Hierarchical Bayesian modelling for the analysis of the lactation of dairy animals

Abstract

English: This thesis was written with the aim of modelling the lactation process in dairy cows and goats by applying a hierarchical Bayesian approach. Information on cofactors that could possibly affect lactation is included in the model through a novel approach using covariates. Posterior distributions of quantities of interest are obtained by means of the Markov chain Monte Carlo methods. Prediction of future lactation cycle(s) is also performed. In chapter one lactation is defined, its characteristics considered, the factors that could possibly influence lactation mentioned, and the reasons for modelling lactation explained. Chapter two provides a historical perspective to lactation models, considers typical lactation curve shapes and curves fitted to the lactation composition traits fat and protein of milk. Attention is also paid to persistency of lactation. Chapter three considers alternative methods of obtaining total yield and producing Standard Lactation Curves (SLAC’s). Attention is paid to methods used in fitting lactation curves and the assumptions about the errors. In chapter four the generalised Bayesian model approach used to simultaneous ly model more than one lactation trait, while also incorporating information on cofactors that could possibly influence lactation, is developed. Special attention is paid not only to the model for complete data, but also how modelling is adjusted to make provision for cases where not all lactation cycles have been observed for all animals, also referred to as incomplete data. The use of the Gibbs sampler and the Metropolis-Hastings algorithm in determining marginal posterior distributions of model parameters and quantities that are functions of such parameters are also discussed. Prediction of future lactation cycles using the model is also considered. In chapter five the Bayesian approach together with the Wood model, applied to 4564 lactation cycles of 1141 Jersey cows, is used to illustrate the approach to modelling and prediction of milk yield, percentage of fat and percentage of protein in milk composition in the case of complete data. The incorporation of cofactor information through the use of the covariate matrix is also considered in greater detail. The results from the Gibbs sampler are evaluated and convergence there-of investigated. Attention is also paid to the expected lactation curve characteristics as defined by Wood, as well as obtaining the expected lactation 254 curve of one of the levels of a cofactor when the influence of the other cofactors on the lactation curve has be eliminated. Chapter six considers the use of the Bayesian approach together with the general exponential and 4-parameter Morant model, as well as an adaptation of a model suggested by Wilmink, in modelling and predicting milk yield, fat content and protein content of milk for the Jersey data. In chapter seven a diagnostic comparison by means of Bayes factors of the results from the four models in the preceding two chapters, when used together with the Bayesian approach, is performed. As a result the adapted form of the Wilmink model fared best of the models considered! Chapter eight illustrates the use of the Bayesian approach, together with the four lactation models considered in this study, to predict the lactation traits for animals similar to, but not contained in the data used to develop the respective models. In chapter nine the Bayesian approach together with the Wood model, applied to 755 lactation cycles of 493 Saanen does collected during either or both of two consecutive year, is used to illustrate the approach to modelling and predicting milk yield, percentage of fat and percentage of protein in milk in the case of incomplete data. Chapter ten provides a summary of the results and a perspective of the contribution of this research to lactation modelling.

Afrikaans: Hierdie tesis is geskryf met die doel om die laktasieproses in suiwelkoeie en suiwelbokke te modelleer deur ‘n hierargiese Bayesbenadering toe te pas. Inligting aangaande kofaktore wat moontlik laktasie kan beïnvloed, is in die model ingesluit deur middel van ‘n unieke benadering wat van koveranderlikes gebruik maak. Posteriorverdelings van hoeveelhede van belang word deur middel van die Mar kovketting Monte Carlo metodes verkry. Voorspelling van toekomstige laktasiesiklus(se) is ook uitgevoer. In hoofstuk een word laktasie gedefinieer, die eienskappe daarvan beskou, die faktore wat moonlik laktasie mag beïnvloed genoem, en die redes vir die modellering van laktasie verduidelik. Hoofstuk twee lewer ‘n historiese perspektief tot laktasiemodelle, beskou tipiese laktasiekurwe vorms, asook kurwes gepas aan die laktasiesamestellingskenmerke vet en proteïen van melk. Aandag word ook aan die volhoubaarheid van laktasie geskenk. Hoofstuk drie beskou alternatiewe metodes om totale opbrengs te verkry en Standaard Laktasiekurwes (SLAC’s) voort te bring. Aandag word geskenk aan metodes wat gebruik word in die passing van laktasiekurwes en die aannames aa ngaande die foute. In hoofstuk vier word die veralgemeende Bayesmodelbenadering ontwikkel om meer as een laktasiekenmerk gelyktydig te modelleer, terwyl inligting aangaande kofaktore wat moonlik laktasie kan beïnvloed ook ingesluit word. Spesiale aandag word nie net aan die model vir volledige data geskenk nie, maar ook aan hoe modellering aangepas moet word om voorsiening te maak vir gevalle waar nie al die laktasiesiklusse vir alle diere waargeneem is nie, wat ook na verwys word as onvolledige data. Die gebruik van Gibbssteekproefneming en die Metropolis-Hastings algoritme in die bepaling van posterior randverdelings van die model parameters en hoeveelhede wat funksies van sulke parameters is, word ook bespreek. Voorspelling van toekomstige laktasiesiklusse deur die model te gebruik word ook beskou. In hoofstuk vyf word die Bayesbenadering saam met die Woodmodel, toegepas op 4564 laktasiesiklusse van 1141 Jerseykoeie, ter illustrasie van die benadering tot modellering en voorspelling van melkopbrengs, pe rsentasie vet en persentasie proteïen in die samestelling van melk in die geval van volledige data gebruik. Die insluiting van kofaktorinligting deur die gebruik van die matriks van koveranderlikes word ook in meer besonderhede beskou. Die resultaat vanaf Gibbssteekproefneming word evalueer en die konvergensie daarvan ondersoek. Aandag word ook geskenk aan die verwagte laktasiekurwe eienskappe soos gedefinieer deur Wood, asook die bepaling van verwagte laktasiekurwes vir een van die vlakke van ‘n kofaktor indien die invloed van die ander kofaktore op die laktasiekurwe uitgeskakel word. Hoofstuk ses beskou die gebruik van die Bayesbenadering saam met die veralgemeende eksponensiaal en 4-parameter Morant model, asook ‘n aanpassing van ‘n model wat deur Wilmink voorgestel is, in die modellering en voorspelling van melkopbrengs, asook die samestellingskenmerke vet en proteïen in melk vir die Jerseydata. In hoofstuk sewe word ‘n diagnostiese vergelyking deur middel van Bayesfaktore uitgevoer op die resultate va n die vier modelle in die voorafgaande twee hoofstukke wanneer dit saam met die Bayesbenadering gebruik word. As resultaat hiervan het die aangepaste vorm van die Wilmink model die beste van die modelle wat oorweeg is, gevaar! Hoofstuk agt illustreer die gebruik van die Bayesbenadering, saam met die vier laktasiemodelle onder beskouing in hierdie studie, om die laktasiekenmerke te voorspel van diere soortgelyk aan, maar nie ingesluit in die data wat gebruik is in die ontwikkelling van die onderskeie modelle nie. In hoofstuk nege word die Bayesbenadering saam met die Wood model toegepas op 755 laktasiesiklusse van 493 Saanenooie om die benadering tot modellering en voorspelling van melkopbrengs, persentasie vet en persentasie proteïene in die samestelling van melk in die geval van onvolledige data te illustreer. Hoofstuk tien lewer ‘n opsomming van die resultate en ‘n perspektief van die bydrae van hierdie navorsing tot laktasiemodellering.