Doctoral Degrees (Mathematical Statistics and Actuarial Science)

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Browsing Doctoral Degrees (Mathematical Statistics and Actuarial Science) by Author "Schall, R."

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    Bayesian non-linear models for the bactericidal activity of tuberculosis drugs
    (University of the Free State, 2015-05) Burger, Divan Aristo; Schall, R.; Van der Merwe, A. J.
    Trials of the early bactericidal activity (EBA) of tuberculosis (TB) treatments assess the decline, during the first few days to weeks of treatment, in colony forming unit (CFU) count of Mycobacterium tuberculosis in the sputum of patients with smear-microscopy-positive pulmonary TB. Profiles over time of CFU data have conventionally been modeled using linear, bilinear or bi-exponential regression. This thesis proposes a new biphasic nonlinear regression model for CFU data that comprises linear and bilinear regression models as special cases, and is more exible than bi-exponential regression models. A Bayesian nonlinear mixed effects (NLME) regression model is fitted jointly to the data of all patients from clinical trials, and statistical inference about the mean EBA of TB treatments is based on the Bayesian NLME regression model. The posterior predictive distribution of relevant slope parameters of the Bayesian NLME regression model provides insight into the nature of the EBA of TB treatments; specifically, the posterior predictive distribution allows one to judge whether treatments are associated with mono-linear or bilinear decline of log(CFU) count, and whether CFU count initially decreases fast, followed by a slower rate of decrease, or vice versa. The fit of alternative specifications of residuals, random effects and prior distributions is explored. In particular, the conventional normal regression models for log(CFU) count versus time profiles are extended to provide a robust approach which accommodates outliers and potential skewness in the data. The deviance information criterion and compound Laplace-Metropolis Bayes factors are calculated to discriminate between models. The biphasic model is fitted to time to positivity data in the same way as for CFU data.
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    Extending the reach of sequential regression multiple imputation
    (University of the Free State, 2015-06) Von Maltitz, Michael Johan; Raghunathan, T. E.; Schall, R.; Van der Merwe, A. J.
    English: The purpose of this thesis is twofold. Firstly, it reviews a signi cant portion of literature concerning multiple imputation and, in particular, sequential regression multiple imputation, and summarises this information, thereby allowing a reader to gain in-depth knowledge of this research eld. Secondly, the thesis delves into one particular novel topic in sequential regression multiple imputation. The latter objective, of course, is not truly possible without the former, since the deeper the review of multiple imputation, the more likely it will be to identify and solve pressing concerns in the sequential regression multiple imputation sub eld. The literature review will show that there is room in imputation research for work on a robust model for the sequential regression multiple imputation algorithm. This thesis pays particular attention to this robust model, formulating its estimation procedure within the context of sequential regression multiple imputation of continuous data, attempting to discover a statistic that would show when to use the robust model over the regular Normal speci cation, and then implementing the robust model in another estimation algorithm that might allow for better imputation of ordinal data. This thesis contributes to `extending the reach of sequential regression multiple imputation' in two ways. Firstly, it is my wish for users of public data sets, particularly in South Africa, to become familiar with the (now internationally standard) topics presented in the rst half of this thesis. The only way to start publicising sequential regression multiple imputation in South Africa is to lay out the evidence for and against this procedure in a logical manner, so that any reader of this thesis might be able to understand the procedures for analysing multiply imputed data, or tackle one of the many research problems uncovered in this text. In this way, this thesis will extend the reach of sequential regression multiple imputation to many more South African researchers. Secondly, by working on a new robust model for use in the sequential regression multiple imputation algorithm, this thesis strengthens the sequential regression multiple imputation algorithm by extending its reach to incomplete data that is not necessarily Normally distributed, be it due to heavy tails, or inherent skewness, or both.