247-
Appendix A: Selective Algorithms for the Gibbs Sampler
df=N-2i
x=(randn(df,l) )2i
v=sum(Xx) i
%%Calculate
numvec=(Y-X*BetaVec-Z*UVec) '*(Y-X*BetaVec-Z*UVec)i
02e=numvec/vi
%%where BetaVec is the pxl vector of fixed effects
%%Simulate BetaVec from a Normal distribution.
%%Calculate
muB=BetaHati
sigmaB=sqrtm(invXtX*o2e)i
BetaVec=sigmaB*randn(p,l)+muB;
%%Simulate CJ: from an Inverse Gamma distribution
%%Calculate
df=q-2i
x=(randn(df,1))2i
v=sum(Xx) i
Ainv=inv (A);
o2u=(UVec'*Ainv*UVec)/v;
%%Simulate UVec from a Normal distribution
%%Calculate
muU=[inv((Z'*Z+(o2e/o2u)*Ainv) ))* [Z'*(Y-X*BetaVec));
sigmaU=sqrtm(02e*inv(Z'*Z + (o2e/o2u) * Ainv));
x=randn(q,l)i
UVec=sigmaU*x+muU;
End %% Update the different model parameters
END .%%'Program
%%Calculate and display the averages of the different model parameters
%%SAVE results.
-248-
Appendix A: Selective Algorithms for the Gibbs Sampler
1.2 Algorithm for Simulating a y; with a certain Probability (see Di:richlet process)
%%Specify the vector with the different probabilities
y1~ProbVec %%e.g. ProbVec = [0.25 0.25 0.45 0.05) with sum(ProbVec)=l
%%Specify the vector to choose from
sires= %%e.g. [12 3 4}, thus sire i can be set equal to sire 2 with
%%probability 0.25, sire 3 with probability 0.45 etc.
%%Then calculate
Oppv= sum(1*y1);
y1=y1./0ppv;
CumSumY=cumsum(y1') ';
kwh=l*csY;
tr=rand (l, 1);
kla=(kwh-tr);
kl=abs(kwh-tr);
klein=min(abs(kwh-tr));
IN=find(kl==klein);
s=size (IN);
if s(l,l»l
.IN=IN(l,l);
end
kleina=kla (IN);
if kleina < 0
ID=I.N+1;
else
ID=IN;
end
.pp=sires(ID);
%%Thus, sire i will have the same breeding value as the sire in the
%%position ID of the vector 'sires'.
-249-
Appendix A: Selective Algorithmsfor the Gibbs Sampler
1.3 Algorithm for Simulating the Variance-covariance Matrix D from an Inverse
Wishart
%%First, simulate A from a Wishart (\[L" A,A,' )-1 ,k-v-l,v) distribution
%%Simulate l:.l,l2 , '" ,lk-v-l form Nv(Q, Iv), i.e. (k-v-1) vectors each of
%%order (vx1) from a multivariate normal distribution
loop=k-v-l
H=[] t-
for q=l:loop
H=[H randn(v,l)];
end
I I
%%Define H [ll,l2'" ·,lk-v-d, then calculate A=(IA,A;tïHH'(IA,A;tï
for i=O:k
sta=i*2+1;
sto=(i+1)*2;
u=UVec(sta:sto);
utu=u*u';
UtU=UtU+utu;
end
mat=sqrtm(inv(UtU) );
A=mat*H*H'*mat;
D=inv (A) ; ,
%%Then 0 = A-I is from an Inverse Wishart distribution. Simulation from
the Wishart distribution can also easily be done by using the algorithm of
Odell and Feiveson (1966) in A Numerical procedure to generate a sample
Covariance matrix, Journal of American Statistical Association, 61, 198 -
203.
-250-
Appendix A: Selective Algorithms/or the Gibbs Sampler
1.4 Algorithm for Simulating the Precision Parameter, M (see Dirichlet process)
%%This algorithm illustrates the simulation of the parameter M and mixing
%%parameter X in the Dirichlet process prior. Gr is the number of
%%groups/clusters for the analysis
%% Initialize starting values for X
XVar=O.S;
%%Simulate two values 21 and 22 from two Chi-square distributions where
%%df= degrees of freedom i.e.
%%Simulate u
df=2*GR;
x=(randn(l,df) )2;
u=sum(x);
2l=u/(-2*log(Xvar));
%%Simulate v
df=2*(GR-l);
x=(randn(l,df) )2;
v=sum(x);
22=v/(-2*log(Xvar) );
%%Set the value of M to any of 21 or 22 with probability 0.5 for each
%%See the algorithm in 1.2
%%Simulate another two values 23 and 24 from two éhi-square distributions,
%%given M
df=2* (M+l);
x=(randn(1,df))2;
v=sum(x) ;
23=v/(2*(M+l));
df=2*.k; .. %%Number of sires k=200;
x=(randn(1,df))2;
v=sum (x);
24=v/(2*k) ;
%%Calculate the following
y=23/24;
Xl=y*(M+l)/k;
X2=1+(y*(M+l)/k);
XVar=Xl/X2;
-251-
Appendix A: Selective Algorithmsfor the Gibbs Sampler
1.5 Algorithm for Plotting the Unconditional Posterior Density of the Precision
Parameter, M (see Dirichlet process)
%%Loid the results
%%Specify the interval for the parameter
M=[lO:lOOO);
%%Calculate the value of the conditional posterior distribution at each M
%%SimTot is the number of simulations, and Gr the vector of resulting
%%groups/clusters from the analysis
for j=l:SimTot
k=Gr (1,j) ;
for i=l:l
m=M (1,i);
y=(mA(k/2)*(m+q)*beta(m+1,q) )*(mA(k/2-2));
Y=[Y yl;
end
YY=YY+Y;
end
%%Calculate the marginal posterior as the average of all the conditional
%%posterior densities
Y=YY/SimTot;
plot(M,Y);
-252-
Appendix A: Selective Algorithmsfor the Gibbs Sampler
1.6 Setup Algorithm for an Animal Breeding Experiment (Chapter 3, § 3.6)
%%Initialize the specifications of the experiment, i.e. the number of
%%observations to sample for each sire for each fixed effect
n=20;
%%Per sire, i.e. n/2 male and n/2 female if sex is included as a fixed
%%effect
%%Number of Sires,
k=200;
%%Total number of observations,
NN=n*k;
%%Specify the true values of the variance components and fixed
%%effect (s), /3i
? ?
0";=4.88; 0";=0.7211; /30 = 0.705,
%%Construct a vector Y (Nxl) of observations distributed N(O,l)
Y=randn(NN,l);
%%Simulate the different random effects according to the Polya Urn scheme
%%and add it to the vectors Y of observations (see Chapter 3, §3.4.3)
%%Add a random residual to each observation
for i=l:k
e=randn (n,1)*sqrt 2(O"c ) ;
Y=[Y (U(i,l)+e));
end
%%Add an overall mean to each observation
Y=Y+12;
%%Add.the effect of the fixed effect to each observation
Y(l: (n/2), :)=Y(l: (n/2), :)+0.705;
%%Con.st:r:ucthe different matrices corresponding to the experiment, i.e. X
%%and Z
%%SAVE the experimental data.
-253-
Appendix B: Dormer Stud Data; Elsenburg College of Agriculture
APPENDIXB.
In this APPENDIX we present the Dormer stud sample used in estimating the breeding values and·
variance components. It refers to 879 weaning weight records from the progeny of 17 sires from the
Elsenburg Dormer sheep stud near Stellenbosch. The sheep used in the analysis were born in the
period 1943 - 1950. The animal ID, sire ID, year (season of birth), age of dam, sex birth status and
weaning weight are presented in this table.
Animal ID refers to the ID of the lamb that was born.
Sire ID and dam ID refers to the parents of the lamb.
Year (season of birth) refers to the year in which the lamb was born. Here 1943 is denoted by 43 and
1944 by 44, ect.
Age of dam refers to the age of the dam used in producing the progeny. Here age 2 is denoted by 2
and age 3 is denoted by 3, eet.
Sex refers to the sex of the lamb. Here male is denoted by I and female by 2.
Birth status refers to the birth status of the lamb. Here single births are denoted by 1, twins are
denoted by 2 and triplets are denoted by 3.
Weaning weight refers to the weaning weight of the lambs in kilogram.
-254-
Appendix B: Dormer Stud Data; Elsenburg College of Agriculture Appendix B: Dormer SUn)Dato. /;·/senburg ('vllege (~fAgriculture
43064 41037 41139 43 2 I I 36.6
43067 41004 41163 43 2 I 2 29.6
ANIMAL ID SIRE ID DAM ID SEASON AGE OF SEX BIRTH WEANING 43068 41004 41163 43 2 I 2 29.6
DAM STATUS WEIGHTS 4307Ó 41004 .41175 43 2 2 I 34.9
,. 43071 41037 41042 43 2 I I 3443002 41037 41076 43 2 I 34.2 43072 41037 41015 43 2 2 I. 35.3
43003 41037 41130 43 2 2 I 33.1 43077 41037 41115 43 2 I I 30.6
43004 41037 41029 43 2 I I 40.1 43081 41037 41093 43 2 2 1 35.5
43005 41004 41134 43 2 I I 32 43083 41037 41053 43 2 2 1 34.5
43006 41037 41096 43 2 I 2 33.6 43084 41004 41030 43 2 2 1 33.8
43008 41037 41007 43 2 I I 38.5 43085 41004 41013 43 2 1 1 41.1
43009 41037 41167 43 2 I I 37.9 43086 41004 41010 43 2 2 1 34.2
43010 41037 41031 43 2 2 I 37.8 43087 41037 41053 43 2 2 1 34.5
43011 41004 41040 43 2 I I 30.6 43088 41037 41058 43 2 2 1 32.1
43012 41004 41165 43 2 2 2 21 43089 41004 41055 43 2 2 1 36.6
43013 41004 41165 43 2 I 2 27. I 43093 41004 41023 43 2 1 1 37.7
43014 41037 41028 43 2 2 I 38.2 43094 41004 41077 43 2 1 2 34.5
43015 41004 41001 43 2 I I 359 43095 41004 41077 43 2 2 2 23.6
43018 41004 41023 43 2 2 I 283 43098 41037 41088 43 2 2 1 35
43019 41004 41104 43 2 I 2 252 43100 41037 41096 43 2 2 2 27.7
43020 41004 41104 43 2 2 2 252 43101 41004 41050 43 2 I 1 39
43021 41004 41112 43 2 I I 36 43102 41037 41138 43 2 1 1 39.4
43022 41004 41181 43 2 2 2 26.8 43104 41037 41151 43 2 I 1 36.4
43023 41004 41181 43 2 I 2 26.8 43110 41037 41044 43 2 2 2 31.8
43024 41004 41187 43 2 2 I 36.3 43111 41037 41060 43 2 2 1 21.4
43025 41004 41068 43 2 2 I 31.8 43117 41037 41057 43 2 2 1 31.8
43026 41004 41061 43 2 2 2 26.7 43118 41037 41035 43 2 I 1 38.2
43027 41004 41061 43 2 I 2 27.4 43136 41004 41174 43 2 2 I 30.2
43028 41004 41033 43 2 I I 36.5 44002 41037 41040 44 3 2 1 29.8
43029 41037 41051 43 2 2 I 34.6 44003 41004 41130 44 3 2 1 26.6
43030 41004 41034 43 2 I I 32.1 44007 41037 41096 44 3 1 2 34.4
43032 41004 4105 43 2 2 2 29.1 44012 41004 41068 44 3 1 1 41.7
43033 41004 4105 43 2 2 2 25.9 44013 41004 41003 44 3 2 1 37.1
43035 41037 41087 43 2 1 I 37.1 44022 41004 41171 44 3 2 2 36.1
43037 41004 41046 43 2 2 I 27.9 44026 41004 42080 44 2 2 1 35
43038 41004 41074 43 2 I 2 36.2 44027 41037 41031 44 3 2 1 33.5
43039 41004 41074 43 2 I 2 30.7 44028 41037 41008 44 3 2 1 38
43043 41037 41018 43 2 I I 37.9 44033 41004 41025 44 3 1 2 36.9
43044 41004 41171 43 2 I 1 44.8 44034 41004 41025 44 3 1 2 33.8
43045 41004 41047 43 2 2 I 39.3 44040 41004 41165 44 3 2 1 35
43049 41004 41080 43 2 2 I 24.7 44041 41037 42071 44 2 1 1 35.1
43053 41037 41172 43 2 2 I 33.6 44042 41004 41187 44 3 2 1 37.7
43058 41037 41069 43 2 I I 41.9 44043 41004 41187 44 3 2 2 29.4
43061 41037 41041 43 2 I I 37.6 44050 41004 41144 44 3 2 1 32.8
43062 41004 41074 43 2 I 2 38.2 44052 41037 41029 44 3 1 2 33.9
43063 41004 41074 43 2 2 2 30.1 44053 41037 41029 44 3 1 2 31.5
-255- -256-
Appendix B: Dormer Stud Dato; Elsenburg CoJ/ege of Agriculture Appendix H: Dormer ."1111"Data; Elsenburg ('vl/exe of Agrscuhure
44054 41004 41135 44 3 1 2 30.5 44192 41004 41043 44 3 1 1 32.1
44055 41004 41135 44 3 1 2 33 44198 41004 41138 44 3 2 1 38.9
44063 41004 41181 44 3 1 2 42.5 44204 41037 41093 44 3 1 1 38.8
44064 .. 41004 41047 44 3 2 2 . 30 44205 41037. 41041 44 3 2 1 379
44066 41004 41047 44 3 1 2 27.2 44210 41004 42069 44 2 1 2 37.8
44072 41037 41167 44 3 2 1 37.3 44212 41037 41174 44 3 1 1 31.2
44076 41037 41029 44 3 1 2 29.4 44213 41004 41005 44 3 2 1 31.8
44077 41037 41029 44 3 2 2 28.8 44217 41004 41175 44 3 2 1 35.9
44082 41004 41015 44 3 2 1 36.4 44220 41037 41104 44 3 2 2 30.8
44083 41004 41087 44 3 2 1 38.7 44222 41037 41013 44 3 2 1 38.9
44084 41004 41035 44 3 1 1 38.1 44224 41037 41055 44 3 2 2 34.4
44093 41004 41051 44 3 2 2 30 44228 41037 41018 44 3 2 1 35
44094 41004 41051 44 3 2 2 29.1 44230 41037 41131 44 3 2 1 37.2
44096 41037 41096 44 3 1 2 32.2 44232 41004 41053 44 3 2 1 36.8
44098 41037 41105 44 3 2 2 34.1 44236 41037 42062 44 2 2 1 34.7
44110 41037 41099 44 3 2 1 37.4 44244 41037 42071 44 2 1 2 30.4
44115 41004 41023 44 3 2 1 32.8 44245 41037 42071 44 2 2 2 27.8
44121 41004 42071 44 2 2 2 23.1 44250 41037 41057 44 3 2 ~ 38.9
44122 41004 42071 44 2 2 2 19.7 44252 41037 41050 44 3 2 2 33
44123 41004 41122 44 3 1 1 34.6 44253 41037 41050 44 3 2 2 31.2
44125 .41037 41058 44 3 1 1 39.8 45001 41004 41015 45 4 2 2 23.4
44127 41004 41042 44 3 1 2 26.4 45002 41004 41015 45 4 1 2 28.5
44128 41004 41042 44 3 1 2 30.3 45003 41004 41122 45 4 2 2 31.9
44129 41004 41139 44 3 1 1 37.6 45004 11004 41122 45 4 1 2 34.4
44130 41004 41172 44 3 2 1 35.5 45005 43002 43136 45 2 1 1 38.4
44134 41037 42115 44 2 2 1 32 45006 43002 43029 45 2 1 1 35.5
44147 41004 42060 44 2 2 2 35.8 45007 41019 41040 45 4 1 1 41
44149 41004 41061 44 2 1 1 41 45008 41004 43003 45 2 1 1 36.9
44157 41037 41028 44 3 1 2 33.6 45009 43002 43045 45 2 1 1 27.5 I
44158 41037 41028 44 3 2 2 29.3 45010 41019 41057 45 4 1 2 30.4 ,
44162 41004 41034 44 3 1 2 30.3 45011 41019 41057 45 4 2 2 30
44163 41004 41034 44 3 2 2 25.9 45015 41004 41010 45 4 2 1 39.9
44165 41037 41033 44 3 2 2 29.6 45019 41004 43100 45 2 1 2 33.4
44166 41037 41001 44 3 2 1 38.6 45020 41004 43100 45 2 2 2 23.9
44167 41004 41074 44 3 1 2 26.1 45022 43002 43024 45 2 2 2 21.2
44168 41004 41074 44 3 2 3 20.7 45026 43002 43095 45 2 2 1 30.8
44169 41004 41136 44 3 1 1 38.6 45027 43002 43049 45 2 2 1 28.8
44170 41004 41077 44 3 1 1 34.8 45031 41019 42071 45 3 2 1 33.5
44172 41037 42065 44 3 1 2 23.4 45032 43002 43026 45 2 2 1 20.3
44173 41037 41063 44 3 2 2 22.8 45033 43002 43063 45 2 1 2 32.6
44174 41004 41010 44 3 1 1 33.2 45034 43002 43063 45 2 2 2 24.9
44178 41004 41115 44 3 1 1 38.5 45036 41019 42115 45 3 1 2 44.3
44179 .41037 42071 44 2 1 1 39.8 45040 41019 41028 45 4 1 1 48.1
44183 41004 41083 44 3 1 1 32.6 45042 41019 41031 45 4 2 2 37.8
44184 41037 41023 44 3 2 1 35.2 45044 43002 43033 45 2 1 1 39.9
44191 41037 42065 44 -- 2 1 1 29 45045 41004 41135 45 ~ 1 2 31.6--
-257- -258-
Appendix H: Dormer Stud Data; Elsenburg Cullege of Agriculture Appendix H: Dormer .,"ud Data, Elsenburg (·ol/f.!-xt! (!/ Agrscuhurr
45046 41004 41135 45 4 2 2 25.5 45122 41004 41136 45 4 1 2 308 I
'45047 41004 41013 45 4 1 1 47.6 45123 41019 41055 45 4 1 1 496
45048 43002 43089 45 2 2 2 28.4 45126 41004 41144 45 4 1 1 475
45049 43002 43089 45 2 2 2 29 .45127 41019 42062 45 3 1 2 . 425
45050 41019 41105 45 4 1 2 41.3 45129 41019 42062 45 3 2 2 393
45051 41019 41105 45 4 2 2 38 45132 41019 41133 45 4 2 1 41.5
45053 41004 42069 45 3 2 2 28.6 45133 41004 41047 45 4 1 2 408 .
