Doctoral Degrees (Mathematical Statistics and Actuarial Science)

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Browsing Doctoral Degrees (Mathematical Statistics and Actuarial Science) by Subject "Bayes"

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    A Bayesian analysis of multiple interval-censored failure time events with application to AIDS data
    (University of the Free State, 2003-05) Mokgatlhe, Lucky; Groenewald, C. N.; De Waal, Daniel J.
    English: The measure of time to event (failure) for units on longitudinal clinical visits cannot always be ascertained exactly. Instead only time intervals within which the event occurred may be recorded. That being the case, each unit's failure will be described by a single interval resulting in grouped interval data over the sample. Yet, due to non-compliance to visits by some units, failure will be described by endpoints within which the event has occurred. These endpoints may encompass several intervals, hence overlapping intervals across units. Furthermore, some units may not realize the event of interest within the preset duration of study, hence are censored. Finally, several events of interest can be investigated on a single unit resulting in several failure times that inevitably are dependent. All these prescribe an interval-censored survival data with multiple-failure times. Three models of analysing interval-censored survival data with two failure times were applied to four sets of data. For the distribution free methods, Cox's hazard with either a log-log transform or logit transform on the baseline conditional survival probabilities was used to derive the likelihood. The Independence assumption model (lW) work under the assumption that the lifetimes are independent and any dependence exists through the use of common covariates. The second model that do not necessarily assume independence, computes the joint failure probabilities for two lifetimes by Bayes' rule of conditioning on the interval of failure for one lifetime, hence Conditional Bivariate model (CB). The use of Clayton and Farley-Morgenstern bivariate Copulas (CC) with inbuilt dependence parameter was the other model. For parametric models the IW and CC methods were applied to the data sets on the assumption that the marginal distribution of the lifetimes is Weibull. The traditional classical estimation method of Newton-Raphson was used to find optimum parameter estimates and their variances stabilized using a sandwich estimator, where possible. Bayesian methods combine the data with prior information. Thus for either transforms, two proper priors were derived, of which their combination with the likelihood resulted in a posterior function. To estimate the entire distribution of a parameter from non-standard posterior functions, two Markov Chain Monte Carlo (MCMC) methods were used. The Gibbs Sampler method samples in turn observations from the conditional distribution of a parameter in question, while holding other parameters constant. For intractably complex posterior functions, the Metropolis-Hastings method of sampling vectors of parameter values in blocks from a Multivariate Normal proposal density was used. The analysis of ACTG175data revealed that increase in levels of HIV RNA precede decline in CD4 cell counts. There is a strong dependence between the two failure times, hence restricting the use of the independence model. The most preferred models are using copulas and the conditional bivariate model. It was shown that ARV's actually improves a patient's lifetime at varying rates, with combination treatment performing better. The worrying issue is the resistance that HIV virus develops against the drugs. This is evidenced by the adverse effect the previous use of ARV's has on patients, in that a new drug used on them has less effect. Finally it is important that patients start therapy at early stages since patients displaying signs of AIDS at entry respond negatively to drugs.
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    ItemOpen Access
    Bayesian control charts based on predictive distributions
    (University of the Free State, 2016-01) Van Zyl, Ruaan; Van der Merwe, A. J.
    English: Control charts are statistical process control (SPC) tools that are widely used in the monitoring of processes, specifically taking into account stability and dispersion. Control charts signal when a significant change in the process being studied is observed. This signal can then be investigated to identify issues and to find solutions. It is generally accepted that SPC are implemented in two phases, Phase I and Phase II. In Phase I the primary interest is assessing process stability, often trying to bring the process in control by locating and eliminating any assignable causes, estimating any unknown parameters and setting up the control charts. After that the process move on to Phase II where the control limits obtained in Phase I are used for online process monitoring based on new samples of data. This thesis concentrate mainly on implementing a Bayesian approach to monitoring processes using SPC. This is done by providing an overview of some non-informative priors and then to specifically derive the reference and probability-matching priors for the common coefficient of variation, standardized mean and tolerance limits for a normal population. Using the Bayesian approach described in this thesis SPC is performed, including derivations of control limits in Phase I and monitoring by the use of runlengths and average run-lengths in Phase II for the common coefficient of variation, standardized mean, variance and generalized variance, tolerance limits for normal populations, two-parameter exponential distribution, piecewise exponential model and capability indices. Results obtained using the Bayesian approach are compared to frequentist results.