45054 41004 42069 45 3 2 2 31.8 45134 41004 41047 45 4 2 2 273
45055 41004 41025 45 4 2 2 35 45135 43002 43117 45 2 1 1 343
45056 41004 41025 45 4 2 2 35.3 45153 41004 41043 45 4 2 1 426
45059 41019 41008 45 4 2 1 35.8 45155 43002 43037 45 2 1 2 393
45061 41019 4100 45 4 2 2 34.1 45156 41019 41023 45 4 2 1 38.6
45062 41019 4100 45 4 2 2 26.9 45190 41004 41171 45 4 2 1 474
45063 41019 41041 45 4 2 1 40.6 45205 41004 42060 45 3 1 1 352
45064 41004 41138 45 4 1 1 51.5 45207 41004 41087 45 4 2 1 32.8
45066 41004 41069 45 4 1 1 39.5 45208 41004 42080 45 3 2 1 42 1
45069 41004 43088 45 2 2 2 23.2 45211 41004 41115 45 4 2 1 465
45070 41004 41044 45 4 1 2 46.3 46001 41019 41040 46 5 1 1 417
45071 41004 41044 45 4 2 2 31.1 46002 41004 41044 46 5 1 2 35.2
45072 43002 43018 45 2 1 1 36.7 46003 41004 41044 46 5 1 2 28.4
45073 43002 43084 45 2 2 1 35.3 46004 41004 41122 46 5 2 1 37.8
45074 41004 42071 45 3 2 2 26.6 46005 41004 41015 46 5 2 2 26.8
45075 41004 42071 45 3 1 2 26.2 46006 41004 41015 46 5 2 2 29.3
45076 41004 41034 45 4 1 2 37.4 46007 41019 41057 46 5 1 1 39.6
45077 41004 41034 45 4 2 2 34.6 46008 41004 41135 46 5 2 1 376
45081 41019 41018 45 4 1 1 38 46009 44170 43084 46 3 2 2 29.2
45082 41004 41051 45 4 2 2 26.2 46010 44170 43084 46 3 2 2 26.3
45083 41004 41051 45 4 1 2 30.6 46012 41004 41139 46 5 1 2 32.2
45084 41004 41074 45 4 2 3 22.2 46013 41004 41139 46 5 2 2 301
45085 41004 41074 45 4 1 3 30 46014 41004 41043 46 5 1 I' 41.9
45086 41004 41074 45 4 1 3 29.2 46015 41004 42087 46 4 1 1 41.1
45094 41004 41172 45 4 2 1 40 46020 41004 41034 46 5 2 2 22.4
45096 41004 42087 45 3 2 1 36.3 46021 41004 41034 46 5 1 2 32.3
45098 41019 41001 45 4 2 1 47.2 46022 44170 43018 46 3 2 1 31.3
45101 41019 41167 45 4 1 1 49.3 46023 41004 41069 46 5 2 2 29.4
,
45102 41019 42065 45 3 2 1 40.8 46024 41004 41069 46 5 2 2 30.3
45103 43002 43086 45 2 2 1 36.4 46025 41004 41138 46 5 2 1 33.2
45111 41004 41083 45 4 1 2 38.9 46026 41019 42071 46 4 2 1 30.4
45112 41004 41083 45 4 2 2 31.2 46031 41019 41167 46 5 1 1 47.6
45113 43002 43072 45 2 2 2 28.2 46032 41004 41172 46 5 1 1 215
45114 43002 43072 45 2 1 2 34.4 46033 41004 41171 46 5 1 2 30.6
45115 41004 43014 45 2 2 1 33.9 46034 41004 41171 46 5 2 2 24.7
"
45116 41019 41093 45 4 2 1 . 41.2 46037 44170 44228 46 2 1 1 40. t
45118 41004 43081 45 2 1 2 30.8 46040 44170 43045 46 3 1 1 31.9
45119 41004 43081 45 2 2 2 26.2 46041 41019 41033 46 5 1 2 35.6
45121 41004 41136 45 4 2 2 27.1 46042 41019 41033 46 5 1 2 26.1 I- -
-259- -260-
Appendix Jl: Dormer Stud Dato: Elsenburg College of Agriculture Appendix 13: Dormer Stud Dato; Elsenburg ('ol/eXl' (dAgriculmrv
-- ----
46044 41004 42080 46 4 1 1 40 46208 44174 43063 46 3 1 2 22.7
46045 41019 42115 46 4 1· 2 34.9 46209 44174 43063 46 3 2 2 23
46046 41019 42115 46 4 2 2 32.4 46210 44042 44063 46 2 2 1 23.1
46047 41004 42060 46 4 .. 2 1 37.3 46214 44042 43081 46 3 2 2 22.6
46048 41019 41041 46 5 2 2 35.8 46215 44042 43081 46 3 2 2 23.5
46050 44170 43033 46 3 1 1 44.1 46217 44042 44082 46 2 1 1 29.1
46054 44170 44158 46 2 2 1 34.6 46222 44042 44121 46 2 1 1 29.8
46055 44170 44098 46 2 1 1 26.4 46224 44174 44134 46 2 2 1 38.3
46057 41004 41051 46 5 1 1 28 46227 44042 44252 46 2 1 1 28.6
46058 41004 41010 46 5 1 1 23.6 46228 44042 44093 46 2 2 1 31
46063 44170 44072 46 2 1 1 27.7 47001 45070 45073 47 2 2 2 21.8
46064 41004 41061 46 5 2 2 32.2 47002 45070 45073 47 2 1 2 246
46065 41004 41061 46 5 1 2 33.6 47003 45070 45049 47 2 1 2 23.2
46070 41004 4100 46 5 1 2 20.9 47004 45135 45211 47 2 1 2 22.1
46072 41004 41144 46 5 2 1 36.2 47006 41019 41040 47 6 1 1 30.9
46073 41004 41025 46 5 2 1 36.5 47007 44042 43003 47 4 1 1 29.2
46076 41019 41008 46 5 1 2 38 47008 45135 45054 47 2 2 2 20.9
46086 44170 44184 46 2 1 1 26.1 47009 45135 45054 47 2 1 2 24.7
46087 41004 41136 46 5 2 2 23.8 47010 45135 43018 47 4 1 2 26
46088 41004 41136 46 5 1 2 33.1 47011 45135 44205 47 3 1 2 26.3
46095 41004 41074 46 5 2 2 24 47012 45135 44205 47 3 1 2 28.5
46096 41004 41074 46 5 1 2 29 47014 45135 43033 47 4 2 1 29:4
46097 41004 41083 46 5 1 2 26.6 47015 44042 44013 47 3 2 1 28.6
46098 41004 41083 46 5 1 2 28 47016 41004 41025 47 6 1 2 37.1
46102 41019 42062 46 4 2 1 33.6 47017 45135 44222 47 3 2 2 17.6
46125 41019 42065 46 4 1 1 42.7 47018 45135 44222 47 3 1 2 28.1
46131 41019 41023 46 5 1 2 30.6 47019 45135 44134 47 3 1 1 38.5
46132 41019 41023 46 5 1 2 31.7 47020 45135 44098 47 3 1 1 33.4
46133 41019 41031 46 5 2 2 31.3 47022 45070 45115 47 2 2 1 29.3
46134 41019 41031 46 5 2 2 31.4 47023 41019 41031 47 6 1 2 32.2
46135 44170 44224 46 2 1 1 34.7 47024 41019 41031 47 6 2 2 27.6
46136 44170 44222 46 2 1 1 36.6 47026 45135 44158 47 3 2 1 28.6
46137 44170 44165 46 2 1 1 23.2 47028 41004 41010 47 6 1 1 35
46147 41019 41093 46 5 2 1 36.5 47029 41004 41015 47 6 2 1 25.4
46149 41004 41047 46 5 1 1 38.8 47030 41004 41015 47 6 1 1 23
46163 44170 44205 46 2 2 1 34.6 47033 45135 45077 47 2 1 1 35.4
46170 41004 42074 46 4 2 1 30.5 47034 44042 44217 47 3 2 1 25
46175 41004 41115 46 5 2 1 29.2 47035 45135 45082 47 2 1 2 21
46177 41019 41001 46 5 2 1 44.5 47036 45135 45082 47 2 1 2 22.2
46181 44174 44166 46 2 1 2 28 47037 41004 4100 47 6 1 1 37.4
46182 44174 44166 46 2 2 2 24 47040 45135 44165 47 3 1 1 32.4
46189 44174 43086 46 3 1 1 23.9 47042 41004 41047 47 6 1 2 36.2
46198 44042 44083 46 2 1 1 26.9 47043 44042 44082 47 3 2 1 28.2
46199 44042 43110 46 3 1 1 31.1 47049 41004 41135 47 6 1 2 28.5
46205 44042 43100 46 3 2 1 27.8 47051 45135 45053 47 2 1 1 30.4
46206 44042 44013 46 2 1 1 33.6 ~55 44042 43110 47 4 1 2 24
-261- -262-
Appendix B: Dormer SlIId DUIa; Elsenburg College ufAgriculture Appendix Jj: Dormer Stud Dato: I:'lscllhurg('u"e~e ofAgrsculture
_----- - .~- ~---- _ ..
47056 44042 43110 47 4 2 2 24.1 47140 44042 43029 47 4 2 2 24.9
47060 44042 44063 47 3 2 2 17.2 47141 44042 43029 47 4' 2 2 25.2
47064 41004 42060 47 5 1 1 33.3 47147 45135 43063 47 4 2 2 23.9
47067 41004 42087 47 5 1 2 33 47148 45135 43063 47 .. 4 2 2 21.6
47068 41004 42087 47 5 2 2 23.2 47150 45135 45056 47 2 2 2 22.1
47074 44042 44213 47 3 2 1 27.6 47151 45135 45056 47 2 2 2 18.7
47076 41019 41008 47 6 2 2 25 47161 44042 44253 47 3 1 2 31.9
47077 41019 41008 47 6 2 2 29.8 47167 45070 45048 47 2 2 2 25.2
47078 45135 45121 47 2 2 2 11.5 47169 45070 45048 47 2 1 2 22.1
47080 45135 45190 47 2 1 2 26.8 47170 45135 43012 47 4 1 1 30.6
47083 45135 44184 47 3 2 1 25.6 47174 45135 45094 47 2 2 1 34
47087 45135 45015 47 2 2 1 25.7 47184 44042 43100 47 4 1 1 22.3
47091 41004 41083 47 6 1 1 27.1 47188 45135 43024 47 4 2 1 27.3
47092 41004 41051 47 6 2 2 16.4 47189 44042 44252 47 3 1 1 32.9
47094 44042 44022 47 3 2 2 18.2 47204 41004 41044 47 6 1 1 32.6
47095 44042 44022 47 3 2 2 22.8 47208 41019 41041 47 6 2 2 30.6
47096 44042 45102 47 2 1 2 24.2 47210 44042 45063 47 2 1 1 23
47097 44042 45102 47 2 2 2 23.2 47211 44042 44064 47 3 1 1 32.6
47098 41019 41033 47 6 1 1 37.7 47212 45135 43020 47 4 2 1 24.9
47101 41004 42071 47 5 2 2 17.1 47213 45135 44027 47 3 1 1 32.2
47102 41004 42071 47 5 1 2 21.1 47215 41004 41061 47 6 2 2 26.1
47105 45135 44072 47 3 2 1 26.8 47216 41004 41061 47 6 1 2 26
47108 41004 41138 47 6 1 2 28.1 48001 46015 45049 48 3 2 2 22.1
47109 41004 41138 47 6 2 2 26.2 48002 46015 45049 48 3 2 2 25.4
47111 41004 45026 47 2 1 2 12.4 48003 41004 41031 48 7 2 2 22.6
47112 41019 42071 47 5 2 2 21.2 48004 41004 41031 48 7 1 2 -33-.6---
47113 41019 42071 47 5 2 2 21 48006 44042 41040 48 7 1 2 33.8
47114 45135 44224 47 3 1 2 26.4 48007 41004 41144 48 7 1 2 24.4
47115 45135 44224 47 3 2 2 23.9 48008 41004 41144 48 7 2 2 24.6
47117 41004 41144 47 6 1 1 28.4 48009 46037 44083 48 4 1 1 39.5
47118 41004 41136 47 6 1 2 27.9 48010 46037 45054 48 3 2 1 35.2
47119 41004 41136 47 6 2 2 22.3 48011 44042 45115 48 3 1 2 20.9
47120 45135 45055 47 2 2 1 26.9 48012 44042 45115 48 3 1 2 29.1
47121 41004 41043 47 6 1 2 38.8 48013 41004 42080 48 6 2 1 42.7
47123 41019 41028 47 6 1 2 28.2 I 48014 45070 45116 48 3 1 1 42.5
47124 41019 41028 47 6 2 2 31.1 48015 46037 44022 48 4 1 1 39.3
47126 41019 41023 47 6 1 2 22.7 48016 46037 46170 48 2 1 1 33.6
47127 41019 41023 47 6 1 2 26.6 48017 46015 44205 48 4 2 2 27.7
47130 45135 43037 47 4 2 2 22.6 48018 46015 44205 48 4 1 2 32.4
47131 45135 43037 47 4 1 2 30.8 48023 44042 41028 48 7 1 2 37.4
47132 44042 45098 47 2 2 1 30.8 48027 41004 46134 48 2 1 1 40
47133 44042 45098 47 2 2 1 30.8 48028 41004 41122 48 7 1 2 28.9
47134 41019 42065 47 5 1 2 38 48029 41004 41122 48 7 2 2 23.8
47135 41019 41057 47 6 1 2 31.4 48030 41004 46177 48 2 2 2 31.1
47136 44042 45061 47 2 1 1 33.3 48031 41004 46177 48 2 2 2 30.9
47139 45135 43088 47 4 1 1 28.3 48032 41004 45036 ~_f!_ __ L. 3 2 2 29.9
-263- -264-
Appendix B: Donner Stud Dato; Elsenburg College aj Agriculture Appendix 8: Dormer Stud Data; tilsenburg ('ol!t.~Keof Agrrcuh ure
48034 41004 41001 48 7 2 2 32.9 48098 46037 44115 48 4 1 1 44.5
48035 41004 41001 48 7 2 2 32.8 48100 44042 41023· 48 7 2 2 36.4
48036 41004 41010 48 7 2 1 39.9 48102 44042 44163 48 4 1 2 30.3
48039 .46015 45027 48 3 1 1 32.1 48i03 44042 44163 48 4 2 2 24.4
48042 46037 45211 48 3 2 2 27.1 48104 45070 43100 48 5 1 1 36 I
48043 46037 45211 48 3 1 2 30.4 48107 46037 44217 48 4 1 1· 35
48045 45070 46006 48 2 1 1 47.3 48110 46037 43089 48 5 1 3 33.2
48046 46015 46064 48 2 1 1 42.6 48111 46037 43089 48 5 1 3 19.7
48047 45070 45063 48 3 2 2 35.6 48112 46037 43089 48 5 2 3 21.8
48049 46037 45082 48 3 1 2 20 48113 45070 46209 48 2 1 2 19.3
48050 46037 45082 48 3 2 2 27.2 48114 45070 46209 48 2 2 2 20.7
48051 41004 46048 48 2 2 1 27.7 48116 45070 43026 48 5 1 1 36.8
48052 44042 41074 48 7 1 3 25.6 48117 45070 45094 48 3 2 1 31.1
48054 44042 41074 48 7 2 3 22.5 481Hi 46015 44098 48 4 2 2 28.8
48056 45070 43033 48 5 2 2 26.2 48119 46015 44098 48 4 1 2 29.2
48057 45070 43033 48 5 2 2 29.6 48120 46015 46008 48 2 1 2 23.3 I
48058 41004 42060 48 6 2 1 37 48121 46015 46008 48 2 2 2 21.7
I
48060 45070 44121 48 4 2 1 381 48122 44042 41105 48 7 1 2 34.7
48062 46037 46076 48 2 1 1 453 48123 44042 41105 48 7 1 2 31.1
48064 46015 46004 48 2 1 2 252 48124 45070 45156 48 3 2 2 20.9
48065 46015 46004 48 2 2 2 255 48125 45070 45156 48 3 1 2 25.7
48066 44042 45073 48 3 2 2 29.5 48126 46015 46072 48 2 2 2 25.2
48067 44042 45073 48 3 2 2 27.6 48127 46015 46072 48 2 1 2 27.6
48068 46015 46175 48 2 2 2 25 48129 45070 43020 48 5 2 1 27.8
,
48069 46015 46175 48 2 2 2 26.1 48130 46037 44064 48 4 1 1 37.5
48070 44042 45102 48 3 1 2 27.8 48131 46037 45015 48 3 2 2 24.2 I
48071 44042 45102 48 3 2 2 29.6 48132 46037 45015 48 3 2 2 25.5
48072 45070 43084 48 5 1 1 39.3 48133 46015 44228 48 4 1 2 30.9
48076 44042 43063 48 5 1 2 25.4 48134 46015 44228 48 4 2 2 25.3
48077 44042 43063 48 5 1 2 25.8 48136 46037 46047 48 2 1 1 33.2
48078 44042 41093 48 7 1 2 23.3 48137 44042 43072 48 5 1 2 29.6
48079 44042 41093 48 7 2 2 25.4 i 48138 44042 43072 48 5 1 2 31.5
48080 41004 42065 48 6 1 1 45.7 48139 44042 41083 48 7 2 1 28
48082 41004 41044 48 7 1 1 43 48140 46015 44184 48 4 1 1 37.6
48084 41004 41057 48 7 1 1 44 48141 46015 44224 48 4 1 2 29.7
48085 46037 45077 48 3 1 1 46.1 48142 46015 44224 48 4 1 2 29
48086 46037 45056 48 3 1 1 35.2 I 48143 46015 46023 48 2 2 2 24.6
48087 44042 43029 48 5 1 2 33.6 48144 46015 46023 48 2 2 2 23.6
48088 44042 43029 48 5 1 2 28.1 48145 41004 46147 48 2 2 1 36.4
48089 44042 44082 48 4 2 1 36.6 48146 45070 43003 48 5 1 1 32.9
48090 46015 44072 48 4 2 1 38.3 48147 46037 44013 48 4 2 1 33.6
48091 46037 45003 48 3 1 1 47.2 48150 45070 45153 48 3 2 2 30.8
48092 45070 46005 48 2 1 1 39.8 48151 45070 45153 48 3 1 2 28.4
48093 46037 43086 48 5 2 1 38.9 48152 45070 45061 48 3 2 1 28.2
48095 44042 44253 48 4 1 2 38.7 48155 45070 46102 48 2 2 1 30.4
48097 45070 43014 48 5_ 2 1 38.5 48156 44042 43083 48 5 1 1 37.5
-265- -266-
Appendix H: Dormer Stud Data; Etsenburg College oJAgriculture Appendix H: Dormer Stud t )UIa; 1:'/Sl!lIhllrx ( 'olie}!. •:. ofAgricutture
----_. __ . - --- ----- - --_._ ..-
.._-----
48158 46015 46025 48 2 2 1 32.9 49026 48014 47034 49 2 1 1 41.9
'48159 45070 43018 48 5 2 1 23.8 49029 45070 41031 49 8 2 2 17.3
48160 44042 43110 48 5 2 2 26.4 49030 45070 41031 49 8 2 2 23.6
48161 44042 43110 48 5 1 2 33.1 . 49032 46037 46013 49 3 1 2. 30.6
48162 45070 41047 48 7 2 1 31.6 49033 45070 46147 49 3 2 2 39.5
48164 46037 44213 48 4 1 1 39.8 49034 45070 46147 49 3 2 2 28.9
48166 46015 4622 48 2 2 2 23 49035 48014 47102 49 2 2 1 348
48167 46015 4622 48 2 2 2 25.2 49036 45070 45156 49 4 2 2 29.7
48170 41004 41043 48 7 1 2 39.7 49037 45070 45156 49 4 1 2 16.8
48171 44042 44252 48 4 2 2 31 49038 48052 47008 49 2 1 1 39.8
48172 44042 44252 48 4 1 2 33.1 49039 46037 43063 49 6 1 2 28
48173 46015 46205 48 2 2 1 27.3 49040 46037 43063 49 6 1 2 26.5
48174 45070 46026 48 2 2 1 29.1 49041 46037 44098 49 5 2 1 37.2
48175 45070 45103 48 3 1 1 26.7 49042 45070 44072 49 5 2 1 42
48177 46037 45208 48 3 2 1 32.2 49043 45070 45116 49 4 1 2 41.4
48179 44042 44158 48 4 2 1 37.5 49044 45070 45116 49 4 1 2 35.9
48181 44042 41033 48 7 2 1 33.5 49045 48052 47029 49 2 1 1 32.3
48184 45070 43025 48 5 1 1 31.3 49046 48052 47068 49 2 1 2 38.3
48185 46037 45053 48 3 1 1 35.9 49047 45070 46048 49 3 2 2 33.5
48186 46037 44220 48 4 2 1 35 49048 45070 46048 49 3 2 2 34.5
48187 45070 43024 48 5 2 1 33.8 49049 46037 44121 49 5 2 1 38.8
48188 45070 46224 48 2 2 1 33.7 49050 46037 47151 49 2 2 1 37.6
48204 46037 45096 48 3 2 1 22.1 49051 48052 41093 49 8 1 2 32.6
48205 4)004 41135 48 7 2 2 30.6 49052 45070 41093 49 8 2 2 24
48206 41004 41135 48 7 2 2 28.5 49053 48014 47141 49 2 1 2 50.3
48218 45070 45132 48 3 2 1 35.6 49054 48014 47"141 49 2 2 2 22.8
49001 45070 47208 49 2 1 2 34.3 49055 45070 41047 49 8 2 2 36.8
49002 45070 45027 49 4 2 2 28.1 49056 45070 41047 49 8 2 2 33.2
49003 45070 45027 49 4 2 2 27 49057 48014 46010 49 3 1 1 39.3
49004 48014 47076 49 2 2 2 18 49058 46037 43084 49 6 1 1 44.9
49005 46037 43089 49 6 2 2 25.1 49059 45070 46177 49 3 1 1 45.7
49006 46037 43089 49 6 2 2 29 49060 46037 45004 49 4 2 2 38.6
49008 46015 44064 49 5 2 1 31 49062 48052 45073 49 4 1 1 37.8
49011 48014 47022 49 2 2 1 35.6 49063 48014 47074 49 2 2 1 35.6
49012 48014 47148 49 2 1 1 38.8 49064 48014 44228 49 5 1 1 42.9
49013 48014 47092 49 2 2 1 36.2 49065 48014 47112 49 2 1 1 43.8
49014 46015 44217 49 5 2 1 33.7 49067 45070 41033 49 8 2 1 37
49017 45070 45102 49 4 2 2 26.6 49068 46015 45115 49 4 1 3 33.8
49018 45070 45102 49 4 2 2 27.4 49069 46015 45115 49 4 2 3 27.1
49019 48052 45103 49 4 2 1 39.8 49070 46015 45115 49 4 2 3 28
49020 46037 43086 49 6 2 1 39 49071 48014 47147 49 2 2 2 29.8
49021 46037 44184 49 5 2 1 37.7 49072 48014 47147 49 2 2 2 34.4
49022 45070 44224 49 5 2 1 41.9 49073 48052 47124 49 2 1 2 36.5
49023 46037 44082 49 5 2 1 34 49074 48052 47124 49 2 2 2 30.4
49024 46037 45054 49 4 1 2 36.3 49075 45070 46134 49 3 1 1 49.4
49025 46037 45054 49 4 2 2 29.6 49076 48014 46073 49 3 1 1 44.6
-267- -268-
Appendix 8; Dormer Stud Data: Elsenburg Co/ll!ge of Agriculture Appendix IJ: Dormer SUiltData: Elsenburg ( 'o/lege oj Agriculture
49077 46015 46072 49 3 2 1 34.7 49139 45070 44253 49 5 1 3 . 40.5
49076 45070 44252 49 5 2 2 41 49141 46015 4622 49 3 1 2 35.1
49079 45070 44252 49 5 1 2 36.4 49142 46015 4622 49 3 2 2 30.9
49060 46037 45056' 49 4 2 2 32.4 49143 45070 41083 49 B 2 2 27.4
49061 46037 45056 49 4 1 2 37.9 49144 45070 41063 49 B 1 2 33
49062 46037 44115 49 5 2 1 43.6 49145 46037 45153 49 4 2 2 33.1
49063 48014 46005 49 3 1 2 35 49146 46037 45153 49 4 2 2 33.5
49084 48014 46005 49 3 1 2 29.2 49149 46015 46025 49 3 1 2 38.7
49085 46037 42065 49 7 2 2 40.8 49150 46015 46025 49 3 1 2 23.6
49088 45070 45063 49 4 1 2 36.8 49151 48014 46009 49 3 1 2 27.1
49089 45070 45063 49 4 2 2 30.5 49152 48014 46009 49 3 2 2 27.3
49090 46015 46175 49 3 2 2 29.3 49154 46015 43110 49 6 2 1 33.2
49091 46015 46175 49 3 1 2 34.4 49155 46015 44205 49 5 2 1 36.3
49092 46037 45055 49 4 2 1 37.2 49156 48014 47113 49 2 1 2 26.3
49094 48014 47043 49 2 2 1 36.6 49157 48014 47113 49 2 2 2 26.1
49095 46015 46004 49 3 2 1 38.8 49158 48014 47132 49 2 1 2 35.4
49097 46037 44022 49 5 1 1 46 49159 48014 47132 49 2 1 2 34.2
49099 45070 46026 49 3 2 2 30.7 49160 46037 45053 49 4 2 2 27.3
49100 45070 46026 49 3 2 2 29.9 49161 46037 45053 49 4 2 2 31.4
49101 46015 45077 49 4 2 1 38.5 49162 45070 41026 49 6 1 2 37.6
49103 48052 44156 49 5 2 1 42 49163 45070 41028 49 8 1 2 35.1
49104 48052 47174 49 2 1 1 43.5 49167 45070 41043 49 8 1 2 28.5
49106 46037 45015 49 4 1 2 33.3 49168 45070 41043 49 8 1 2 29.7
49107 46037 45015 49 4 1 2 33.9 49169 46015 46064 49 3 2 1 43.4
49108 46037 46006 49 3 1 2 44.5 49171 46015 43072 49 6 1 1 44.4
49109 46037 46006 49 3 2 2 30.1 49172 46015 47120 49 2 2 1 39.1
49110 48052 47119 49 2 1 2 35.3 49175 46052 47167 49 2 1 ~ 28
49111 48052 47119 49 2 2 2 27.8 49177 46037 45208 49 4 1 1 45.8
49112 46015 43100 49 6 1 1 32.1 49183 48052 47001 49 2 1 1 38.4
49113 48014 47094 49 2 2 2 26.7 49187 48014 47188 49 2 2 2 26
49115 45070 41055 49 8 2 1 45 49188 46014 47188 49 2 2 2 19.2
49117 46015 45211 49 4 2 1 37 49193 46037 44213 49 5 1 1 47.1
49118 46015 43014 49 6 2 1 35.1 49196 48014 47014 49 2 1 1 41.6
49119 48052 47215 49 2 1 1 31.6 49197 46015 46205 49 3 2 1 35.7
49121 46015 43003 49 6 2 1 32.6 49198 46037 43033 49 6 1 1 39
49122 46015 43029 49 6 1 1 45 49201 46015 47115 49 2 2 1 39
49123 48052 47080 49 2 2 1 32.2 49202 46015 47130 49 2 2 1 32.7
49125 46037 44163 49 5 1 1 41.8 49206 46015 43026 49 6 1 1 45.1
49126 46015 43020 49 6 1 1 36.5 49208 48052 47087 49 2 2 1 35
49127 46037 46023 49 3 1 2 37 49212 45070 45132 49 4 2 1 39.7
49126 46037 46023 49 3 2 2 26.6 49216 46037 46224 49 3 1 3 37
49129 46015 44220 49 5 2 1 40.5 49219 46037 46224 49 3 ? 3 23.4
49132 45070 42060 49 7 1 1 43.1 49220 46037 46224 49 3 2 3 33
49134 46037 45082 49 4 1 1 43.9 49225 48014 47140 49 2 1 3 40.7
49135 46015 44013 49 5 2 1 39.1 49227 48014 47015 49 2 1 3 26.2
49136 48014 47056 49 2 2 1 35.5 _ 49228 46037 46024 49 6 2 2 33.4
-
-269- -270-
Appendix IJ: Dormer Stud Data; Elsenburg Cullege of Agriculture Appendix H: Dormer Stud /Jata; J:Jsl!nhurx ('ollexe of Agncutture
49229 46037 43024 49 6 2 2 34.9 50070 48140 47130 50 3 2 1 34.1
49231 45070 41057 49 8 1 1 45.7 50072 49053 48165 50 2 2 2 20.5
50005 48014 45208 50 5 1 2 21.1 50074 45070 44205 50 6 2 1 34.1
50006 48014 45208 50 .5- 2 2 32.1 50075 45070 44072 50 6 2 1 35.7
50007 48014 45103 50 5 1 1 36.5 50076 46037 44022 50 6 2 1 30.6
50010 45070 42060 50 8 2 1 33.5 50077 48014 48068 50 2 2 1 28.7
50011 48140 47008 50 3 2 3 22.2 50078 48052 44217 50 6 2 1 3D"
50012 48140 47008 50 3 2 3 25.3 50079 49053 48058 50 2 1 1 36.7
50014 48014 48118 50 2 1 2 21.9 50080 48052 48060 50 2 1 1 34.5
50015 48014 48118 50 2 2 2 25.1 50081 48014 46205 50 4 2 1 34"
50018 48014 43033 50 7 1 2 25.3 50083 45070 47077 50 3 2 1 15.8
50019 48014 43033 50 7 2 2 22.1 50084 46037 43018 50 7 2 1 26.3
50021 48140 47087 50 3 2 1 20.3 50087 48014 48134 50 2 2 1 30.2
50023 48140 48160 50 2 1 2 29.9 50088 49134 48036 50 2 1 2 33.6
50024 48140 48160 50 2 2 2 25.2 50089 49134 48036 50 2 2 2 24.5
50027 48140 47141 50 3 2 1 28.6 50090 48014 47097 50 3 2 2 23.4
50028 48014 47140 50 3 2 1 34.2 50091 48014 47097 50 3 2 2 15.4
50030 48014 48144 50 2 1 1 33.1 50093 46037 46004 50 4 1 2 30.4
50031 48140 47080 50 3 1 2 29.1 50094 46037 46004 50 4 1 2 35.6
50032 48140 47080 50 3 2 2 21.2 I 50095 48052 48047 50 2 2 1 32.6
50033 48140 48089 50 2 1 1 29.4 50096 48014 48143 50 2 2 2 29.1
50034 48014 47056 50 3 2 1 31.1 50098 45070 47112 50 3 1 1 37.2
50036 45070 42065 50 8 2 1 31.6 50099 48014 47132 50 3 1 2 31.8
50040 49053 48054 50 2 2 2 36.1 i 50100 48014 47132 50 3 2 2 30.6
50042 48140 48079· 50 2 1 1 38 50101 48140 48171 50 2 1 2 20.9
50043 48014 43089 50 7 1 2 27.4 50102 48140 48171 50 2 2 2 24.9
50044 48014 43089 50 7 2 2 26.5 50103 48052 48162 50 2 2 2 21.2
50045 48014 44064 50 6 2 ·1 29 50104 48052 48162 50 2 2 2 22.5
50047 48140 47148 50 3 1 2 29.3 50105 48014 47043 50 3 2 1 28.2
50048 48140 47148 50 3 2 2 21.2 50106 48140 43110 50 7 2 1 18.9
50052 48140 43014 50 7 1 1 38.3 50108 46037 47215 50 3 2 1 31.1
50053 48140 47105 50 3 1 1 37.8 50109 49053 48158 50 2 1 2 29.6
50054 45070 44098 50 6 1 1 34.7 50110 49053 48158 50 2 2 2 23.1
50055 48014 45056 50 5 1 1 35.4 50111 48014 4622 50 4 1 2 30.5
50056 46037 46170 50 4 2 1 22.9 50112 48014 4622 50 4 2 2 32.9
50057 46037 44013 50 6 1 2 32.8 50113 49053 48166 50 2 1 2 33.3
50058 46037 44013 50 6 1 2 25.9 50114 49053 48166 50 2 2 2 26.7
50060 46037 45003 50 5 2 1 36.4 50115 46037 45015 50 5 2 1 32
50061 48014 48001 50 2 1 2 31.1 50116 48140 47113 50 3 2 1 24.7
50062 48014 48001 50 2 1 2 27.6 50117 46037 46064 50 4 2 1 37.3
50063 48014 45094 50 5 2 1 37.1 50118 48014 46009 50 4 2 1 34.5
50065 45070 45132 50 5 2 1 37.1 50119 49134 48030 50 2 1 1 40
50066 46037 46006 50 4 2 2 31.9 50120 48014 48090 50 2 1 2 21.7
50067 46037 46006 50 4 2 2 28.1 50121 48014 48090 50 2 2 2 28.3
50068 46037 45116 50 5 2 1 31.6 50122 46037 45153 50 5 2 2 29.7
50069 46037 45077 50 5 2 1 33.3 50123 46037 45153 50 5 2 2 23.1
---------
-271- -272-
Appendix B: Dormer Stud Duw; Hlsenburg College of Agriculture Appendix JJ: /Jurmer Stud Data; Elsenburg ('ullege of Agriculture
- ------
50124 48052 46026 50 4 1 2 31.7 50192 48052 47174 50 3 2 2 25.1
50125 48052 46026 ·50 4 2 2 27.6 50193 48052 47174 50 3 2 2 31.9
50128 46037 43003 50 7 2 2 23 50194 48014 44184 50 6 2 - 2
26.9
50129 46037 46013 50 4 2 2 20.7 50195 48014 44184 .50 6 2 2 25.7
50130 46037 46073 50 4 1 2 25.4 50196 48052 43024 50 7 I 1 38.9
50131 46037 46025 50 4 1 1 36.1 50197 48052 48187 50 2 2 I 27.3
50132 48140 44224 50 6 2 1 33.5 50200 48052 44213 50 6 I I 50.6
50133 48052 48002 50 2 1 1 31.6 50201 48052 43072 50 7 I 1 42.7
50134 46037 43086 50 7 1 2 32.2 50202 46037 41028 50 9 1 2 36.6
50135 46037 43086 50 7 1 2 36.5 50203 46037 41028 50 9 1 2 30.1
50138 48014 48069 50 2 2 1 25.6
50139 49134 48179 50 2 2 1 40.8
50140 48140 47115 50 3 2 2 35.9
50142 46037 45211 50 5 2 1 Jo
50143 49134 48003 50 2 1 2 28.1
50144 49134 48003 50 2 2 2 21.1
50145 48014 ~&121 50 2 I 2 28.1
50146 48014 48121 50 2 2 2 25.6
50149 48140 48139 50 2 2 1 25.9
50150 46037 44121 50 6 2 I 32.4
50151 49053 48051 50 2 1 I 34.1
50152 46037 44163 50 6 2 2 20.5
50153 46037 44163 50 6 2 2 19.6
50154 48052 48017 50 2 2 2 17.2
50155 48052 48017 50 2 2 2 21.6
50158 46037 44082 50 6 1 2 28.7
50159 46037 44083 50 6 2 2 23.9
50160 49046 48050 50 2 2 I 34.4
50163 48140 48100 50 2 2 I 35.3
50164 48052 46048 50 4 1 2 38.7
50167 46037 46175 50 4 2 I 31.1
50168 46037 47119 50 3 2 I 28.9
50169 48052 48056 50 2 2 2 26.7
50170 48052 48056 50 2 2 2 23.1
50171 48140 47014 50 3 2 I 30
50172 48140 48067 50 2 2 1 35.1
50173 48140 46224 50 4 2 1 37.4
50174 48140 48177 50 2 1 I 32.8
50175 49046 48167 50 2 2 2 27.6
50176 49046 48167 50 2 2 2 25.7
50178 48052 44228 50 6 I 2 33
50179 48052 44228 50 6 2 2 32.9
50185 49046 48173 50 2 1 I 34.2
50186 48052 44115 50 6 2 1 38.6
50188 48052 47022 50 3 2 I 28.3
50191 46037 43026 50 7 I 1 36.7
-273- -274-
Appendix C: Datasetfrom Duchateau, et al., 1998; International Livestock Research Institute (ILRI), Kenya
APPENDIX C
The example used for illustrative purposes are based on an experiment undertaken at the
International Livestock Research Institute (ILRI) at the University of Nairobi, Kenya in the early
90's (Duchateau, et al., 1998). The goal of the research was to select for improved helminth
resistance in sheep.
The female sheep used in the experiment are from three different breeds (br), whereas the males
are from two breeds. In each of the six crosses, there are at least 25 and at most 42 different sires,
. .~"
and each sire within a crossbreed has on average offspring of 6.4 lambs. ''Fhe weaning weight is
"..........
measured for each lamb (YWW), '...'_
Although the same sire is mated to ewes from different breeds, the sire nested in breed is taken as
a single random effect and it is assumed that these random effects are independent. A total of
n = 1277 weaning weight records, from the progeny of q = 200 sires are available after editing, and
breed, sex and age are included as fixed effects in the final model.
-275-
~---------------------------------------------------------------------------,
Appendix C: Dataset/rom Duchateau, et al .. 199'1: Internottonat Livestock Research Institute (ILR/). Kenya Appendix C: Datoset from Duchcueou, et al., 191)X; Iruernanonal Livestock Research lnstit ute (fLUJ), Kenya
1 1999 M 149 14 1 7 1 1 0 0 0 0 1 149 I
1 1999 F 144 18.3 0 7 1 1 0 0 0 0 0 144
Brl Br2 Br3 Br4 Br5 Br6 Age 1 1999 M 141 17.3 1 7 1 1 0 0 0 0 1 141
Breed Sire 10 Sex1B6) Age YIWW) Sex Sire_Nr BO Bl B2 B3 54 es B6 B7 1 .1999 M 134 13.2 1 7 1 1 0 Ó 0 0 1 134
1 1971 F 145 11.2 0 1 I 1 0 0 0 0 0 145 1 4907 M 155 18 1 8 1 1 0 0 0 0 1 155
1 1971 F 140 15.4 0 1 1 1 0 0 0 0 0 140 1 4907 F 153 13.1 0 8 1 1 0 0 0 0 0 153
1 1971 F 140 10.9 0 1 1 1 0 0 0 0 0 140 1 4907 F 152 13.1 0 8 1 1 0 0 0 0 0 152
1 1971 M 122 11.4 1 1 1 1 0 0 0 0 1 122 1 4907 M 152 9.5 1 8 1 1 0 0 0 0 1 152
1 1972 F 152 16 0 2 1 1 0 0 0 0 0 152 1 4907 M 135 13.8 1 8 1 1 0 0 0 0 1 135
1 1972 M 151 13.2 1 2 1 1 0 0 0 0 1 151 I 4907 M 132 15.7 1 8 1 1 0 0 0 0 1 132
1 1972 F 146 14.9 0 2 1 1 0 0 0 0 0 146 1 4907 M 129 12.6 1 8 1 1 0 0 0 0 1 129
1 1972 F 139 7.9 0 2 1 1 0 0 0 0 0 139 1 4907 M 115 12 1 8 1 1 0 0 0 0 1 115
1 1972 M 132 15.7 1 2 1 1 0 0 0 0 1 132 1 4908 F 162 19 0 9 1 1 0 0 0 0 0 162
1 1972 F 131 13.1 0 2 I 1 0 0 0 0 0 131 1 4908 M 148 15.3 1 9 1 1 0 0 0 0 1 148
I 1972 F 128 12.5 0 2 1 1 0 0 0 0 0 128 1 4908 M 147 18.8 1 9 1 1 0 0 0 0 1 147
1 1972 M 124 12.2 I 2 1 1 0 0 0 0 I 124 I 4908 F 139 15.1 0 9 1 1 0 0 0 0 0 139
1 1972 M 122 14.6 1 2 1 1 0 0 0 0 1 122 1 4908 M 134 16.1 1 9 1 1 0 0 0 0 1 134
1 1972 F 115 13.6 0 2 1 1 0 0 0 0 0 lIS 1 4908 F 118 13.2 0 9 1 1 0 0 0 0 0 118
1 1972 M 110 12.6 1 2 1 1 0 0 0 0 1 110 1 4908 M 132 14.2 1 9 1 1 0 0 0 0 1 132
1 1973 F 157 11.9 0 3 1 1 0 0 0 0 0 157 1 4908 F 126 13.5 0 9 1 1 0 0 0 0 0 126
1 1973 M 157 20 1 3 1 I 0 0 0 0 1 157 1 4908 M 125 13.8 1 9 1 1 0 0 0 0 1 125
1 1973 M 156 17.1 I 3 1 1 0 0 0 0 1 156 1 4908 F 125 14.2 0 9 1 1 0 0 0 0 0 125
1 1973 M 143 10.8 1 3 1 1 0 0 0 0 I 143 1 4908 F 122 18.9 0 9 1 1 0 0 0 0 0 122
1 1973 M 136 12.5 I 3 1 1 0 0 0 0 1 136 1 4908 F 122 12.7 0 9 1 1 0 0 0 0 0 122
1 1974 M 159 15.8 1 4 1 1 0 0 0 0 1 159 1 4908 M 94 11.4 1 9 1 1 0 0 0 0 1 94
1 1974 F 156 12.8 0 4 1 I 0 0 0 0 0 156 1 4908 F 169 12.8 0 9 1 1 0 0 0 0 0 169
1 1974 F 156 20.2 0 4 1 1 0 0 0 0 0 156 1 4908 M 156 16.4 1 9 1 1 0 0 0 0 1 156
1 1974 M 155 18.7 1 4 1 1 0 0 0 0 1 155 1 4909 F 145 6.3 0 10 1 1 0 0 0 0 0 145
1 1974 M 144 18.5 1 4 1 1 0 0 0 0 1 144 1 4909 M 140 11.5 1 10 1 1 0 0 0 0 1 140
1 1974 M 143 18.8 I 4 1 1 0 0 0 0 1 143 1 4909 F 124 13.2 0 10 1 1 0 0 0 0 0 124
1 1974 F 136 11.1 0 4 1 I 0 0 0 0 0 136 1 4909 F 117 9.6 0 10 1 1 0 0 0 0 0 117
1 1974 M 135 16.8 1 4 1 1 0 0 0 0 1 135 1 4910 F 157 12.1 0 11 1 1 0 0 0 0 0 157
I 1974 M 134 17.5 1 4 1 1 0 0 0 0 1 134 1 4910 F 154 13 0 11 1 1 0 0 0 0 0 154
1 1974 F 126 14.2 0 4 1 1 0 0 0 0 0 126 1 4910 M 150 13.2 1 11 1 1 0 0 0 0 1 150
1 1980 M 175 18.5 1 5 1 1 0 0 0 0 1 175 1 4910 F 143 19.1 0 11 1 1 0 0 0 0 0 143
1 1980 M 155 17.9 1 5 1 1 0 0 0 0 1 155 1 4910 F 121 12.3· 0 11 1 1 0 0 0 0 0 121
1 1980 M 151 12.7 1 5 I I 0 0 0 0 1 151 1 4910 F 129 9.3 0 11 1 1 0 0 0 0 0 129
1 1980 F 148 13.4 0 5 I 1 0 0 0 0 0 148 1 4910 F 129 14.3 0 11 1 1 0 0 0 0 0 129
1 1980 M 145 12.7 1 5 1 1 0 0 0 0 1 145 1 4910 F 127 15 0 11 1 1 0 0 0 0 0 127
1 1980 M 141 19 1 5 1 1 0 0 0 0 1 141 1 4910 M 121 11.4 1 11 1 1 0 0 0 0 1 121
1 1980 M 124 13.4 1 5 1 1 0 0 0 0 1 124 1 4910 F 115 12.4 0 11 1 1 0 0 0 0 0 115
1 1991 F 134 10.6 0 6 I 1 0 0 0 0 0 134 1 4910 M 102 14.3 1 11 1 1 0 0 0 0 1 102
1 1991 M 128 13.1 1 6 1 1 0 0 0 0 1 128 1 4910 F ·84 10.2 0 11 1 1 0 0 0 0 0 84
1 1991 F 121 14.3 0 6 1 I 0 0 0 0 0 121 1 4910 M 97 9.1 1 11 1 1 0 0 0 0 1 97
1 1991 M 112 15.6 1 6 1 1 0 0 0 0 1 112 1 4911 M 158 17.3 1 12 1 1 0 0 0 0 1 158
1 1991 M 98 10.4 I 6 1 1 0 0 0 0 1 98 1 4911 F 156 17.3 0 12 1 1 0 0 0 0 0 156
-276- -277-
Appendix (': Duwsetfrom Duohateau. et uI., 1998; lruernanonal Livestock Research institute (IL/U), Kenya Appendix C: Dataset from Duohuteau. et ul., 19YH; tntemononot Livestock Research lnstu ute (ILIU), Kenya
1 4911 M 155 12.4 1 12 1 1 0 0 0 0 1 155 1 5009 M 115 13.4 1 23 1 1 0 0 0 0 1 115
1 4911 M 141 11.5 1 12 1 1 0 0 0 0 1 141 1 5009 M 110 9.8 1 23 1 1 0 0 0 0 1 110
1 4911 M 141 16.5 1 12 1 1 0 0 0 0 1 141 1 5010 F 169 13.5 0 24 1 1 0 0 0 0 0 169
.1 4911 M 144 16.9 1 12 1 1 0 0 0 O. 1 144 1 5010 M 143 15.6 1 24 1 1 0 ei 0 0 1 143
1 4911 F 138 12 0 12 1 1 0 0 0 0 0 138 1 5010 F 140 11.6 0 24 1 1 0 0 0 0 0 140
1 4911 M 131 13.3 1 12 1 1 0 0 0 0 1 131 1 SOll M 175 10.5 1 25 1 1 0 0 0 0 1 175
1 4912 F 155 11 0 13 1 1 0 0 0 O· 0 155 1 5011 F 166 11.9 0 25 1 1 0 0 0 0 0 166
1 4912 M 144 10.8 1 13 1 1 0 0 0 0 1 144 1 S011 F 112 10.9 0 25 1 1 0 0 0 0 0 112
1 4912 F 140 8.1 0 13 1 1 0 0 0 0 0 140 1 5012 M 168 17 1 26 1 1 0 0 0 0 1 168
1 4912 F 131 11.3 0 13 1 1 0 0 0 0 0 131 1 S012 M 156 12.6 1 26 1 1 0 0 0 0 1 156
1 4915 F 128 10.1 0 14 1 1 0 0 0 0 0 128 1 S013 M 172 13.7 1 27 1 1 0 0 0 0 1 172
1 4915 F 121 14.2 0 14 1 1 0 0 0 0 0 121 1 S013 F 168 13.5 0 27 1 1 0 0 0 0 0 168
1 4915 M 113 15.6 1 14 1 1 0 0 0 0 1 113 1 S013 F 157 11.5 0 27 1 1 0 0 0 0 0 157
1 4915 M 101 9.6 1 14 1 1 0 0 0 0 1 101 1 S071 F 116 9.7 0 28 1 1 0 0 0 0 0 116
1 4916 F 112 10.4 0 15 1 1 0 0 0 0 0 112 1 S011 F 110 12.7 0 28 1 1 0 0 0 0 0 110
1 4916 M 98 10.2 1 15 1 1 0 0 0 0 1 98 1 S071 M 133 16 1 28 1 1 0 0 0 0 1 133
1 4916 M . 111 10.1 1 15 1 1 O· 0 0 0 1 111 1 S071 F 132 13.1 0 28 1 1 0 0 0 0 0 132
1 4916 F 109 13.2 0 15 1 1 0 0 0 0 0 109 1 S073 M 90 8.9 1 29 1 1 0 0 0 O. 1 90
1 5001 F 119 1 0 16 1 1 0 0 0 0 0 119 1 5016 M 117 11 1 30 1 1 0 0 0 0 1 117
1 5001 F 172 15 0 16 1 1 0 0 0 0 0 112 1 5205 F 108 1.5 0 31 1 1 0 0 0 0 0 108
1 5002 M 113 13.4 1 11 1 1 0 0 0 0 1 113 1 5324 F 136 6.2 0 32 1 1 0 0 0 0 0 136
1 5002 F 101 11.1 0 11 1 1 0 0 0 0 0 101 1 5326 F 93 8.5 0 33 1 1 0 0 0 0 0 93
1 5002 F 100 1.9 0 11 1 1 0 0 0 0 0 100 1 5326 M 135 10.2 1 33 1 1 0 0 0 0 1 135
1 S002 F 99 10.5 0 11 1 1 0 0 0 0 0 99 1 5328 M 126 13.9 1 34 1 1 0 0 0 0 1 126
1 5002 M 152 9.2 1 11 1 1 0 0 0 0 1 152 1 5328 F 121 10.7 0 34 1 1 0 0 0 0 0 121
1 5003 M 134 15.1 1 18 1 1 0 0 0 0 1 134 1 5328 M 118 7.4 1 34 1 1 0 0 0 0 1 118
1 S003 F 121 12.4 0 18 1 1 0 0 0 0 0 121 1 5328 F 138 14 0 34 1 1 0 0 0 0 0 138
1 5004 M 111 15.5 1 19 1 1 0 0 0 0 1 111 1 5328 M 135 12.8 1 34 1 1 0 0 0 0 1 135
1 5004 M 110 13.2 1 19 1 1 0 0 0 0 1 110 1 5329 F 122 12.4 0 35 1 1 0 0 0 0 0 122
1 5004 F 108 10.4 0 19 1 1 0 0 0 0 0 108 1 5329 F 98 6.2 0 35 1 1 0 0 (] 0 0 98
1 5004 F 101 10.1 0 19 1 1 0 0 0 0 0 101 1 5329 M 137 12.6 1 35 1 1 0 0 0 0 1 131
1 5004 F 106 14.8 0 19 1 1 0 0 0 0 0 106 1 5329 F 132 11.3 0 35 1 1 0 0 0 0 0 132
1 5005 M 130 13.1 1 20 1 1 0 0 0 0 1 130 1 5329 M 132 9.1 1 35 1 1 0 0 0 0 1 132
1 5005 M 104 11.6 1 20 1 1 0 0 0 0 1 104 1 5329 F 128 13 0 35 1 1 0 0 0 0 0 128
1 5005 F 91 9.5 0 20 1 1 0 0 0 0 0 91 1 5330 F 122 8.8 0 36 1 1 0 0 0 0 0 122
1 5005 F 88 15.1 0 20 1 1 0 0 0 0 0 88 1 5330 M 141 8.3 1 36 1 1 0 0 0 0 1 141
1 5005 F 163 9.3 0 20 1 1 0 0 0 0 0 163 1 5330 F 139 7.9 0 36 1 1 0 0 0 0 0 139
1 5001 M 113 11.1 1 21 1 1 0 0 0 0 1 113 1 5330 F 131 8.3 0 36 1 1 0 0 0 0 0 131
1 5001 F 166 12.5 0 21 1 1 0 0 0 0 0 166 1 5330 M 115 10.5 1 36 1 1 0 0 0 0 1 115
1 5001 M 163 16.5 1 21 1 1 0 0 0 0 1 163 1 5331 F 129 10.3 0 37 1 1 0 0 0 0 0 129
1 S007 F 121 6.1 0 21 1 1 0 0 0 0 0 121 1 5337 M 127 12.5 1 37 1 1 0 0 0 0 1 127
1 5007 M 115 10.1 1 21 1 1 0 0 0 0 1 115 1 5331 M 116 10.9 1 37 1 1 0 0 0 0 1 116
1 5008 F 115 13.8 0 22 1 1 0 0 0 0 0 175 1 5337 F 115 7.8 0 37 1 1 0 0 0 0 0 115
1 5008 F 158 10.8 0 22 1 1 0 0 0 0 0 158 1 5338 M 121 9.2 1 38 1 1 0 0 0 0 1 121
1 5008 M lSO 8.1 1 22 1 1 0 0 0 0 1 150 1 5338 F 139 11 0 38 1 1 0 0 0 0 0 139
1 5008 M 111 14.1 1 22 1 1 0 0 0 0 1 111 1 5338 M 112 10.1 1 38 1 1 0 0 0 0 1 112
-278- -279-
Appendix C: Datasetfrom Duohateau. et ul., /998; International Livestock Research Institute (JL/U). Kenya Appendix C: Dataset/rom Duohuteau. et al .. 1998; tnternationat Ltvestock Research Institute (IL/U), Kenya
2 1975 F 147 11.2 0 39 1 0 1 0 0 0 0 147 2 1988 F 145 13.4 0 47 1 0 1 0 0 0 0 145
2 1975 M 145 12.8 1 39 1 0 1 0 0 0 1 145 2 1988 'F 141 16.9 0 47 1 0 1 0 0 0 0 141
2 1975 M 136 12.5 1 39 1 0 1 0 0 0 1 136 2 1988 M 139 17.1 1 47 1 0 1 0 0 0 1 139
.2 1975 M 134 12.4 1 39 1 Ó 1 0 0 .0. 1 134 2 .1988 F 137 13.2 0 47 1 0 1 Ó 0 0 0 .137
2 1975 M 133 16.6 1 39 1 0 1 0 0 0 1 133 2 4901 F 153 17.2 0 48 1 0 1 0 0 0 0 153
2 1975 F 126 8.9 0 39 1 0 1 0 0 0 0 '26 2 4901 F 139 15.7 0 48 1 0 1 0 0 0 0 139
2 1976 M 154 '5.5 1 40 , 0 ,
2 1976 F 147 9.5 0 40 ,, 0 ,,
0 0 0 , 154 2 4901 F 139 10.8 0 48 1 0 1 0 0 0 0 139
0 0 0 0 '47 2 4901 M 139 9.8 1 48 1 0 1 0 0 0 1 139
2 1976 F '42 15.3 0 40 0 0 0 0 0 '42 2 4901 F 135 13.5 0 48 1 0 1 0 0 0 0 135
2 1976 M '39 '6.2 , 40 , 0 , 0 0 0 ,, '39 2 4901 M 131 13.3 1 48 1 0 1 0 0 0 1 1312 1976 F '23 1'.7 0 40 ,1 0 , 0 0 0 0 '23 2 4901 M 123 15.8 1 48 1 0 1 0 0 0 1 1232 '979 M 153 16.3 1 41 0 0 0 0 1 '53 2 4901 M 122 14.8 1 48 1 0 1 0 0 0 1 122
2 '979 F '53 15.9 0 4' 1 0 1 0 0 0 0 '53 2 4901 F 120 10.7 0 48 1 0 1 0 0 0 0 120
2 1979 F 152 18.7 0 41 1 0 , 0 0 0 0 152 2 4901 M 81 11.2 1 48 1 0 1 0 0 0 r 81
2 1979 F 148 15.7 0 41 1 0 1 0 0 0 0 148 2 4902 M 148 17.5 1 49 1 0 1 0 0 0 1 148
2 1979 F 146 13.1 0 41 1 0 1 0 0 0 0 146 2 4902 M 147 14.5 1 49 1 0 1 0 0 0 1 147
2 1979 M 146 15 1 41 1 0 ,. 0 0 0 1 146 2 4902 M 146 17.1 1 49 1 0 1 0 0 0 1 146
2 1979 F 132 12.5 0 41 1 0 1 0 0 0 0 132 2 4902 M 145 14.4 1 49 1 0 1 0 0 0 1 145
2 1979 F 120 11 0 41 1 0 1 0 0 0 0 120 2 4902 M 139 8.8 1 49 1 0 1 0 0 0 1 139
2 1979 M 111 10.3 1 41 1 0 1 0 0 0 1 111 2 4902 M 139 15.5 1 49 1 0 1 0 0 0 1 139
2 1979 M 109 10.8 1 41 1 0 1 0 0 0 1 109 2 4902 M 132 14.3 1 49 1 0 1 0 0 0 1 132
2 1981 M 157 15.6 1 42 1 0 1 0 0 0 1 157 2 4902 M 125 9.3 1 49 1 0 1 0 0 0 1 125
2 1981 F 153 16.2 0 42 1 0 1 0 0 0 0 153 2 4902 F 119 12 0 49 1 0 1 0 0 0 0 119
2 1981 F 153 11.6 0 42 1 0 1 0 0 0 0 153 2 4902 M 116 10.4 1 49 1 0 1 0 0 0 1 116
2 1981 M 140 11 1 42 1 0 1 0 0 0 1 140 2 4903 F 150 15.4 0 50 1 0 1 0 0 0 0 150
2 1981 M 136 12.2 1 42 1 0 1 0 0 0 1 136 2 4903 F 146 12.1 0 50 1 0 1 0 0 0 0 146
2 1981 M 135 11.9 1 42 1 0 1 0 0 0 1 135 2 4903 M 144 15.5 1 50 1 0 1 0 0 0 1 144 i
2 1981 M 131 12.5 1 42 1 0 1 0. 0 0 1 '31 2 4903 F 141 16.8 0 50 1 0 1 0 0 0 0 141 I
2 1981 M 117 14.5 1 42 1 0 1 0 0 0 1 117 2 4903 F 137 13.9 0 50 1 0 1 0 0 0 0 137
2 1982 M 130 8.1 1 43 1 0 1 0 0 0 1 130 2 4903 F 135 12.7 0 50 1 0 1 0 0 0 0 135
2 1982 M 125 14.6 1 43 1 0 1 0 0 0 1 125 2 4903 M 125 13 1 50 1 0 1 0 0 0 1 125
2 1982 M 118 10.6 1 43 1 0 1 0 0 0 1 118 2 4903 M 115 8.5 1 50 1 0 1 0 0 0 1 115
2 1982 F 84 10.1 0 43 1 0 1 0 0 0 0 84 2 4903 M 114 7.1 1 50 1 0 1 0 0 0 1 114
2 1983 M 156 18.1 1 44 1 0 1 0 0 0 1 156 2 4903 F 113 8.5 0 50 1 0 1 0 0 0 0 113
2 1983 F 156 17.1 0 44 1 0 1 0 0 0 0 156 2 4903 M 107 12.2 1 50 1 0 1 0 0 0 1 107
2 1983 F '53 15.9 0 44 1 0 1 0 0 0 0 153 2 4903 M 100 12.3 1 50 1 0 1 0 0 0 1 100
2 1983 F 152 15 0 44 1 0 1 0 0 0 0 152 2 4905 M 155 17.2 1 51 1 0 1 0 0 0 1 155
2 1983 F 147 17.2 0 44 1 0 1 0 0 0 0 147 2 4905 M 149 16.4 1 51 1 0 1 0 0 0 1 149
2 1984 F 151 16.9 0 45 1 0 1 0 0 0 0 151 2 4905 M 143 10.1 1 51 1 0 1 0 0 0 1 143
2 1984 F 150 12.1 0 45 1 0 1 0 0 0 0 150 2 4905 M 138 10.2 1 51 1 0 1 0 0 0 1 138
2 1984 F 147 8.2 0 45 1 0 1 0 0 0 0 147 : 2 4905 M 134 15.3 1 51 1 0 1 0 0 0 1 134
2 1984 M 145 15.2 1 45 1 0 1 0 0 0 1 145 I 2 4905 F 125 10.9 0 51 1 0 1 0 0 0 0 125
2 1984 M 141 15.3 1 45 1 0 1 0 0 0 1 141 2 4905 M 127 13.1 1 51 1 0 1 0 0 0 1 127
2 1984 F 135 13.5 0 45 1 0 1 0 0 0 0 135 2 4905 M 121 13.7 1 51 1 0 1 0 0 0 1 121
2 1986 M 124 15.7 1 46 1 0 1 0 0 0 1 124 2 4905 M 117 7.6 1 51 1 0 1 0 0 0 1 117
2 1986 M 121 9.9 1 46 1 0 1 0 0 0 1 121 2 4905 F 117 14 0 51 1 0 1 0 0 o_
-- -- o _~
-280- -281-
Appendix C: Datasetfrom Duchateau, et at., 1998; lnternauonol Livestock Research Institute (11.lU), Kenya Appendix C: Datasetfrom Duohoteau. et ol., IY9N; Imemauonal l.ivestock Research Insnnne (JL/lj). A('''YD
------ --_._--- ----- ----- --
2 4905 F 114 15.2 0 51 1 0 1 0 0 0 0 114 2 4921 M 156 11.3 1 57 1 0 1 0 0 0 1 156
2 4905 F 161 13 0 51 1 0 1 0 0 0 0 161 2 4923· M 126 11.7 1 58 1 0 1 0 0 0 1 126
2 4905 F 150 10 0 51 1 0 1 0 0 0 0 150 2 4923 M 113 14.5 1 58 1 0 1 0 0 0 1 113
2 4905 F 149 12.1 0 51 1 0 1 0 IJ 0 0 149 2. 4923 M 106 12.7 1 58 1 0 1 0 0 0 .1 106
2 4905 F 146 8.3 0 51 1 0 1 0 0 0 0 146 2 4923 F 102 10.7 0 58 1 0 1 0 0 0 0 102
2 4906 F 145 12 0 52 1 0 1 0 0 0 0 145 2 4923 M 168 11.8 1 58 1 0 1 0 0 0 1 168
2 4906 M 143 16.2 1 52 1 0 1 0 0 0 1 143 2 4923 M 163 14.5 1 58 1 0 1 0 0 0 1 163
2 4906 F 140 13.7 0 52 1 0 1 0 0 0 0 140 2 4923 M 159 10.6 1 58 1 0 1 0 0 0 1 159
2 4906 F 138 15.9 0 52 1 0 1 0 0 0 0 138 2 5015 M 167 14 1 59 1 0 1 0 0 0 1 167
2 4906 M 115 11.9 1 52 1 0 1 0 0 0 1 115 2 5015 M 166 14.8 1 59 1 0 1 0 0 0 1 166
2 4906 M 112 13.8 1 52 1 0 1 0 0 0 1 112 2 5015 F 161 12.1 0 59 1 0 1 0 0 0 0 161
2 4906 F 117 10.7 0 52 1 0 1 0 0 0 0 117 2 5015 M 139 12.6 1 59 1 0 1 0 0 0 1 139
2 4906 M 107 14.2 1 52 1 0 1 0 0 0 1 107 2 5015 M 128 14.9 1 59 1 0 1 0 0 0 1 128
2 4906 M 96 12.2 1 52 1 0 1 0 0 0 1 96 2 5016 F 175 11.6 0 60 1 0 1 0 0 0 0 175
2 4906 F 163 14.7 0 52 1 0 1 0 0 0 0 163 2 5016 M 161 11.1 1 60 1 0 1 0 0 0 1 161
2 4906 M 161 11.5 1 52 1 0 1 0 0 0 1 161 2 5016 M 160 13 1 60 1 0 1 0 0 0 1 160
2 4906 F 151 11.7 0 52 1 0 1 0 0 0 0 151 2 5016 M 156 12 1 60 1 0 1 0 0 0 1 156
2 4906 F 148 13 0 52 1 0 1 0 0 0 0 148 2 5016 F 147 13.2 0 60 1 0 1 0 0 0 0 147
2 4906 M 146 15.4 1 52 1 0 1 0 0 0 1 146 2 5016 M 102 7.4 1 60 1 0 1 0 0 0 1 102
2 4913 F 130 14.1 0 53 1 0 1 0 0 0 0 130 2 5017 F 171 11.1 0 61 1 0 1 0 0 0 0 171
2 4913 F 120 11.7 0 53 1 0 1 0 0 0 0 120 2 5017 M 162 16.6 1 61 1 0 1 0 0 0 1 162
2 4913 F 117 11.6 0 53 1 0 1 0 0 0 0 117 2 5017 F 161 12 0 61 1 0 1 0 0 0 0 161
2 4913 F 111 10.3 0 53 1 0 1 0 0 0 0 111 2 5017 M 156 10.4 1 61 1 0 1 0 0 0 1 156
2 4914 M 129 14.3 1 54 1 0 1 0 0 0 1 129 2 5017 F 117 6.1 0 61 1 0 1 0 0 0 0 117
2 4914 F 126 6.8 0 54 1 0 1 0 0 0 0 126 2 5017 F 126 6.5 0 61 1 0 1 0 0 0 0 126
2 4914 F 117 10.5 0 54 1 0 1 0 0 0 0 117 2 5018 M 166 12.3 1 62 1 0 1 0 0 0 1 166
2 4914 M 168 15.1 1 54 1 0 1 0 0 0 1 168 2 5018 F 113 11.1 0 62 1 0 1 0 0 0 0 113
2 4914 F 164 10.5 0 54 1 0 1 0 0 0 0 164 2 5019 F 168 17.2 0 63 1 0 1 0 0 0 0 168
2 4914 F 151 7 0 54 1 0 1 0 0 0 0 151 2 5019 M 162 13.6 1 63 1 0 1 0 0 0 1 162
2 4914 F 150 10.1 0 54 1 0 1 0 0 0 0 150 2 5019 F 159 10.5 0 63 1 0 1 0 0 0 0 159
2 4914 F 122 10.1 0 54 1 0 1 0 0 0 0 122 2 5019 M 159 9.2 1 63 1 0 1 0 0 0 1 159
2 4918 M 129 18.8 1 55 1 0 1 0 0 0 1 129 2 5019 F 134 11.9 0 63 1 0 1 0 0 0 0 134
2 4918 M 110 11.2 1 55 1 0 1 0 0 0 1 110 2 5019 M 118 8.7 1 63 1 0 1 0 0 0 1 118
2 4918 M 107 12.8 1 55 1 0 1 0 0 0 1 107 2 5019 F 98 11.3 0 63 1 0 1 0 0 0 0 98
2 4919 M 170 10.6 1 56 1 0 1 0 0 0 1 170 2 5020 F 164 12.5 0 64 1 0 1 0 0 0 0 164
2 4919 M 168 12.2 1 56 1 0 1 0 0 0 1 168 2 5020 M 161 16.9 1 64 1 0 1 0 0 0 1 161
2 4919 F 165 10.5 0 56 1 0 1 0 0 0 0 165 2 5020 M 159 10.5 1 64 1 0 1 0 0 0 1 159
2 4919 F 161 11.7 0 56 1 0 1 0 0 0 0 161 2 5020 M 154 13.3 1 64 1 0 1 0 0 0 1 154
2 4921 M 118 9.7 1 57 1 0 1 0 0 0 1 118 2 5020 F 132 11.4 0 64 1 0 1 0 0 0 0 132
2 4921 F 113 13.6 0 57 1 0 1 0 0 0 0 113 2 5204 F 109 9 0 65 1 0 1 0 0 0 0 109
2 4921 F 112 9 0 57 1 0 1 0 0 0 0 112 2 5207 F 121 10.2 0 66 1 0 1 0 0 0 0 121 i
2 4921 M 108 7.3 1 57 1 0 1 0 0 0 1 108 2 5331 F 121 11.3 0 67 1 0 1 0 0 0 0 121
2 4921 M 97 13.5 1 57 1 0 1 0 0 0 1 97 2 5334 F 125 9.3 0 68 1 0 1 0 0 0 0 125
2 4921 M 168 12 1 57 1 0 1 0 0 0 1 168 2 5336 F 129 12.8 0 69 1 0 1 0 0 0 0 129
2 4921 F 159 13.9 0 57 1 0 1 0 0 0 0 159 2 5336 M 122 8.9 1 69 1 0 1 0 0 0 1 122
2 4921 F 159 10.3 0 57 1 0 1 0 0 0 0 159 3 1971 M 160 14.5 1 70 1 0 0 1 0 0 1 160 ;
-282- -283-
Appendix (.': Datoserfrom Duohateau. et al., 199H; lnternationat Livestock Research Institute (IL/U), Kenya Appendix C: J)UIW"el from Duchuteuu, et al .. 199H: International Livestock Research institute (//./U). Kenya
3 ,'97' F '59 '2.' 0 70 , 0 0 , 0 0 0 '59 3 4907 M '48 11.7 , 77 , 0 0 , 0 0 '48
·3 '97' M '54 '3.9 , 70 ,, 0 0
, ,
, 0 ·0 '54 3 ·4907 F '46 11.6 0, 77
, 0 0 , 0 0 0 '46
3 '97' F '5' '0.5 0 70 , 0 0 , 0 0 0 '5' 3 4907 M
, , ,
'3' '2.8 77 0 0 0 0 '3'
3 '97' F '48 11.7 0 7Ó , 0 O. , 0 0 0 '48 3 4907 M 127 '3.9 , 77 , 0 0 , 0 0 , '273 '97' F '48 '2.6 0 7Q 0 0 0 0 0 '48 3 4907 M '32 11.6 , 77 , 0 0 , 0 0 , '32
3 '97' M '47 8.7 , 70 ,, 0 0 ,, 0 0 , '47 3 4907 F '24 '3.7 0 77 ,, 0 0 ,, 0 0 0 '243 '97' F '44 6.5 0, 70 , 0 0 , 0 0 0, '44 3 4907 F '20 '3.8 0 77 0 0 0 0 0 '203 '97' M '39 17.2 , 70 , 0 0 , 0 0 ., '39 3 4907 M 112 9.9 , ,
, ,
3 '972 M '52 '8.6 7' , 0 0 , 0 0 '52 3 4907 M '08 11.7 ,
77 0 0 0 0 112
77 , 0 0 , 0 0 , 'OB
3 '972 M '49 '5.2 , 7' , 0 0 , 0 0 , '49 3 4907 F '06 '2.4 0 77 , 0 0 , 0 0 0 '063 '972 F '49 '6 0, 7' 0 0 0 0 0 '49 3 4908 M '43 '5 , 78 , 0 0 , 0 0
, '43
3 '972 M '48 '6.' 7'
3 '972 F '45 8.3 0 7' ,
, 0 0 ,, 0 0
, '48 3 4908 F '39 '2.6 0 78 , 0 0 , 0 0 0 '39
, , 0 0 0 0 0
, ,
3 '972 M '43 '5.7 7' 0 0 , 0 0 ,
'45 3 4908 F '38 '5.4 0 78 0 0 0 0 0 '38
'43 3 4908 F '33 '5.9 0 78 , 0 0 , 0 0 0 '33
3 '972 F '26 '6.7 0 7' ,, 0 0
, 0 0 0 '26 3 4908 M '28 11.3 , 78 , 0 0 , 0 0 , '28
3 '972 F '22 10.5 0 71 0 0 1 0 0 0 122 3 4908 F '32 '2.6 0 78 , 0 0 1 0 0 0 '32
3 1972 F 122 18.1 0 71 1 0 0 1, 0 0 0 122 3 4908 M 129 '2.4
, 78 , 0 0 , 0 0 , ,_'2_9_
3 1972 F 119 8.8 0 71 1 0 0 0 0 0 119 3 4908 M '29 '5.4 , 78 , 0 0 , 0 0 , '29
3 1972 F 114 12.8 0 71 1 0 0 1, 0 0 0 11. 3 4908 F 118 '2.' 0
,
, 78 , 0 0
,
, 0 0 0 1183 '972 F 110 8.8 0 71 1 0 0 0 0 0 110 3 4908 M 9' '5.4 78 ,
3 1972 F 107 11.7 0 71 1 0 0 1 0 0 0 '07 3 4908 M 173 '3.2 , 78 ,
0 0 , 0 0 , .9'-0 0 0 0 '73
3 1973 M 152 12.5 1 72 1 0 0 1 0 0 , '52 3 4908 F 170 '3.9 0 78 , 0 0 , 0 0 0 17~
3 1973 F 150 15.4 0, 72 ,1 0 0 ,1 0 0 0, '50 3 4908 M '59 '2.9 , 78 , 0 0 ,, 0 0 1 1593 '973 M 149 2'.2 72 , 0 0 , 0 0 '49 3 4908 F '58 15.2 0 78 0 0 1 0 0 0 '583 '973 F '44 9.9 0 72 0 0 0 0 0 '44 3 4908 F '46 '2.' 0 78 , 0 0 , 0 0 0 '46
3 '973 F '40 '3.3 0 72 , 0 0 , 0 0 0 '40 3 4908 F '36 10.' 0 78 , 0 0 , 0 0 0 '36
3 '973 F '27 '4.4 0 72 1 0 0 ,1 0 0 0 '27 3 4909 F '53 14.9 0 79
, 0 0 , 0 0 0 153
3 '974 F 159 11.6 0 73 ,1 0 0 , 0 0 0 '59 3 4909 F '52 17.3 0 79 1 0 0 1 0 0 0 '523 '974 F '52 '5.2 0 73 0 0 0 0 0 '52 3 4909 M '52 '5.6 , 79 , 0 0 , 0 0 , '52
3 '974 M 141 11.5 1 73 1 0 0 , 0 0 , '4' 3 4909 F '52 '6.7 0 79 , 0 0 , 0 0 0 '52
3 '974 M 140 12.1 1 73 1 0 0 1 0 0 1 140 3 4909 F '49 8.7 0 79 , ,, 0 0 , 0 0 0 '493 '980 M 158 15.2 1 74 1 0 0 1 0 0 1 158 3 4909 F '48 11.5 0 79 0 0 , 0 0 0 1483 1980 F 153 8.8 0 74 1 0 0 1 0 0 0 153 3 4909 F 144 '2.6 0 79 1 0 0 0 0 0 '44
3 1980 M 151 18.6 1 74 1 0 0 1 0 0 1 151 3 4909 M '44 '4.3
, 79 , 0 0 , 0 0 , '44
3 1980 M 147 16.3 1 74 1 0 0 1 0 0 1 147 3 4909 F '38 13.5 0 79 ,, 0 0
, 0 0 0 '38
3 '980 M 138 14.2 1 74 1 0 0 1 0 0 1 138 3 49'0 F '49 '4.9 0 80 0 0 , 0 0 0 149
3 1980 M 126 9.2 1 74 1 0 0 1 0 0 1 126 3 49'0 M '40 '3.4 , 80 , 0 0 , 0 0 , '40
3 '991 M 128 15.4 1 75 1 0 0 1 0 0 1 128 3 49'0 F '40 11.9 0 80 , 0 0 , 0 0 0 '40
3 1991 F 122 15.1 0 75 1 0 0 1 0 0 0 122 3 49'0 F '28 '3.' 0 80 ,, 0 0
, 0 0 0 128
3 1991 F 118 11.8 0 75 1 0 0 1 0 0 0 118 3 49'0 F '25 '2.5 0 80 0 0 , 0 0 0 '25
3 1991 F 113 8.4 0 75 1 0 0 1 0 0 0 113 3 49'0 M '24 9.2 , 80 , 0 0 , 0 0 1 124
3 '991 F 92 10.9 0 75 1 0 0 1 0 0 0 92 3 49'0 M '24 7
, 80 , 0 0 , 0 0 , '24
3 1999 M 155 14.7 1 76 1 0 0 1 0 0 1 155 3 49'0 M 119 '4.3 , 80 , ,, , 0 0 , 0 0
, 119
3 1999 F 136 11.5 0 76 1 0 0 1 0 0 0 136 3 4910 M 117 '4 , 80 , 0 0 , 0 0
, 117
3 1999 M 136 16 1 76 ,1 0 0 ,1 0 0 1 136 3 49'0 M '06 11.6 80 0 0 0 0 , '063 '999 F '24 ,1.7 0 76 0 0 0 0 0 '24 3 49'0 M 99 ,0.3 , 80 _ , ._(l_ O , 0 ~-.-i,,___!l!l__
-284- -285-
r---------------------------------------------------------------------------------------------------~---- ---------~--.---------------------.-------------------- __-------- __
Appendix C: Datasetfrom Duohateau. et al., 1998; International Livestock Research institute (IL/U). Kenya Appendix C: Dataset frum Ducbateau, et al .. /9<)/'/; International Livestock teeseoreb Insnune (II.IU), Kenya
3 4910 F 93 7.3 0 80 1 0 0 1 0 0 0 93 3 5002 M 172 11.9 1 86 1 0 0 1 0 0 1 172
3 4911 M 155 18.1 1 81 1 0 0 1 '0 0 1 155 3 5002 M 170 14.8 1 86 1 0 0 1 0 0 '1 170
3 4911 M 155 12.9 1 81 1 0 0 1 0 0 1 155 3 5002 F 166 12.9 0 86 1 0 0 1 0 0 0 166
3 4911 M 155 17 1 81 1 O. 0 1 0 0 1 155 3 5003 F 131 16.7 0 87 1 0 0 1 0 0 0 131
3 4911 F 150 16.7 0 81 1 0 0 1 0 0 0 150 3 5003 M 126 14.9 1 87 1 0 0 1 0 0 1 126
3 4911 F 149 12.7 0 81 1 0 0 1 0 0 0 149 3 5003 F 119 12.6 0 87 1 0 0 1 0 0 0 119
3 4912 F 156 16.8 0 82 1 0 0 1 0 0 0 156 3 5003 M 95 13.5 1 87 1 0 0 1 0 0 1 95
3 4912 M 155 13.9 1 82 1 0 0 1 0 0 1 155 3 5003 M 92 11.7 1 87 1 0 0 1 0 0 1 92
3 4912 M 155 17.9 1 82 1 0 0 1 0 0 1 155 3 5003 F 83 11.3 0 87 1 0 0 1 0 0 0 83
3 4912 F 153 14.3 0 82 1 0 0 1 0 0 0 153 3 5003 F 170 15.1 0 87 1 0 0 1 0 0 0 170
3 4912 M 150 11.4 1 82 1 0 0 1 0 0 1 150 3 5003 M 166 14.6 1 87 1 0 0 1 0 0 1 166
3 4912 M 144 10.5 1 82 1 0 0 1 0 0 1 144 3 5003 M 159 10.3 1 87 1 0 0 1 0 0 1 159
3 4912 F 144 11.9 0 82 1 0 0 1 0 0 0 144 3 5003 F 151 13.6 0 87 1 0 0 1 0 0 0 151
3 4912 F 143 12.8 0 82 1 0 0 1 0 0 0 143 3 5004 F 114 15.2 0 88 1 0 0 1 0 0 0 114
3 4912 F 140 16 0 82 1 0 0 1 0 0 0 140 3 5004 F 111 6.4 0 88 1 0 0 1 0 0 0 111
3 4912 F 126 12.3 0 82 1 0 0 1 0 0 0 126 3 5004 M 110 12.7 1 88 1 0 0 1 0 0 1 110
3 4912 M 115 15.1 1 82 1 0 0 1 0 0 1 115 3 5004 F 106 8.6 0 88 1 0 0 1 0 0 0 106
3 4915 F 119 8.4 0 83 1 0 0 1 0 0 0 119 3 5004 F 105 11.3 0 88 1 0 0 1 0 0 0 105
3 4915 M 112 8.1 1 83 1 0 0 1 0 0 1 112 3 5004 M 94 12 1 88 1 0 0 1 0 0 1 94
3 4915 F 110 12.5 0 83 1 0 0 1 0 0 0 110 3 5005 F 118 12.4 0 89 1 0 0 1 0 0 0 118
3 4915 F 120 10.3 0 83 1 0 0 1 0 0 0 120 3 5005 F 108 16.4 0 89 1 0 0 1 0 0 0 108
3 4915 F 109 11.9 0 83 1 0 0 1 0 0 0 109 3 5005 M 102 9.9 1 89 1 0 0 1 0 0 1 102
3 4915 F 99 7.6 0 83 1 0 0 1 0 0 0 99 3 5005 F 96 12.6 0 89 1 0 0 1 0 0 0 96
3 49)6 M 127 16.4 1 84 1 0 0 1 0 0 1 127 3 5005 F 83 12.9 0 89 1 0 0 1 0 0 0 83
3 4916 F 126 12.8 0 84 1 0 0 1 0 0 0 126 3 5005 F 83 11.8 0 89 1 0 0 1 0 0 0 83
3 4916 F 125 6.8 0 84 1 0 0 1 0 0 0 125 3 5005 F 171 14 0 89 1 0 0 1 0 0 0 171
3 4916 F 121 12.7 0 84 1 0 0 1 0 0 0 121 3 5005 M 163 16.5 1 89 1 0 0 1 0 0 1 163
3 4916 M 114 12.3 1 84 1 0 0 1 0 0 1 114 3 5005 M 162 13.7 1 89 1 0 0 1 0 0 1 162
3 4916 M 111 10.7 1 84 1 0 0 1 0 0 1 111 3 5005 ,F 159 9.2 0 89 1 0 0 1 0 0 0 159
3 4916 M 99 10.7 1 84 1 0 0 1 0 0 1 99 3 5005 M 138 12.2 1 89 1 0 0 1 0 0 1 138
3 4916 M 98 12.2 1 84 1 0 0 1 0 0 1 98 3 5006 M 124 12 1 90 1 0 0 1 0 0 1 124
3 4916 M 113 12 1 84 1 0 0 1 0 0 1 113 3 5007 M 170 7.6 1 91 1 0 0 1 0 0 1 170
3 4916 M 110 6.7 1 84 1 0 0 1 0 0 1 110 3 5007 F 169 15.2 0 91 1 0 0 1 0 0 0 169
3 4916 F 87 12.2 0 84 1 0 0 1 0 0 0 87 3 5007 F 161 13.2 0 91 1 0 0 1 0 0 0 161
3 5001 F 167 15.5 0 85 1 0 0 1 0 0 0 167 3 5007 F 157 12 0 91 1 0 0 1 0 0 0 157
3 5001 F 166 15.6 0 85 1 0 0 1 0 0 0 166 3 5007 M 151 14.1 1 91 1 0 0 1 0 0 1 151
3 5001 M 165 13.8 1 85 1 0 0 1 0 0 1 165 3 5007 M 141 10.8 1 91 1 0 0 1 0 0 1 141
3 5001 M 159 12.2 1 85 1 0 0 1 0 0 1 159 3 5007 M 113 9.5 1 91 1 0 0 1 0 0 1 113
3 5001 F 154 11.2 0 85 1 0 0 1 0 0 0 154 3 5007 M 110 8.5 1 91 1 0 0 1 0 0 1 110
3 5002 M 120 11.9 1 86 1 0 0 1 0 0 1 120 3 5007 M 107 12.6 1 91 1 0 0 1 0 0 1 107
3 5002 F 111 15.2 0 86 1 0 0 1 0 0 0 111 3 5008 F 170 14.8 0 92 1 0 0 1 0 0 0 170
3 5002 F 109 11 0 86 1 0 0 1 0 0 0 109 3 5008 M 164 16.1 1 92 1 0 0 1 0 0 1 164
3 5002 F 104 12.8 0 86 1 0 0 1 0 0 0 104 3 5008 F 162 12.3 0 92 1 0 0 1 0 0 0 162
3 5002 M 102 10.6 1 86 1 0 0 1 0 0 1 102 3 5008 M 160 14 1 92 1 0 0 1 0 0 1 160
3 5002 M 97 12.1 1 86 1 0 0 1 0 0 1 97 3 5008 F 110 10.5 0 92 1 0 0 1 0 0 0 110
3 5002 M 97 9~_1 86 1 0 0 1 0 0 1 97 3 5008 M 99 7.5 1 92 1 0 0 1 0 0 1 99
·286· -287-
r---------------------------------------------------------------------------------------------------~--~------------~~--------------------------------------------.---------------------------------------------------
Appendix (': Datosetfrom Duohateau. et al .. 1998: International Livestock Research Institute (II.Rl). Kenya Appendix C: Datase/from lluchateau, ef al., /99"; [nternanonal Livestock Research Institute (II./U). Kenya
3 500B M 96 12.1 1 92 1 0 0 1 0 0 1 96 3 5205 M 121 11.7 1 101 1 0 0 1 o 0 1 121
• 3 500B M 92 7.4 1 92 1 0 0 1 0 0 1 92 3 5205 M 111 13.B 1 101 1 0 0 1 0 0 1 111
3 5009 M 172 17.2 1 93 1 0 0 1 0 0 1 172 3 5205 M lO. 7 1 101 1 0 0 1 0 0 1 lO.
3 5009 F 16B 10.6 0 93 1 0 D 1 0 0 0 16B 3 5206 F 120 12.1 0 102 1 0 0 1 D 0 0 120
3 5009 M 165 13.6 1 93 1 0 0 1 0 0 1 165 3 5206 F 120 10.2 0 102 1 0 0 1 0 0 0 120
3 5009 M 161 10.2 1 93 1 0 0 1 0 0 1 161 3 5206 M 117 15.7 1 102 1 0 0 1 0 0 1 117
3 5009 M 115 B.6 1 93 1 0 0 1 0 0 1 115 3 5206 M 116 11 1 102 1 0 0 1 0 0 1 116
3 5009 F 109 14.3 0 93 1 0 0 1 0 0 0 109 3 5206 M 109 9.6 1 102 1 0 0 1 0 0 1 109
3 5009 M 9B B.9 1 93 1 0 0 1 0 0 1 9B 3 5206 F 105 9.1 0 102 1 0 0 1 0 0 0 105
3 5009 F 93 9 0 93 1 0 0 1 0 0 0 93 3 5206 M 102 12.9 1 102 1 0 0 1 0 0 1 102
3 5010 F 165 15.2 0 94 1 0 0 1 0 0 0 165 3 5321 F 139 10.9 0 103 1 0 0 1 0 0 0 139
3 5010 M 163 12.4 1 94 1 0 0 1 0 0 1 163 3 5321 M 136 15.3 1 103 1 0 0 1 0 0 1 136
3 5010 F 160 15 0 94 1 0 0 1 0 0 0 160 3 5322 M 121 9.1 1 104 1 0 0 1 0 0 1 121
3 5011 F 172 12.4 0 95 1 0 0 1 0 0 0 172 3 5322 M 121 10.1 1 104 1 0 0 1 0 0 1 121
3 5011 F 168 13.4 0 95 1 0 0 1 0 0 0 16B 3 5322 M 119 "" 1 104 1 0 0 1 0 0 1 1193 5011 F 162 11.4 0 95 1 0 0 1 0 0 0 162 3 5322 M 117 11.7 1 104 1 0 0 1 0 0 1 117
3 5011 M 159 12.2 1 95 1 0 O· 1 0 0 1 159 3 5322 F 113 7.7 0 104 1 0 0 1 0 0 0 113
3 5011 F 159 13.3 0 95 1 0 0 1 0 0 0 159 3 5322 F 103 8.7 0 104 1 0 0 1 0 O. 0 103
3 5011 M 15B 14.6 1 95 1 0 0 1 0 0 1 15B 3 532. F 110 11.7 0 105 1 0 0 1 0 0 0 110
3 5011 F 146 11.6 0 95 1 0 0 1 0 0 0 146 3 5324 F 106 ".B 0 105 1 0 0 1 0 0 0 106
3 5011 F 117 14.1 0 95 1 0 0 1 0 0 0 117 3 5326 M 116 12 1 106 1 0 0 1 0 0 1 116
3 5011 M 115 11.5 1 95 1 0 0 1 0 0 1 115 3 5326 M 132 9.2 1 106 1 0 0 1 0 0 1 132
3 5011 F 114 7.B 0 95 1 0 0 1 0 0 0 114 3 5326 M 127 11 1 106 1 0 0 1 0 O. 1 127
3 5011 M 111 9.9 1 95 1 0 0 1 0 0 1 111 3 5328 M 129 11.3 1 107 1 0 0 1 0 0 1 129
3 5011 M 111 9.2 1 95 1 0 0 1 0 0 1 111 3 5328 M 127 12.5 1 107 1 0 0 1 0 0 1 127
3 5012 M 169 17.6 1 96 1 0 0 1 0 0 1 169 3 532B F 121 13.4 0 107 1 0 0 1 0 0 0 121
3 5012 F 166 14.5 0 96 1 0 0 1 0 0 0 166 3 532B M 117 B.' 1 107 1 0 0 1 0 0 1 117
3 5012 F 159 13.1 0 96 1 0 0 1 0 0 0 159 3 532B M 113 7 1 107 1 0 0 1 0 0 1 113
3 5013 M 16B 11.6 1 97 1 0 0 1 0 0 1 16B 3 532B M ,.3 13.6 1 107 1 0 0 1 0 0 1 ,.3
3 5013 F 162 10.7 0 97 1 0 0 1 0 0 0 162 3 532B M 127 11.1 1 107 1 0 0 1 0 0 1. 127
3 5013 M 154 13.1 1 97 1 0 0 1 0 0 1 154 3 5329 M 103 8.1 1 108 1 0 0 1 0 0 1 103
3 5013 M 150 9.4 1 97 1 0 0 1 0 0 1 150 3 5329 F 100 11.5 0 108 1 0 0 1 0 0 0 100
3 5013 F 14B 11.9 0 97 1 0 0 1 0 0 0 148 3 5329 F 100 8.3 0 lOB 1 0 0 1 0 0 0 100
3 5071 F 111 10.3 0 98 1 0 0 1 0 0 0 111 3 5329 M 99 7.' 1 108 1 0 0 1 0 0 1 99
3 5071 F 104 12.8 0 98 1 0 0 1 0 0 0 104 3 5329 F 94 11.1 0 lOB 1 0 0 1 0 0 0 9.
3 5071 F 96 12.5 0 98 1 0 0 1 0 0 0 96 3 5329 F 144 5.5 0 108 1 0 0 1 0 0 0 144
3 5071 F 93 5.8 0 98 1 0 0 1 0 0 0 93 3 5329 F 140 13.4 0 108 1 0 0 1 0 0 0 140
3 5071 F 85 10.2 0 98 1 0 0 1 0 0 0 85 3 5329 F 136 13.4 0 108 1 0 0 1 0 0 0 136
3 5071 F 132 10.7 0 98 1 0 0 1 0 0 0 132 3 5329 M 133 12 1 lOB 1 0 0 1 0 0 1 133
3 5071 F 125 9.4 0 98 1 0 0 1 0 0 0 125 3 5329 F 120 9.1 0 108 1 0 0 1 0 0 0 120
3 5071 F 122 11.7 0 98 1 0 0 1 0 0 0 122 3 5330 F 129 10.2 0 109 1 0 0 1 0 0 0 129
3 5071 M 121 11.7 1 98 1 0 0 1 0 0 1 121 3 5330 M 128 8 1 109 1 0 0 1 0 0 1 12B
3 5071 F 120 14 0 98 1 0 0 1 0 0 0 '120 3 5330 M 120 6.7 1 109 1 0 0 1 0 0 1 120
3 5073 M lIB ".B 1 99 1 0 0 1 0 0 1 lIB 3 5330 M 116 12.1 1 109 1 0 0 1 0 0 1 116
3 5076 F 105 11.7 0 100 1 0 0 1 0 0 0 105 3 5330 M 137 12.2 1 109 1 0 0 1 0 0 1 137
3 5205 F 121 9.3 0 101 1 0 0 1 0 0 0 121 3 5330 F 137 9.7 0 109 1 0 0 1 0 0 0 137
-288- -289-
Appendix C: Datasetfrom Duchateau, et al., 1998: tnternational Ltvestock Research institute (ILlU), Kenya Appendix C: Dataset from Duchateou, et al .. 199X; International l.ivestock Research Institute (IL/U), Kenyo
3 5330 F 135 10.7 0 109 1 0 0 1 0 0 0 135 4 1979 M 132 10 1 114 1 0 0 0 1 0 1 132
3" 5330 F 134 13.4 0 109 1 0 0 1 0 O' 0 134 4 1979 M 123 9.1 1 114 1 0 0 0 1 0 1 123
3 5330 F 129 6.9 0 109 1 0 0 1 0 0 0 129 4 1979 M 114 11.3 1 114 1 0 0 0 1 0 1 114
3 5337 M 134 8.7 1 110 1 0 0 .1. 0 0 1 134 .A 1979 M 119 12 1 114 1 Ó 0 0 1 O. 1 119
3 5337 F 119 16.2 0 110 1 0 0 1 0 0 0 119 4 1979 F 116 12.5 0 114 1 0 0 0 1 0 0 116
3 5337 F 117 11.2 0 110 1 0 0 1 0 0 0 117 4 1979 M 99 8.5 1 114 1 0 0 0 1 0 1 99
3 5337 F 121 13 0 110 1 0 0 1 0 0 0 121 4 1981 M 154 18.2 1 115 1 0 0 0 1 0 1 154
3 5337 F 116 9.8 0 110 1 0 0 1 0 0 0 116 4 1981 F 148 9 0 115 1 0 0 0 1 0 0 148
3 5338 F 129 10.7 0 111 1 0 0 1 0 0 0 129 4 1981 F 147 12.3 0 115 1 0 0 0 1 0 0 147
3 5338 M 121 10.6 1 111 1 0 0 1 0 0 1 121 4 1981 F 145 13.4 0 115 1 0 0 0 1 0 0 145
3 5338 M 121 9.7 1 111 1 0 0 1 0 0 1 121 4 1981 F 144 15 0 115 1 0 0 0 1 0 0 144
3 5338 M 117 9.2 1 111 1 0 0 1 0 0 1 117 4 1981 M 136 11.9 1 115 1 0 0 0 1 0 1 136
3 5338 M 116 15.1 1 111 1 0 0 1 0 0 1 116 4 1982 F 121 10.3 0 116 1 0 0 0 1 0 0 121
3 5338 M 115 8.5 1 111 1 0 0 1 0 0 1 115 4 1983 F 156 12.7 0 117 1 0 0 0 1 0 0 156
3 5338 F 143 12.8 0 111 1 0 0 1 0 0 0 143 4 1983 F 144 12 0 117 1 0 0 0 1 0 0 144
3 5338 M 142 13.4 1 111 1 0 0 1 0 0 1 142 4 1983 F 143 11.7 0 117 1 0 0 0 1 0 0 143
3 5338 F 141 10.3 0 111 1 0 0 1 0 0 0 141 4 1983 M 142 11.6 1 117 1 0 0 0 1 0 1 142
3 5338 M 139 15.1 1 111 1 0 0 1 0 0 1 139 4 1983 M 136 12.8 1 117 1 0 0 0 1 0 1 136
3 5338 F 136 9.5 0 111 1 0 0 1 0 0 0 136 4 1983 M 133 11.6 1 117 1 0 0 0 1 0 1 133
3 5338 M 123 12.7 1 111 1 0 0 1 0 0 1 123 4 1983 F 120 13.2 0 117 1 0 0 0 1 0 0 120
4 1975 F 153 14.4 0 112 1 0 0 0 1 0 0 153 4 1984 M 150 15.2 1 118 1 0 0 0 1 0 1 150
4 1975 F 153 15.9 0 112 1 0 0 0 1 0 0 153 4 1984 M 147 16.3 1 118 1 0 0 0 1 0 1 147
4 1975 M 151 12.4 1 112 1 0 0 0 1 0 1 151 4 1984 F 147 11.3 0 118 1 0 0 0 1 0 0 147
4 1975 M 142 16.4 1 112 1 0 0 0 1 0 1 142 4 1984 F 146 12.5 0 118 1 0 0 0 1 0 0 146
4 1975 M 142 14.1 1 112 1 0 0 0 1 0 1 142 4 1984 M 145 16.3 1 118 1 0 0 0 1 0 1 145
4 1975 M 141 14.8 1 112 1 0 0 0 1 0 1 141 4 1984 F 140 17.8 0 118 1 0 0 0 1 0 0 140
4 1975 F 141 15 0 112 1 0 0 0 1 0 0 141 4 1984 F 139 14.1 0 118 1 0 0 0 1 0 0 139
4 1975 M 128 11.9 1 112 1 0 0 O· 1 0 1 128 4 1984 F 133 6.8 0 118 1 0 0 0 1 0 0 133
4 1975 M 127 12.8 1 112 1 0 0 0 1 0 1 127 4 1986 M 128 11.9 1 119 1 0 0 0 1 0 1 128
4 1975 F 126 14.9 0 112 1 0 0 0 1 0 0 126 4 1986 M ;26 16.5 1 119 1 0 0 0 1 0 1 126
4 1975 F 125 9 0 112 1 0 0 0 1 0 0 125 4 1986 M 123 11.5 1 119 1 0 0 0 1 0 1 123
4 1975 M 121 13.5 1 112 1 0 0 0 1 0 1 121 4 1986 M 122 12.6 1 119 1 0 0 0 1 0 1 122
4 1975 M 114 5.9 1 112 1 0 0 0 1 0 1 114 4 1986 F 119 10.8 0 119 1 0 0 0 1 0 0 119
4 1976 F 151 12.7 0 113 1 0 0 0 1 0 0 151 4 1988 F 156 14.9 0 120 1 0 0 0 1 0 0 156
4 1976 M 151 14.9 1 113 1 0 0 0 1 0 1 151 4 1988 F 153 11.5 0 120 1 0 o 0 1 0 0 153
4 1976 M 151 10.7 1 113 1 0 0 0 1 0 1 151 4 1988 F 150 12.4 0 120 1 0 0 0 1 0 0 150
4 1976 M 146 17.8 1 113 1 0 0 0 1 0 1 146 4 1988 F 148 11.2 0 120 1 0 0 0 1 0 0 148
4 1976 M 145 15 1 113 1 0 0 0 1 0 1 145 4 1988 F 143 12.3 0 120 1 0 0 0 1 0 0 143
4 1976 M 144 17 1 113 1 0 0 0 1 0 1 144 4 1988 F 138 13.5 0 120 1 0 0 0 1 0 0 138
4 1979 M 154 15.5 1 114 1 0 0 0 1 0 1 154 4 1988 M 131 10.3 1 120 1 0 0 0 1 0 1 131
4 1979 M 149 8.8 1 114 1 0 0 0 1 0 1 149 4 1988 F 127 6.5 0 120 1 0 0 0 1 0 0 127
4 1979 M 147 6.6 1 114 1 0 0 0 1 0 1 147 4 1988 M 127 9.9 1 120 1 0 0 0 1 0 1 127
4 1979 F 145 12.7 0 114 1 0 0 0 1 0 0 145 4 4901 M 155 14.4 1 121 1 0 0 0 1 0 1 155
4 1979 F 144 13.2 0 114 1 0 0 0 1 0 0 144 4 4901 M 151 15.1 1 121 1 0 0 0 1 0 1 151
4 1979 M 143 14.5 1 114 1 0 0 0 1 0 1 143 4 4901 M 150 16.9 1 121 1 0 0 0 1 0 1 150
4 1979 M 135 12.8 1 114 1 0 0 0 1 0 1 135 4 4901 F 141 13.3 0 121 1 0 0 0 1 0 0 141
-290- -291-
r-------------------------~---------------------------------------------------------------------------.------------------~---------------------------------------------------------------------------------------------~
Appendix (.': Dataset from Duohateau. et al. 199H;International Livestock Research Institute (ILRJ), Kenya Appendix C: Datoset from Duchuteau, et at, 19YN;lnternationul Livestock Research Institute (ILIU). Kenya
4 4901 M 137 13.6 1 121 1 0 0 0 1 0 1 137 4 4905 M 146 14.2 1 124 1 0 0 0 1 0 1 146
4 ·4901 F 133 12.8 0 121 1 0 0 0 1 0 0 133 4 4905 M 142 11.6 1 124 1 0 0 0 1 0 1 142
4 4901 F 127 10.5 0 121 1 0 0 0 1 0 0 127 4 4905 M 140 11.6 1 124 1 0 0 0 1 0 1 140
4 4901 M 130 12.6 1 121 1 0 0 0 1 .0 1 130 4 .4905 F 123 11.6 0 124 1 0 0 0 1 0 O.. 123
4 4901 F 130 13.3 0 121 1 0 0 0 1 0 0 130 4 4905 F 117 15.4 0 124 1 0 0 0 1 0 0 117
4 4901 M 128 16.8 1 121 1 0 0 0 1 0 1 128 4 4905 M 106 10.3 1 124 1 0 0 0 1 0 1 106
4 4901 F 128 7.6 0 121 1 0 0 0 1 0 0 128 4 4905 M 173 14.2 1 124 1 0 0 0 1 0 1 173
4 4901 M 125 10.1 1 121 1 0 0 0 1 0 1 125 4 4905 M 168 12.6 1 124 1 0 0 0 1 0 1 168
4 4901 F 119 12.8 0 121 1 0 0 0 1 0 0 119 4 4905 F 165 10.4 0 124 1 0 0 0 1 0 0 165
4 4901 M 113 13 1 121 1 0 0 0 1 0 1 113 4 4905 M 165 13.1 1 124 1 0 0 0 1 0 1 165
4 4902 F 150 14 0 122 1 0 0 0 1 0 0 150 4 4905 M 159 13.4 1 124 1 0 0 0 1 0 1 159
4 4902 M 146 10.6 1 122 1 0 0 0 1 0 1 146 4 4905 M 157 14.3 1 124 1 0 0 0 1 0 1 157
4 4902 F 146 11.7 0 122 1 0 0 0 1 0 0 146 4 4905 F 154 12.8 0 124 1 0 0 0 1 0 0 154
4 4902 F 138 11.2 0 122 1 Ó 0 0 1 0 0 138 4 4905 F 153 14 0 124 1 0 0 0 1 0 0 153
4 4902 F 137 12 0 122 1 0 0 0 1 0 0 137 4 4906 M 153 14.4 1 125 1 0 0 0 1 0 1 153
4 4902 M 137 12.8 1 122 1 0 0 0 1 0 1 137 4 4906 M 144 11.2 1 125 1 0 0 0 1 0 1 144
4 4902 F 134 12.8 0 122 1 0 O· 0 1 0 0 134 4 4906 F 144 13.1 0 125 1 0 0 0 1 0 0 144
4 4902 F 127 12.5 0 122 1 0 0 0 1 0 0 127 4 4906 M 138 10.1 1 125 1 0 0 0 1 0 1 138
4 4902 M 124 7.6 1 122 1 0 0 0 1 0 1 124 4 4906 F 136 13.9 0 125 1 0 0 0 1 0 0 136
4 4902 M 123 15.2 1 122 1 0 0 0 1 0 1 123 4 4906 F 136 13.4 0 125 1 0 0 0 1 0 0 136
4 4902 M 113 10 1 122 1 0 0 0 1 0 1 113 4 4906 M 132 12.7 1 125 1 0 0 0 1 0 1 132
4 4902 M 113 13.1 1 122 1 0 0 0 1 0 1 113 4 4906 M 129 13.1 1 125 1 0 0 0 1 0 1 129
4 4902 F 142 6.6 0 122 1 0 0 0 1 0 0 142 4 4906 F 121 12.8 0 125 1 0 0 0 1 0 0 121
4 4902 M 140 12.5 1 122 1 0 0 0 1 0 1 140 4 4906 M 83 11.4 1 125 1 0 0 0 1 0 1 83
4 4902 F 134 10.5 0 122 1 0 0 0 1 0 0 134 4 4906 F 167 14.3 0 125 1 0 0 0 1 0 0 167
4 4902 F 131 9.1 0 122 1 0 0 0 1 0 0 131 4 4906 M 165 16.3 1 125 1 0 0 0 1 0 1 165
4 4902 M 127 13.4 1 122 1 0 0 0 1 0 1 127 4 4906 F 148 10.8 0 125 1 0 0 0 1 0 0 148
4 4902 F 127 11.3 0 122 1 0 0 0 1 0 0 127 4 4906 F 139 9.4 0 125 1 0 0 0 1 0 0 139
4 4902 M 125 10.3 1 122 1 0 0 0 1 0 1 125 4 4906 M 137 9.7 1 125 1 0 0 0 1 0 1 137
4 4902 M 118 12.2 1 122 1 0 0 0 1 0 1 118 4 4913 M 124 14 1 126 1 0 0 o . 1 0 1 124
4 4902 M 109 11.9 1 122 1 0 0 0 1 0 1 109 4 4913 M 124 12.4 1 126 1 0 0 0 1 0 1 124
4 4903 M 154 15.4 1 123 1 0 0 0 1 0 1 154 4 4913 M 116 12.3 1 126 1 0 0 0 1 0 1 116
4 4903 M 148 13.8 1 123 1 0 0 0 1 0 1 148 4 4913 M 116 11.1 1 126 1 0 0 0 1 0 1 116
4 4903 M 147 16.3 1 123 1 0 0 0 1 0 1 147 4 4913 M 114 9.9 1 126 1 0 0 0 1 0 1 114
4 4903 M 145 14.9 1 123 1 0 0 0 1 0 1 145 4 4913 M 113 15.1 1 126 1 0 0 0 1 0 1 113
4 4903 M 145 13.1 1 123 1 0 0 0 1 0 1 145 4 4913 F 106 8.6 0 126 1 0 0 0 1 0 0 106
4 4903 F 142 13.6 0 123 1 0 0 0 1 0 0 142 4 4913 M 100 10.1 1 126 1 0 0 0 1 0 1 100
4 4903 M 123 14 1 123 1 0 0 0 1 0 1 123 4 4914 F 127 13.4 0 127 1 0 0 0 1 0 0 127
4 4903 F 121 12.6 0 123 1 0 0 0 1 0 0 121 4 4914 M 124 16.8 1 127 1 0 0 0 1 0 1 124
4 4903 F 118 11.3 0 123 1 0 0 0 1 0 0 118 4 4914 F 123 14 0 127 1 0 0 0 1 0 0 123
4 4903 F 116 13.1 0 123 1 0 0 0 1 0 0 116 4 4914 F 119 15.9 0 127 1 0 0 0 1 0 0 119
4 4903 F 112 13.2 0 123 1 0 0 0 1 0 0 112 4 4914 M 112 10.2 1 127 1 0 0 0 1 0 1 11-1-2-
4 4905 M 157 6.6 1 124 1 0 0 0 1 0 1 157 4 4914 F 173 14.7 0 127 1 0 0 0 1 0 o 173
4 4905 F 151 12.6 0 124 1 0 0 0 1 0 0 151 4 4914 M 168 17 1 127 1 0 0 0 1 0 1 168.-
4 4905 F 148 9.5 0 124 1 0 0 0 1 0 0 148 4 4914 F 162 11.8 0 127 1 0 0 0 1 0 0 162
4 4905 M 148 14.5 1 124 1 0 0 0 1 0 1 148 4 4914 M 159 10.3 1 127 1 0 0 0 1 0 1 ï591
·292- -293-
Appendix C: Datasecfrom Duchateau, et ot., 1998; International Livestock Research Institute (ILIU), Kenya Appendix (.': Datosetfrom Duchateou. ef al., lC)t)I( Internunonul Livestock tteseurch JnSII/Ule (II.IU), Kenya
4 4914 M 157 12 1 127 1 0 0 0 1 0 1 157 4 4921 F 129 9.7 0 130 1 0 0 0 1 0 0 129
4 4914 M 155 15.1 1 127 1 0 0 0 1 0 1 155 4 4921 ·F 121 9.1 0 130 1 0 0 0 1 0 0 121
4 4914 F 154 12.6 0 127 1 0 0 0 1 0 0 154 4 4921 M 119 8.7 1 130 1 0 0 0 1 0 1 119
. 4 4914 F 124 7.6 0 127 1 Ó 0 0 1 D. 0 124 4 .4921 M 117 10 1 130 1 0 0 Ó 1 0 1 .117
4 4914 M 123 12.4 1 127 1 0 0 0 1 0 1 123 4 4921 F 114 9.3 0 130 1 0 0 0 1 0 0 114
4 4914 M 122 12.7 1 127 1 0 0 0 1 0 1 122 4 4921 M 109 13.3 1 130 1 0 0 0 .1 0 1 109
4 4914 M 119 11.1 1 127 1 0 0 0 1 0 1 119 4 4921 M 107 12.3 1 130 1 0 0 0 1 0 1 107
4 4914 M 118 12.4 1 127 1 0 0 0 1 0 1 118 4 4921 M 106 12.1 1 130 1 0 0 0 1 0 1 106
4 4914 M 117 11.6 1 127 1 0 0 0 1 0 1 117 4 4921 M 93 11.5 1 130 1 0 0 0 1 0 1 93
4 4914 F 114 7.7 0 127 1 0 0 0 1 0 0 114 4 4921 M 171 10.7 1 130 1 0 0 0 1 0 1 171
4 4914 F 143 13.3 0 127 1 0 0 0 1 0 0 143 4 4921 F 168 13.1 0 130 1 0 0 0 1 0 0 168
4 4914 F 128 11.1 0 127 1 0 0 0 1 0 0 128 4 4921 F 160 10.8 0 130 1 0 0 0 1 0 0 160
4 4914 M 126 11.2 1 127 1 0 0 0 1 0 1 126 4 4921 M 160 10.3 1 130 1 0 0 0 1 0 1 160
4 4918 F 114 12.7 0 128 1 0 0 0 1 0 0 114 4 4921 M 155 11.4 1 130 1 0 0 0 1 0 1 155
4 4918 F 111 9.6 0 128 1 0 0 0 1 0 0 111 4 4921 M 155 11.6 1 130 1 0 0 0 1 0 1 155
4 4918 M 108 14.1 1 128 1 0 0 0 1 0 1 108 4 4923 M 128 12.9 1 131 1 0 0 0 1 0 1 128
4 4918 M 106 9.7 1 128 1 0 0 0 1 0 1 106 4 4923 F 122 10.9 0 131 1 0 0 0 1 0 0 122
4 4918 M 106 10.7 1 128 1 0 0 0 1 0 1 106 4 4923 M 120 11.8 1 131 1 0 0 0 1 0 1 120
4 4918 F 104 9.8 0 128 1 0 0 0 1 0 0 104 I 4 4923 M 119 13.8 1 131 1 0 0 0 1 0 1 119
4 4918 F 96 10.3 0 128 1 0 0 0 1 0 0 96: 4 4923 F 98 10.4 0 131 1 0 0 0 1 0 0 98
4 4918 F 120 13 0 128 1 0 0 0 1 0 0 120 I 4 4923 M 93 9.4 1 131 1 0 0 0 1 0 1 93
4 4918 M 119 12.1 1 128 1 0 0 0 1 0 1 119 4 4923 M 172 16.1 1 131 1 0 0 0 1 0 1 172
4 4918 F 110 11.8 0 128 1 0 0 0 1 0 0 110 4 4923 F 165 13.5 0 131 1 0 0 0 1 0 0 165
4 4918 M 101 7.2 1 128 1 0 0 0 1 0 1 101 4 4923 M 164 16 1 131 1 0 0 0 1 0 1 164
4 4918 F 98 9.4 0 128 1 0 0 0 1 0 0 98 4 4923 F 162 13.5 0 131 1 0 0 0 1 0 0 162
4 4919 F 168 12.2 0 129 1 0 0 0 1 0 0 168 4 4923 M 159 9.7 1 131 1 0 0 0 1 0 1 159
4 4919 F 167 10.6 0 129 1 0 0 0 1 0 0 167 4 4923 M 159 13 1 131 1 0 0 0 1 0 1 159
4 4919 F 167 13.2 0 129 1 0 0 0 1 0 0 167 4 4923 F 156 10.7 0 131 1 0 0 0 1 0 0 156
4 4919 F 164 10.8 0 129 1 0 0 0 1 0 0 164 4 4923 M 154 12.3 1 131 1 0 0 0 1 0 1 154
4 4919 F 155 10.2 0 129 1 0 0 0 1 0 0 155 4 5015 M 174 16.1 1 132 1 0 0 0 1 0 1 174
4 4919 F 154 9 0 129 1 0 0 0 1 0 0 154 4 5015 F 164 9.7 0 132 1 0 0 0 1 0 0 164
4 4919 M 151 9.1 1 129 1 0 0 0 1 0 1 151 4 5015 F 158 12.8 0 132 1 0 0 0 1 0 0 158
4 4919 F 121 8.7 0 129 1 0 0 0 1 0 0 121 4 5015 M 155 13.5 1 132 1 0 0 0 1 0 1 155
4 4919 F 114 9.3 0 129 1 0 0 0 1 0 0 114 4 5015 F 154 12.1 0 132 1 0 0 0 1 0 0 154
4 4919 M 103 9.7 1 129 1 0 0 0 1 0 1 103 4 5015 M 151 12.5 1 132 1 0 0 0 1 0 1 151
4 4919 F 133 8.8 0 129 1 0 0 0 1 0 0 133 4 5015 M 146 16 1 132 1 0 0 0 1 0 1 146
4 4919 M 129 12.3 1 129 1 0 0 0 1 0 1 129 4 5015 F 139 6 0 132 1 0 0 0 1 0 0 139
4 4919 M 117 9.1 1 129 1 0 0 0 1 0 1 117 4 5015 M 135 10 1 132 1 0 0 0 1 0 1 135
4 4919 M 116 13.2 1 129 1 0 0 0 1 0 1 116 4 5015 F 122 10.7 0 132 1 0 0 0 1 0 0 122
4 4919 F 144 8.8 0 129 1 0 0 0 1 0 0 144 4 5015 F 120 13.4 0 132 1 0 0 0 1 0 0 120
4 4919 F 141 13.9 0 129 1 0 0 0 1 0 0 141 4 5016 M 172 11.5 1 133 1 0 0 0 1 0 1 172
4 4919 F 136 13 0 129 1 0 0 0 1 0 0 136 4 5016 F 168 10.5 0 133 1 0 0 0 1 0 0 168
4 4919 M 133 9.8 1 129 1 0 0 0 1 0 1 133 4 5016 M 168 12.7 1 133 1 0 0 0 1 0 1 168
4 4919 F 130 11.8 0 129 1 0 0 0 1 0 0 130 4 5016 M 163 13.5 1 133 1 0 0 0 1 0 1 163
4 4919 F 129 11.6 0 129 1 0 0 0 1 0 0 129 4 5016 M 119 6.1 1 133 1 0 0 0 1 0 1 119
4 4919 M 115 7.6 1 129 1 0 0 0 1 0 1 115 4 5016 F 118 9.3 0 133 1_. 0 0 0 1 0 0 118j
-294- -295-
Appendix C: Datoset from Duchateou, et ot., 19Y8; International Livestock Research Institute (ILIU), Kenya Appendix C: Datosetfrom Duchoteau, et al., I(jYH:International Livestock Research Insnnue (II.IU), Kenya
4 5016 M 117 13.7 1 133 1 0 0 0 1 0 1 117 4 5020 F 165 14.5 0 137 1 0 0 0 1 0 0 165
4 5016 F 108 8.5 0 133 1 0 0 0 1 0 0 108 4 5020 M 162 12.7 1 137 1 0 0 0 1 0 1 162
4 5016 F 104 12.3 0 133 1 0 0 0 1 0 0 104 4 5020 F 157 11.7 0 137 1 0 0 0 1 0 0 157
.. 4 5017 F 163 13.8 0 134 1 0 0 0 1 D 0 163 4 .. 5020 F 153 10.8 0 137 1 0 0 0 1 0 0 l53
4 5017 F 162 13.5 0 134 1 0 0 0 1 0 0 162 4 5020 M 152 12.1 1 137 1 0 0 0 1 0 1 152
4 5017 F 161 11.5 0 134 1 0 0 0 1 0 0 161 4 5020 F 118 13.2 0 137 1 0 0 0 1 0 0 118
4 5017 F 157 12 0 134 1 0 0 0 1 0 0 157 4 5020 M 106 7.6 1 137 1 0 0 0 1 0 1 106
4 5017 M 152 14.2 1 134 1 0 0 0 1 0 1 152 4 5020 F 102 8.5 0 137 1 0 0 0 1 0 0 102
4 5017 F 151 13.1 0 134 1 0 0 0 1 0 0 151 4 5020 F 102 10.8 0 137 1 0 0 0 1 0 0 102
4 5017 F 108 11.5 0 134 1 0 0 0 1 0 0 108 4 5020 M 140 14.4 1 137 1 0 0 0 1 0 1 140
4 5017 M 106 8.3 1 134 1 0 0 0 1 0 1 106 4 5020 M 137 11.4 1 137 1 0 0 0 1 0 1 137
4 5017 F 142 9.4 0 134 1 0 0 0 1 0 0 142 , 4 5020 F 133 11.3 0 137 1 0 0 0 1 0 0 133
4 5017 M 142 15.1 1 134 1 0 0 0 1 0 1 142 4 5020 M 132 8.9 1 137 1 0 0 0 1 0 1 132
4 5017 M 136 9.3 1 134 1 0 0 0 1 0 1 136 4 5020 F 132 11.1 0 137 1 0 0 0 1 0 0 132
4 5017 F 133 10.5 0 134 1 0 0 0 1 0 0 133 4 5020 M 113 6.7 1 137 1 0 0 0 1 0 1 113
4 5017 M 117 10.2 1 134 1 0 0 0 1 0 1 117 4 5204 M 118 12.2 1 138 1 0 0 0 1 0 1 118
4 5017 M 100 12.9 1 134 1 0 0 0 1 0 1 100 4 5204 M 115 10.2 1 138 1 0 0 0 1 0 1 115
4 5018 M 167 16 1 135 1 0 0 0 1 0 1 167 i 4 5204 M 113 10.1 1 138 1 0 0 0 1 0 1 113
4 5018 F 166 14.2 0 135 1 0 0 0 1 0 0 166 4 5204 F 110 9 0 138 1 0 0 0 1 0 0 110
4 5018 F 164 8.6 0 135 1 0 0 0 1 0 0 164 4 5204 F 103 6.2 0 138 1 0 0 0 1 0 0 103
4 5018 F 162 13 0 135 1 0 0 0 1 0 0 162 4 5207 M 116 9 1 139 1 0 0 0 1 0 1 116
4 5018 M 158 12.1 1 135 1 0 0 0 1 0 1 15~ 4 5207 F 114 11.6 0 139 1 0 0 0 1 0 0 114
4 5018 F 146 14.5 0 135 1 0 0 0 1 0 0 146 4 5207 M 110 10.1 1 139 1 0 0 0 1 0 1 110
4 5018 F 115 12.1 0 135 1 0 0 0 1 0 0 115 4 5207 M 108 8.6 1 139 1 0 0 0 1 0 1 108
4 5018 F 114 7.8 0 135 1 0 0 0 1 0 0 114 4 5208 M 110 12.1 1 140 1 0 0 0 1 0 1 110
4 5018 F 110 11.9 0 135 1 0 0 0 1 0 0 110 4 5208 M 110 9.4 1 140 1 0 0 0 1 0 1 110
4 5018 F 110 12 0 135 1 0 0 0 1 0 0 110 I 4 5208 F 106 8.3 0 140 1 0 0 0 1 0 0 106
4 5018 M 98 8.5 1 135 1 0 0 0 1 0 1 98 4 5208 F 105 7.6 0 140 1 0 0 0 1 0 0 105
4 5019 F 168 12.7 0 136 1 0 0 0 1 0 0 168 4 5331 .M 128 13.5 1 141 1 0 0 0 1 0 1 128
4 5019 F 166 14.6 0 136 1 0 0 0 1 0 0 166 4 5331 M 121 9.8 1 141 1 0 0 0 1 0 1 121
4 5019 F 166 13.5 0 136 1 0 0 0 1 0 0 166 4 5331 F 120 9.5 0 141 1 0 0 0 1 0 0 120
4 5019 M 152 7.5 1 136 1 0 0 0 1 0 1 152 4 5331 F 118 8.9 0 141 1 0 0 0 1 0 0 118
4 5019 F 146 12.7 0 136 1 0 0 0 1 0 0 146 I 4 5331 M 114 7.7 1 141 1 0 0 0 1 0 1 114
4 5019 M 140 14.2 1 136 1 0 0 0 1 0 1 140 4 5331 M 96 11 1 141 1 0 0 0 1 0 1 96
4 5019 M 139 10.4 1 136 1 0 0 0 1 0 1 139 4 5331 F 116 13.2 0 141 1 0 0 0 1 0 0 116
4 5019 F 121 13.6 0 136 1 0 0 0 1 0 0 121 4 5331 M 115 8.7 1 141 1 0 0 0 1 0 1 115
4 5019 F 114 9.1 0 136 1 0 0 0 1 0 0 114 4 5331 F 114 10.4 0 141 1 0 0 0 1 0 0 114
4 5019 F 109 10.4 0 136 1 0 0 0 1 0 0 109 4 5331 M 108 12.2 1 141 1 0 0 0 1 0 1 108
4 5019 F 108 7.8 0 136 1 0 0 0 1 0 0 108 4 5331 F 97 11.2 0 141 1 0 0 0 1 0 0 97
4 5019 M 108 7.9 1 136 1 0 0 0 1 0 1 108 4 5332 M 129 11.8 1 142 1 0 0 0 1 0 1 129
4 5019 M 140 14.5 1 136 1 0 0 0 1 0 1 140 I 4 5332 M 122 10.1 1 142 1 0 0 0 1 0 1 122
4 5019 F 129 9.3 0 136 1 0 0 0 1 0 0 129 ' 4 5332 M 119 9.9 1 142 1 0 0 0 1 0 1 119
4 5019 F 129 9.5 0 136 1 0 0 0 1 0 0 129 4 5332 F 118 8.2 0 142 1 0 0 0 1 0 0 118
4 5020 F 167 16.1 0 137 1 0 0 0 1 0 0 167 4 5332 F 116 11.2 0 142 1 0 0 0 1 0 0 116
4 5020 M 167 12.5 1 137 1 0 0 0 1 0 1 167 4 5332 F 112 8.9 0 142 1 0 0 0 1 0 0 112
4 5020 M 165 12.6 1 137 1 0 0 0 1 0 1 165 4 5332 M 140 11.4 1 142 1 0 0 0 1 0 1 140-- -_.-
-296- -297-
r---------------------~-------------------------------------------------------------------------
Appendix C: Datasetfrom Duchuteau, et al .. 199X: International Livestock Research Institute (JL/U), Kenya Appendix C: Dataset from Duchoteau. et al .. /99X: lnternanonul Ltvestuck Research Instnute (ILIU), Kenya
4 5332 M 135 13.2 1 142 1 0 0 0 1 0 1 135 5 4902 F 132 11.5 0 149 1 0 0 0 0 1 0 132
4 5332 F 134 13.5 0 142 1 0 0 0 1 0 O' 134 5 4902 M 132 11.4 1 149 1 0 0 0 0 1 1 132
4 5332 F 126 10.4 0 142 1 0 0 0 1 0 0 126 5 4902 F 131 10.2 0 149 1 0 0 0 0 1 0 131
4 5332 M 113 8.4 1 142 '1 0 0 0 .t 0 1 113 5 4905 M 123 11.2 1 150 1 0 0 0 0 1 . t 123
4 5333 M 125 11 1 143 1 0 0 0 1 0 1 125 5 4905 F 146 10.9 0 150 1 0 0 0 0 1 0 146
4 5333 F 122 11.6 0 143 1 0 0 0 1 0 0 122 5 4906 F 109 9.7 0 151 1 0 0 0 0 1 0 109
4 5333 M 121 10.7 1 143 1 0 0 0 1 0 1 121 5 4906 M 166 8 1 151 1 0 0 0 0 1 1 166
4 5333 M 119 7.8 1 143 1 0 0 0 1 0 1 119 5 4906 F 164 10.4 0 151 1 0 0 0 0 1 0 164
4 5333 F 110 7.9 0 143 1 0 0 0 1 0 0 110 5 4906 M 137 9 1 151 1 0 0 0 0 1 1 137
4 5333 F 108 10 0 143 1 0 0 0 1 0 0 108 5 4914 F 122 12.1 0 152 1 0 0 0 0 1 0 122
4 5333 F 107 9.9 0 143 1 0 0 0 1 0 0 107 5 4914 M 165 9.5 1 152 1 0 0 0 0 1 1 165
4 5334 M 121 9.1 1 144 1 0 0 0 1 0 1 121 5 4914 F 162 12 0 152 1 0 0 0 0 1 0 162
4 5334 F 121 9.9 0 144 1 0 0 0 1 0 0 121 5 4914 F 127 11.2 0 152 1 0 0 0 0 1 0 127
4 5334 .F 118 11.6 0 144 1 0 0 0 1 0 0 118 5 4914 F 123 10.2 0 152 1 0 0 0 0 1 0 123
4 5334 F 106 13.8 0 144 1 0 0 0 1 0 0 106 5 4914 F 121 7.3 0 152 1 0 0 0 0 1 0 121
4 5334 F 100 10.4 0 144 1 0 0 0 1 0 0 100 5 4914 M 119 10.4 1 152 1 0 0 0 0 1 1 119
4 5336 F '134 12.2 0 145 1 0 O· 0 1 0 0 134 5 4914 M 116 9.6 1 152 1 0 0 0 0 1 1 116
4 5336 F 131 9.4 0 145 1 0 0 0 1 0 0 131 5 4914 M 115 10.5 1 152 1 0 0 0 0 1 1 115
4 5336 F 128 11.6 0 145 1 0 0 0 1 0 0 128 5 4914 F 140 12.3 0 152 1 0 0 0 0 1 0 140
4 5336 F 126 9.7 0 145 1 0 0 0 1 0 0 12.6 5 4914 M 139 13.4 1 152 1 0 0 0 0 1 1 139
4 5336 M 121 11.6 1 145 1 0 0 0 1 0 1 121 5 4914 F 139 10 0 152 1 0 0 0 0 1 0 139
4 5336 M 120 13.9 1 145 1 0 0 0 1 0 1 120 5 4914 F 134 11.6 0 152 1 0 0 0 0 1 0 134
4 5336 M 120 8.6 1 145 1 0 0 0 1 0 1 120 5 4914 F 131 11.1 0 152 1 0 0 0 0 1 0 131
4 5336 M 118 11.4 1 145 1 0 0 0 1 0 1 118 5 4918 F 125 9 0 153 1 0 0 0 0 1 0 125
4 5336 F 139 9.6 0 145 1 0 0 0 1 0 0 139 5 4918 F 119 8.6 0 153 1 0 0 0 0 1 0 119
4 5336 F 129 10.6 0 145 1 0 0 0 1 0 0 129 5 4918 F 118 6.8 0 153 1 0 0 0 0 1 0 118
4 5336 F 127 13.9 0 145 1 0 0 0 1 0 0 127 5 4918 F 115 9.3 0 153 1 0 0 0 0 1 0 115
4 5336 M 118 15.2 1 145 1 0 0 0 1 0 1 118 5 4919 M 164 10 1 154 1 0 0 0 0 1 1 164
4 5336 F 109 11.6 0 145 1 0 0 0 1 0 0 109 5 4919 F 163 10.2 0 154 1 0 0 0 0 1 0 163
5 1975 M 120 11.3 1 146 1 0 0 0 0 1 1 120 5 4919 F 153 9.8 0 154 1 0 0 0 O' 1 0 153
5 1975 M 118 10.1 1 146 1 0 0 0 0 1 1 118 5 4919 M 113 9.4 1 154 1 0 0 0 0 1 1 113
5 1975 M 114 6.6 1 146 1 0 0 0 0 1 1 114 5 4919 M 109 5.2 1 154 1 0 0 0 0 1 1 '09
5 1975 F 113 6.9 0 146 1 0 0 0 0 1 0 113 5 4919 F 101 8.7 0 154 1 0 0 0 0 1 0 101
5 1975 F 99 7.3 0 146 1 0 0 0 0 1 0 99 5 4919 F 129 9.2 0 154 1 0 0 0 0 1 0 129
5 1975 M 97 5.3 1 146 1 0 0 0 0 1 1 97 5 4919 M 126 8.2 1 154 1 0 0 0 0 1 1 126
5 1979 F 119 5.3 0 147 1 0 0 0 0 1 0 119 5 4919 F 123 7.7 0 154 1 0 0 0 0 1 0 123
5 1979 M 117 10.3 1 147 1 0 0 0 0 1 1 117 5 4919 M 119 8.9 1 154 1 0 0 0 0 1 1 119
5 1979 M 114 8.7 1 147 1 0 0 0 0 1 1 114 5 4919 M 118 10.6 1 154 1 0 0 0 0 1 1 118
5 1979 F 111 8.1 0 147 1 0 0 0 0 1 0 111 5 4919 F 117 10.1 0 154 1 0 0 0 0 1 0 117
5 1979 M 102 9.4 1 147 1 0 0 0 0 1 1 102 5 4919 M 114 7.2 1 154 1 0 0 0 0 1 1 114
5 1982 F 118 11.4 0 148 1 0 0 0 0 1 0 118 5 4919 M 134 12 1 154 1 0 0 0 0 1 1 134
5 4902 F 130 9.9 0 149 1 0 0 0 0 1 0 130 5 4919 M 132 12.5 1 154 1 0 0 0 0 1 1 132
5 4902 M 138 12.3 1 149 1 0 0 0 0 1 1 138 5 4919 F 125 9.8 0 154 1 0 0 0 0 1 0 125
5 4902 M 137 13 1 149 1 0 0 0 0 1 1 137 5 4921 M 125 9.3 1 155 1 0 0 0 0 1 1 125
5 4902 M 134 11.7 1 149 1 0 0 0 0 1 1 134 5 4921 M 164 11.8 1 155 1 0 0 0 0 1 1 164
5 4902 M 132 11.7 1 149 1 0 0 0 0 1 1 132 5 4921 M 143 9.4 1 155 1 0 0 0 0 1 1 143-
-298- -299-
Appendix C: Datasetfrom IJuchUIt!l.IU, et al .. 1998: International Livestock Research institute (IL/U), Kenya Appendix C: Dataset from /JUclWIt!CJU. et al., 199N; lnternanonal Livestock Research In.W'tUII! (II.Rl). Kenya
-----------_. - ------_._-
5 4921 F 142 9.3 0 155 1 0 0 0 0 1 0 142 5 5020 M 126 10.9 1 162 1 0 0 0 0 1 1 126
5· 4923 M 160 12.7 1 156 1 0 0 0 0 1 1 160 5 5020 F 116 9.2 0 162 1 0 0 0 0 I 0 116
5 4923 M 136 9.6 1 156 1 0 0 0 0 1 1 136 5 5020 M 116 10.6 1 162 1 0 0 0 0 1 1 116
5 5015 M 161 9 1 157 1 0 0 O. 0 1 1 161 .5 5020 M 116 11.5 1 162 1 0 0 0 0 1. 1 116
5 5015 M 155 13.4 1 157 1 0 0 0 0 1 1 155 5 5204 M 116 12.6 1 163 1 0 0 0 0 1 1 116
5 5015 F 153 12.6 0 157 1 0 0 0 0 1 0 153 5 5204 M 115 9.7 1 163 1 0 0 0 0 1 1 115
5 5015 F 131 11.6 0 157 1 0 0 0 0 1 0 131 5 5204 M 114 11.5 1 163 1 0 0 0 0 1 I 114
5 5015 M 123 6.6 1 157 1 0 0 0 0 1 1 123 5 5204 M 112 9.1 1 163 1 0 0 0 0 1 1 112
5 5015 M 121 9.1 1 157 1 0 0 0 0 1 1 121 5 5204 F lOl 6 0 163 1 0 0 0 0 I 0 lOl
5 5015 F 120 9.4 0 157 1 0 0 0 0 1 0 120 5 5207 M 121 11.4 1 164 I 0 0 0 0 1 1 121
5 5015 F 94 6.6 0 157 1 0 0 0 0 1 0 94 5 5207 M 115 12.3 1 164 1 0 0 0 0 1 1 115
5 5016 F 116 10.6 0 156 1 0 0 0 0 1 0 116 5 5207 M 114 11.9 1 164 1 0 0 0 0 1 1 114
5 5016 F 115 6.5 0 156 1 0 0 0 0 1 0 115 5 5207 F 96 9.6 0 164 1 0 0 0 0 1 0 96
5 5016 M 114 10.1 1 156 1 0 0 0 0 1 1 114 5 5206 M 116 16.4 1 165 1 0 0 0 0 1 1 116
5 5016 F 112 9 0 156 1 0 0 0 0 1 0 112 5 5206 M 112 12 1 165 1 0 0 0 0 1 1 112
5 5017 M 166 11.9 1 159 1 0 0 0 0 1 1 166 5 5206 F 106 10.5 0 165 1 0 0 0 0 1 0 106
5 5017 M 163 11.3 1 159 1 0 O· 0 0 1 1 163 5 5331 F 132 7 0 166 1 0 0 0 0 1 0 132
5 5017 M 119 6.4 1 159 1 0 0 0 0 1 1 119 5 5331 M 115 9.7 1 166 1 0 0 0 0 1. 1 115
5 5017 M 112 10.7 1 159 1 0 0 0 0 1 1 112 5 5331 M 102 6.5 1 166 1 0 0 0 0 1 1 102
5 5017 F 141 12.3 0 159 1 0 0 0 0 1 0 141 5 5331 M 139 11.4 1 166 1 0 0 0 0 1 1 139
5 5017 M 137 9.6 1 159 1 0 0 0 0 1 1 137 5 5331 F 127 6.6 0 166 1 0 0 0 0 1 0 127
5 5017 M 137 10.2 1 159 1 0 0 0 0 1 1 137 5 5331 F 119 10.8 0 166 1 0 0 0 0 1 0 119
5 5017 M 116 10.6 1 159 1 0 0 0 0 1 1 116 5 5331 M 116 9.3 I 166 1 0 0 0 0 I 1 116
5 5016 F 164 10.6 0 160 1 0 0 0 0 1 0 164 5 5331 M III 9.7 1 166 1 0 0 0 0 1 1 III
5 5016 M 136 11.6 1 160 1 0 0 0 0 1 1 136 5 5332 M 118 II 1 167 1 0 0 0 0 1 1 116
5 5016 F 119 10.1 0 160 1 0 0 0 0 1 0 119 5 5332 M 117 11.9 1 167 1 0 0 0 0 1 1 117
5 5016 F 116 6 0 160 1 0 0 0 0 1 0 116 5 5332 F 116 9.4 0 167 1 0 0 0 0 1 0 116
5 5016 F 113 7.4 0 160 1 0 0 0 0 1 0 113 5 5332 M 115 9.1 1 167 1 0 0 0 0 I 1 115
5 5016 M 112 9 1 160 1 0 0 0 0 1 1 112 5 5332 F 133 6.3 0 167 1 0 0 0 0 1 0 133
5 5019 F 152 14.5 0 161 1 0 0 0 0 1 0 152 5 5332 F 133 10.5 0 167 1 0 0 0 0 1 0 133
5 5019 M 119 13.5 1 161 1 0 0 0 0 1 1 119 5 5332 F 130 9.9 0 167 1 0 0 0 0 1 0 130
5 5019 F 99 10.7 0 161 1 0 0 0 0 1 0 99 5 5332 F 127 8.7 0 167 1 0 0 0 0 1 0 127
5 5019 F 135 7.9 0 161 1 0 0 0 0 1 0 135 5 5332 M 126 7.9 1 167 1 0 0 0 0 1 1 126
5 5019 M 135 13.6 I 161 1 0 0 0 0 I I 135 5 5332 F 125 12.1 0 167 1 0 0 0 0 I 0 125
5. 5019 M 133 6.6 I 161 1 0 0 0 0 1 1 133 5 5332 M 116 11.3 1 167 1 0 0 0 0 1 1 116
5 5019 M 131 10.9 1 161 1 0 0 0 0 1 I 131 5 5333 M 134 12.3 1 166 1 0 0 0 0 1 1 134
5 5019 F 123 8.9 0 161 1 0 0 0 0 1 0 123 5 5333 M 125 12.4 1 168 I 0 0 0 0 1 1 125
5 5020 F 166 11.1 0 162 1 0 0 0 0 I 0 166 5 5333 M 124 7.7 1 166 1 0 0 0 0 1 1 124
5 5020 F 165 12.2 0 162 I 0 0 0 0 I 0 165 5 5333 M 119 13.3 1 168 1 0 0 0 0 I 1 119
5 5020 F 159 10 0 162 1 0 0 0 0 1 0 159 5 5334 M 133 10.9 1 169 1 0 0 0 0 1 1 133
5 5020 M 116 12.8 I 162 1 0 0 0 0 1 I 118 5 5334 F 128 9.9 0 169 1 0 0 0 0 1 0 128
5 5020 F 106 7.9 0 162 1 0 0 0 0 1 0 106 5 5334 F 122 11.2 0 169 1 0 0 0 0 I 0 122
5 5020 M 65 6.7 I 162 I 0 0 0 0 I 1 65 5 5334 F 114 10.6 0 169 1 0 0 0 0 1 0 114
5 5020 F 138 9.7 0 162 I 0 0 0 0 1 0 136 5 5334 F 114 7.4 0 169 1 0 0 0 0 1 0 114
5 5020 M 135 9.3 1 162 1 0 0 0 0 I 1 135 5 5334 M 107 11.7 1 169 I 0 0 0 0 I 1 107
5 5020 F 135 6.1 0 162 1 0 0 0 0 I 0 135 5 5334 F 96 6.6 0 169 1 0 0 0 0 1 0 96
-300- -301-
.~--~
Appendix C: Dutuselfrom Duchoteau, et al., 1998: lnternatlonal Livestock Research Institute (ILIU), Kenya Appendix C: Dataset from Duchuteuu. et ol., 1998: International Livestock Research institute (ILRl), Kenya
5 5336 M 135 9.6 1 170 1 0 0 0 0 1 1 135 6 5012 M 162 16.2 1 186 1 0 0 0 0 0 1 162
5 5336 M 119 11.9 1 170 1 0 0 0 0 i : 1 119 6 5013 F 148 12.9 0 187 1 0 0 0 0 0 0 148
5 5336 M 116 11.5 1 170 1 0 0 0 0 1 1 116 6 5071 M 104 9.6 1 188 1 0 0 0 0 0 1 104
5 5336 F 103 7 0 170 1 0 0 .0 0 1 0 103 . .6 5073 M 113 13.8 1 189 1 Ó 0 0 0 .0 1 113
5 5336 F 141 12.2 0 170 1 0 0 0 0 1 0 141 6 5076 M 112 11.1 1 190 1 0 0 0 0 0 1 112
5 5336 F 137 7.9 0 170 1 0 0 0 0 1 0 137 6 5205 M 120 12.2 1 191 1 0 0 0 0 0 1 120
5 5336 M 134 11.3 1 170 1 0 0 0 0 1 1 134 6 5205 F 110 6.2 0 191 1 0 0 0 0 0 0 110
5 5336 M 114 12.7 1 170 1 0 0 0 0 1 1 114 6 5206 M 119 14.2 1 192 1 0 0 0 0 0 1 119
5 5336 F 98 10.4 0 170 1 0 0 0 0 1 0 98 6 5206 F 117 11.2 0 192 1 0 0 0 0 0 0 117
6 1972 M 101 4.7 1 171 1 0 0 0 0 0 1 101 6 5322 F 115 11.5 0 193 1 0 0 0 0 0 0 115
6 1991 F 121 8.9 0 172 1 0 0 0 0 0 0 121 6 5322 F 98 8.7 0 193 1 0 0 0 0 0 0 98
6 4908 F 166 11.5 0 173 1 0 0 0 0 0 0 166 6 5324 M 130 11.8 1 194 1 0 0 0 0 0 1 130
6 4908 F 156 11 0 173 1 0 0 0 0 0 0 156 6 5324 F 128 7.9 0 194 1 0 0 0 0 0 0 128
6 4910 F 125 10.2 0 174 1 0 0 0 0 0 0 125 6 5324 M 111 13 1 194 1 0 0 0 0 0 1 111
6 4910 M 98 10.1 1 174 1 0 0 0 0 0 1 98 6 5324 F 106 10.3 0 194 1 0 0 0 0 0 0 106
6 4915 M 110 10.1 1 175 1 0 0 0 0 0 1 110 6 5326 M 100 9.8 1 195 1 0 0 0 0 0 1 100
6 4915 M 104 10 1 175 1 0 0 0 0 0 1 104 6 5326 F 139 11.5 0 195 1 0 0 0 0 0 0 139
6 5001 F 169 11.2 0 176 1 0 0 0 0 0 0 169 6 5326 F 131 9.9 0 195 1 0 0 0 0 O. 0 131
6 5002 F 170 11.5 0 177 1 0 0 0 0 0 0 170 6 5328 M 130 14.8 1 196 1 0 0 0 0 0 1 130
6 5002 F 165 10.3 0 177 1 0 0 0 0 0 0 165 6 5328 F 140 11.2 0 196 1 0 0 0 0 0 0 140
6 5003 M 119 7.5 1 178 1 0 0 0 0 0 1 119 6 5328 F 130 11.3 0 196 1 0 0 0 0 0 0 130
6 5003 F 158 9.3 0 178 1 0 0 0 0 0 0 158 6 5328 F 127 7.3 0 196 1 0 0 0 0 0 0 127
6 5004 F 102 7.2 0 179 1 0 0 0 0 0 0 102 6 5329 M 123 9.9 1 197 1 0 0 0 0 0 1 123
6 5005 M 154 9.7 1 180 1 0 0 0 0 0 1 154 6 5329 M 118 7.5 1 197 1 0 0 0 0 0 1 118
6 5007 F 168 12.1 0 181 1 0 0 0 0 0 0 168 6 5329 F 146 14.4 0 197 1 0 0 0 0 0 0 146
6 5007 M 167 11.6 1 181 1 0 0 0 0 0 1 167 6 5329 F 133 7 0 197 1 0 0 0 0 0 0 133
6 5007 F 161 11.3 0 181 1 0 0 0 0 0 0 161 6 5329 F 130 9.7 0 197 1 0 0 0 0 0 0 130
6 5007 M 149 10.4 1 181 1 0 0 0 0 0 1 149 6 5329 F 100 11.5 0 197 1 0 0 0 0 0 0 100
6 5007 M 120 13.9 1 181 1 0 0 0 0 0 1 120 6 5330 M 123 10.9 1 198 1 0 0 0 0 0 1 123
6 5008 M 172 12.7 1 182 1 0 0 0 0 0 1 172 6 5330 F 137 10.1 0 198 1 0 0 0 0 0 0 137
6 5008 F 142 10.8 0 182 1 0 0 0 0 0 0 142 6 5330 F 134 11.8 0 198 1 0 0 0 0 0 0 134
6 5008 M 112 11.8 1 182 1 0 0 0 0 0 1 112 6 5337 F 129 11.3 0 199 1 0 0 0 0 0 0 129
6 5008 M 108 8.5 1 182 1 0 0 0 0 0 1 108 6 5337 M 125 13.5 1 199 1 0 0 0 0 0 1 125
6 5008 F 96 5.6 0 182 1 0 0 0 0 0 0 96 6 5337 M 119 9.7 1 199 1 0 0 0 0 0 1 119
6 5009 F 174 11.8 0 183 1 0 0 0 0 0 0 174 I 6 5337 M 116 9.6 1 199 1 0 0 0 0 0 1 116
6 5009 M 167 12 1 183 1 0 0 0 0 0 1 167 6 5337 F 103 8.3 0 199 1 0 0 0 0 0 0 103
6 5009 F 158 10.8 0 183 1 0 0 0 0 0 0 158 6 5338 F 127 11.4 0 200 1 0 0 0 0 0 0 127
6 5010 F 165 10.8 0 184 1 0 0 0 0 0 0 165 6 5338 M 123 10 1 200 1 0 0 0 0 0 1 123
6 5010 M 164 14.4 1 184 1 0 0 0 0 0 1 164 6 5338 F 137 12.9 0 200 1 0 0 0 0 0 0 137
6 5010 M 146 12.2 1 184 1 0 0 0 0 0 1 146
6 5011 F 164 12.6 0 185 1 0 0 0 0 0 0 164
6 5011 F 159 11.6 0 185 1 0 0 0 0 0 0 159
6 5011 M 157 11.1 1 185 1 0 0 0 0 0 1 157
6 5011 M 121 12.3 1 185 1 0 0 0 0 0 1 121
6 5011 F 107 8.1 0 185 1 0 0 0 0 0 0 107
6 5012 M 166 15.2 1 186 1 0 0 0 0 0 1 166
-302- -303-
Appendix D: Datasetfrom Duchateau, et al., 1998; International Livestock Research Institute (ILRI), Kenya
APPENDIXD
Two drugs against trypanosomosis (parasitic disease transmitted by tsetse flies), Berenil and,
Samorin, are studied in a situation where there is widespread evidence of high levels of drug
resistance. Herds of N'Dama breeds are randomly assigned to Berenil and herds of Boran breeds to
Samorin treatment. The aim of the study was then to see whether there are differences in the change
in pev between the two breeds following a trypanosome infection.
pev is a variable often measured to evaluate the severity of the diseases is packed cell volume
(peV), which is the percentage of the volume of the blood serum taken up by the red blood cells.
Low pev corresponds to anaemia and can indicate infection with the disease. That determined at the
time of treatment was designated pevo, that a month later following treatment was designated
peV35 (with 14 measurements in between), Animals belonging to a herd to which Berenil has been
assigned, however, are randomly assigned to receive a high or a low dose of Berenil when detected
parasitaemicwith trypanosomes.
-304-
Appendix D: Datasetfrom Duchateau, et al., 1998; International Livestock Research Institute (ILRI), Kenya
Animal
Breed Days 1 2 3 4 5 6
Boran 0 36.2 35.9 29.5 28.5 30.4 337
2 35.9 38.5 33.3 27.6 29.5 36.2
4 35.3 35.9 29.2 27.9 28.8 33.3
7 35.4 36 29.9 27.7 28.7 32.2
9 35.4 36.3 29 29.3 28.7 30.9
14 31.5 36.3 29.9 26.7 27.1 29.6
17 25.5 25.2 21.3 21.3 20.7 22.6
18 34.4 31.5 27.4 26.7 25.2 30.3
21 34.1 30.6 25.5 25.2 22.9 28.3
23 25.8 28.7 .25.5 23.6 23.2 25.8
25 28.7 29 24.8 23.6 22.9 24.5
29 21.6 23.9 23.6 20 20.7 21.6
31 21.3 21.3 22.6 19.4 19.1 17.5
35 17.8 18.1 20.4 17.2 18.5 15.9
N'Dama 0 30.4 37.5 32.4 34.3 30.4 40.4
2 33 37.8 30.4 33 32.1 37.5
4 33.3 36.5 31.7 37.5 32.1 38.8
7 31.9 35.7 31.2 36.3 30.9 37
9 30.6 35.7 31.5 34.1 30.6 38.9
14 31.2 33.8 27.7 30.6 . 29.6 31.9
17 27.7 33.4 27.1 27.7 23.6 27.1
18 28 31.5 27.4 29.9 29 31.5
21 28.3 32.5 29 28 29.6 31.2
23 27.7 34.7 28 27.1 28.7 33.1
25 25.8 30.6 27.4 26.7 27.1 35.4
29 26.1 31.5 28.3 28.7 25.8 30.6
31 24.5 25.2 26.1 23.9 26.1 28,7
35 22.6 28.7 22.9 22.6 24.2 26.1
% pev measured at 0,2,4, ..,35 days following infection
-305-
Appendix E: Estimated Breeding Values for the Data Setfrom Duchateau, et al., 1998;
APPENDIXE
Estimated breeding values for the complete data set used in Chapter 3.
DPP = Dirichlet Process when M is simulated given the data
REML Trad. Bayes DPP
SIRE ID BREED Estimate SE Estimate SimM
1971 I -0.1061 0.6654 -0.1024 -0.1097
1972 1 0.5241 0.5349 0.5689 0.5081
1973 1 0.3888 0.6399 0.4464 0.3934
1974 1 1.9339 0.5486 1.9627 1.9026
t 1980 1 0.9299 0.5975 0.9635 0.9193
r 1991 1 0.3611 0.6396 0.3741 0.3438
I 1999 I 0.9266 0.6654 0.9939 0.8936
4907 1 0.2289 0.5790 0.2676 0.2293
4908 1 1.6614 0.4921 1.6662 1.649
4909 1 -0.7628 0.6653 -0.7645 -0.7711
4910 1 0.4627 0.5123 0.4396 0.4511
4911 1 0.6681 0.5795 0.6867 0.653
4912 1 -0.3258 0.6655 -0.3581 -0.3125
4915 1 0.2351 0.6655 0.1955 0.224
4916 I -0.1544 0.6658 -0.1383 -0.1681
5001 I -0.3829 0.7309 -0.3286 -0.3592
5002 I -0.4267 0.6398 -0.4600 -0.4385
5003 1 0.2938 0.7307 0.2753 0.2474
5004 I 0.6637 0.6401 0.6546 0.6468
5005 1 0.1155 0.6397 0.1555 0.0971
5007 I -0.7786 0.6396 -0.7543 -0.7531
5008 I -0.4999 0.6655 -0.5137 -0.5016
5009 1 -0.0700 0.7309 -0.0531 -0.0297
5010 1 0.1004 0.6956 0.1027 0.0818
5011 1 -0.6332 0.6956 -0.6175 -0.5763
5012 1 0.1259 0.7311 0.1063 0.1
5013 1 -0.3030 0.6961 -0.2939 -0.3329
5071 I 0.3608 0.6655 0.3945 0.3812
5073 I -0.2421 0.7732 -0.2776 -0.2338
5076 1 -0.1386 0.7730 -0.1264 -0.1087
5205 1 -0.4303 0.7731 -0.4454 -0.3821
-306-
Appendix E: Estimated Breeding Values for the Data Set from Duchateau, et al., 1998;
5324 1 -0.7496 0.7730 -0.7590 -0.7039
5326 1 :0.5001 0.7308 -0.5171 -0.4887
5328 1 -0.2392 0.6393 -0.2737 -0.232
5329 1 -0.5741 0.6167 -0.6062 -0.5698
5330 1 -1.4541 0.6393 -1.4747 -1.3737
5337 1 -0.5825 0.6654 -0.5811 -0.5837
5338 1 -0.6268 0.6954 -0.6131 -0.5882
1975 2 -0.1147 0.6173 -0.0893 -0.0912
1976 2 0.4043 0.6399 0.4477 0.3964
1979 2 0.8504 0.5495 0.8168 0.8382
1981 2 0.1681 0.5799 0.1769 0.1699
1982 2 -0.2903 0.6663 -0.2016 -0.3346
1983 2 1.4790 0.6404 1:5350 1.4576
1984 2 0.3396 0.6175 0.4279 0.3094
1986 2 0.1270 0.7310 0.1526 0.142
1988 2 0.9403 0.6658 0.9514 0.98
4901 2 0.6692 0.5497 0.6900 0.6585
4902 2 0.3625 0.5502 0.3446 0.3446
4903 2 0.1536 0.5249 0.1349 0.1255
4905 2 -0.0434 0.4942 -0.0279 -0.0769
r 4906 2 0.6141 0.5035 0.6523 0.57174913 2 0.1971 0.6664 0.2227 0.2353
4914 2 -1.0541 0.5800 -1.1073 -1.0039
4918 2 0.7037 0.6962 0.6847 0.6331
4919 2 -0.9475 0.6666 -0.9576 -0.9186
4921 2 -0.6580 0.5638 -0.6768 -0.6184
4923 2 -0.1434 0.5978 -0.1706 -0.1427
5015 2 0.0903 0.6404 0.0550 0.0552
5016 2 -0.8617 0.6177 -0.8988 -0.8042
5017 2 -1.1521 0.6176 -1.1860 -1.1127
5018 2 -0.2094 0.7308 -0.2098 -0.2311
5019 2 -0.4845 0.5975 -0.4910 -0.458
5020 2 -0.1989 0.6403 . -0.2144 -0.1768
5204 .2 -0.2315 0.7732 -0.2585 -0.2016
5207 2 -0.1528 0.7731 -0.1341 -0.1835
5331 2 -0.0178 0.7731 -0.0770 -0.0006
5334 2 -0.2860 0.7731 -0.2829 -0.2741
5336 2 -0.2528 0.7309 -0.2626 -0.2231
1971 3 -0.5916 0.5597 -0.6159 -0.5838
1972 3 1.0193 0.5062 1.0300 1.0054
1973 3 0.8188 0.6145 0.8021 0.8213
1974 3 -0.1343 0.6642 -0.1161 -0.1457
1980 3 0.2847 0.6149 0.2463 0.314
1991 3 0.4573 0.6380 0.4638 0.4214
1999 3 0.3507 0.6639 0.3162 0.3172
4907 3 0.1848 0.5438 0.1809 0.1652
-307-
Appendix E: Estimated Breeding Values for the Data Setfrom Duchateau, et al., /998;
4908 3 0.6656 0.4767 0.6633 0.6427
4909 3 0.6248 0.5598 0.6198 0.6078
4910 3 -0.1241 0.5179 -0.1408 -0.1418
4911 3 0.9102 0.6381 0.9922 0.8869
4912 3 0.7009 0.5306 0.6865 0.6699
4915 3 -0.5662 0.6152 -0.6052 -0.5554
4916 3 0.0111 0.5315 0.0232 0.007
5001 3 0.0392 0.6388 0.0851 0.0638
5002 3 0.20 Il 0.5439 0.2123 0.1804
5003 3 0.7758 0.5437 0.7462 0.7322
5004 3 0.0456 0.6154 0.0304 0.0351
5005 3 0.6248 0.5302 0.6826 0.5889
5006 3 -0.0257 0.7729 -0.0039 -0.0546
5007 3 -0.7342 0.5592 -0.7001 -0.7081
5008 3 -0.2444 0.5753 -0.2629 -0.2602
5009 3 -0.4822 0.5753 -0.4978 -0.4396
5010 3 0.1952 0.6951 0.2524 0.1997
5011 3 -0.5054 0.5179 -0.5245 -0.5325
5012 3 0.4238 0.6952 0.3924 0.4272
5013 3 -0.8623 0.6384 -0.8832 -0.8186
5071 3 -0.0328 0.5459 -0.0596 -0.0237
5073 3 -0.0160 0.7729 -0.0171 -0.0068
5076 3 0.1324 0.7730 0.1270 0.1257
5205 3 -0.4053 0.6643 -0.3947 -0.3858
5206 3 0.0652 0.5944 0.0957 0.0482
5321 3 0.1343 0.7302 0.1409 0.1009
5322 3 -0.8229 0.6147 -0.8742 -0.7943
5324 3 0.2162 0.7306 0.2634 0.2084
5326 3 -0.4500 0.6947 -0.5162 -0.4068
5328 3 -0.5569 0.5941 -0.5849 -0.5516
5329 3 -0.8199 0.5446 -0.8603 -0.7923
5330 3 -1.1606 0.5587 -1.2002 -1.0854
5337 3 0.1048 0.6377 0.1210 0.1019
5338 3 -0.4520 0.5177 -0.4651 -0.4289
1975 4 0.7474 0.5061 0.7256 0.7081
1976 4 0.8951 0.6149 0.9307 0.8653
1979 4 -0.3871 0.5064 -0.3926 -0.3982
1981 4 0.4741 0.6146 0.4465 0.4463
1982 4 -0.0710 0.7729 -0.0575 -0.0396
1983 4 0.1010 0.5937 0.1088 0:1 157
1984 4 0.8469 0.5756 0.8823 0.805
1986 4 0.4516 0.6376 0.4461 0.4418
1988 4 -0.3328 0.5592 -0.3485 -0.3107
4901 4 0.7953 0.4953 0.8141 0.7761
4902 4 -0.1491 0.4385 -0.1395 -0.1696
4903 4 1.1514 0.5300 1.2001 1.1224
-308-
Appendix E: Estimated Breeding Values for the Data Set from Duchateau, et al., 1998;
4905 4 -0.1528 0.4615 -0.1144 -0.1808
4906 4 0.2633 0.4856 0.2961 0.2459
4913 4 0.2709 0.5765 0.2676 0.2398
4914 4 0.4437 0.4323 0.4366 0.4123
4918 4 0.1952 0.5199 0.2266 0.1741
4919 4 -0.9511 0.4390 -0.9531 -0.8958
4921 4 -0.6402 0.4856 -0.6476 -0.6073
4923 4 0.1171 0.4958 0.1508 0.0728
5015 4 -0.2059 0.5308 -0.1867 -0.2039
5016 4 -0.6323 0.5587 -0.5925 -0.6073
5017 4 -0.1042 0.4955 -0.1211 -0.1358
5018 4 0.0329 0.5304 0.0047 0.035
5019 4 -0.4168 0.4857 -0.3861 -0.4184
5020 4 -0.3328 0.4600 -0.3295 -0.3613
5204 4 -0.5484 0.6380 -0.5679 -0.5169
5207 4 -0.4170 0.6644 -0.3610 -0.3791
5208 4 -0.4533 0.6645 -0.4677 -0.4507
5331 4 -0.2136 0.5312 -0.2106 -0.2016
5332 4 -0.4640 0.5302 -0.4453 -0.4668
5333 4 -0.5464 0.5941 -0.5769 -0.5264
5334 4 0.1250 0.6381 0.0967 0.0945
r 5336 4 0.1079 0.5063 0.1247 0.0842
I 1975 5 -0.7175 0.6191 -0.7509 -0.7053
1979 5 -0.4920 0.6411 -0.4991 -0.5028
1982 5 0.2473 0.7732 0.2446 0.2412
4902 5 0.4902 0.5818 0.5047 0.4392
4905 5 0.1193 0.7311 0.1769 0.1122
4906 5 -0.5996 0.6668 -0.6063 -0.5691
4914 5 0.3284 0.5074 0.3184 0.3123
4918 5 -0.3650 0.6670 -0.3851 -0.3673
4919 5 -0.6073 0.4894 -0.6402 -0.5904
4921 5 -0.4122 0.6669 -0.4361 -0.4205
4923 5 -0.0513 0.7314 -0.0619 -0.0329
5015 5 -0.0328 0.5817 -0.0313 -0.019
5016 5 0.0768 0.6667 0.0967 0.0806
5017 5 -0.1279 0.5823 -0.1145 -0.1349
5018 5 -0.4122 0.6187 -0.4317 -0.3841
5019 5 0.4651 0.5816 0.4822 0.4183
5020 5 -0.0474 0.5169 -0.0868 -0.0788
5204 5 0.0700 0.6413 0.0693 0.0713
5207 5 0.6159 0.6667 0.6394 0.5962
5208 5 1.0117 0.6963 1.0674 1.0026
5331 5 -0.4083 0.5817 -0.3608 -0.3985
5332 5 -0.0742 0.5391 -0.1131 -0.0521
5333 5 0.3543 0.6668 0.3541 0.3415
5334 5 0.2657 0.5994 0.2868 0.2627
-309-
Appendix E: Estimated Breeding Values for the Data Setfrom Duchateau, et al., 1998;
5336 5 0.3029 0.5660 0.3 118 0.2618
1972 6 70.6139 0.7735 -0.6616 -0.6009
1991 6 -0.1264 0.7734 -0.1488 -0.119
4908 6 -0.1184 0.7325 -0.0705 -0.0998
4910 6 0.0678 0.7322 0.0918 0.0199
4915 6 0.0147 0.7325 -0.0133 -0.0129
5001 6 -0.1183 0.7736 -0.1402 -0.0864
5002 6 -0.2610 0.7326 -0.2281 -0.2262
5003 6 -0.5894 0.7321 -0.6160 -0.5468
5004 6 -0.2265 0.7735 -0.2923 -0.2258
5005 6 -0.3030 0.7735 -0.3106 -0.2561
5007 6 0.0075 0.6456 0.0763 0.0034
5008 6 -0.2902 0.6450 -0.3131 -0.3151
5009 6 -0.2190 0.6988 ~0.1699 -0.234
5010 6 0.0975 0.6985 0.1176 0.0747
5011 6 -0.0125 0.6451 -0.0613 -0.0067
5012 6 0.6699 0.7326 0.7915 0.6654
5013 6 0.2102 0.7735 0.1985 0.2045
5071 6 -0.0298 0.7735 -0.0459 -0.0133
5073 6 0.4341 0.7735 0.5039 0.4147
5076 6 0.1085 0.7735 0.0970 0.1196
5205 6 -0.1755 0.7322 -0.1493 -0.1477
5206 6 0.5590 0.7322 0.6348 0.5343
5322 6 0.1846 0.7325 0.1772 0.1723
5324 6 0.2054 0.6696 0.2060 0.2179
5326 6 0.0374 0.6981 0.0459 0.0101
5328 6 0.1950 0.6695 0.2470 0.1955
5329 6 -0.1601 0.6239 -0.1136 -0.1856
5330 6 0.0850 0.6980 0.0255 0.0713
5337 6 0.1022 0.6452 0.0809 0.104
5338 6 0.2649 0.6981 0.2788 0.2485
-310-
Appendix F' Milk production records, kg (305 days) of full-sib daughters
APPENDIXF
The data set consists of 44 age-adjusted milk production record (305 days) obtained in the same
year and herd from cows whose sires and dams were considered randomly representative of a large
population. The records were taken from full-sib daughters.
Sires Dams Production Records
4379 6560
2 5560 7733 7198
3 4637 5639 8072
4 5726 5576
5 4968 4574
2 6 5355 7057 7052
7 4605 4180
8 4393 4530
3 9 5195
10 6137 4748 7351
11 6253
12 5553 6026 6666
4 13 6268 7575 7024
14 7112
15 5840 7316 6382
16 6246 5595
17 5400 6440
18 7301 6615
19 5453
20 7374 6693 6592
-311-