The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.

The purpose of this study is to investigate and understand data which are grouped into categories. Various statistical methods was studied for categorical binary responses to investigate the causes of death from diarrhoea in South Africa. Data collected included death type, sex, marital status, province of birth, province of death, place of death, province of residence, education status, smoking status and pregnancy status. The objective of this thesis is to investigate which of the above explanatory variables was most affected by diarrhoea in South Africa. To achieve this objective, different sample survey data analysis techniques are investigated. This includes sketching bar graphs and using several statistical methods namely, logistic regression, surveylogistic, generalised linear model, generalised linear mixed model, and generalised additive model. In the selection of the fixed effects, a bar graph is applied to the response variable individual profile graphs. A logistic regression model is used to identify which of the explanatory variables are more affected by diarrhoea. Statistical applications are conducted in SAS (Statistical Analysis Software). Hosmer and Lemeshow (2000) propose a statistic that they show, through simulation, is distributed as chi‐square when there is no replication in any of the subpopulations. Due to the similarity of the Hosmer and Lemeshow test for logistic regression, Parzen and Lipsitz (1999) suggest using 10 risk score groups. Nevertheless, based on simulation results, May and Hosmer (2004) show that, for all samples or samples with a large percentage of censored observations, the test rejects the null hypothesis too often. They suggest that the number of groups be chosen such that G=integer of {maximum of 12 and minimum of 10}. Lemeshow et al. (2004) state that the observations are firstly sorted in increasing order of their estimated event probability. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Malawi’s economy relies heavily on agriculture which is threatened by declines in soil fertility. Measures to ensure increased crop productivity at household level include the increased use of inorganic fertilizers. To supplement the Government’s effort in ensuring food security, Rural Livelihood Diversification Project (RLDP) was implemented in Kasungu and Lilongwe Districts in Malawi. The RLDP Project was aimed at increasing accessibility and utilisation of inorganic fertilizers. We used the data collected by the International Center for Tropical Agriculture (CIAT), to investigate if there could be any significant impacts of the interventions carried out by the project. A general linear model was initially used to model the data. Terms in the model were selected using the automatic stepwise procedure in GLMSELECT procedure of SAS. Other models that were used included a transformed response general linear model, gamma model based on log link and its alternative inverse link, and quantile regression procedures were used in modelling the amount of fertilizer use per acre response given a set of fixed effect predictors where households were only sampled at baseline or impact assessment study. The general linear model failed to comply with the model assumption of normality and constant variance. The gamma model was affected by influential observations. Quantile regression model is robust to outliers and influential observations. Quantile regression provided that number of plots cultivated, timeline, household saving and irrigation interaction, and the interaction between plots and timeline significantly affected the amounts of fertilizers applied per acre amongst the 25% of the households who apply lower levels of fertilizer per acre. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Just over half (50:6%) of the population of South Sudan lives on less than one US Dollar a day. Three quarters of the population live below the poverty line (NBS, Poverty Report, 2010). Generally, effective government policy to reduce unemployment and eradicate poverty focuses on stimulating new businesses. Micro and small enterprises (MSEs) are the major source of employment and income for many in under-developed countries. The objective of this study is to identify factors that determine business success and failure in South Sudan. To achieve this objective, generalized linear models, survey logistic models, the generalized linear mixed models and multiple correspondence analysis are used. The data used in this study is generated from the business survey conducted in 2010. The response variable, which is defined as business success or failure was measured by profit and loss in businesses. Fourteen explanatory variables were identified as factors contributing to business success and failure. A main effect model consisting of the fourteen explanatory variables and three interaction effects were fitted to the data. In order to account for the complexity of the survey design, survey logistic and generalized linear mixed models are refitted to the same variables in the main effect model. To confirm the results from the model we used multiple correspondence analysis. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

Wadley’s problem frequently emerges in dosage-mortality data and is one in which the number of surviving organisms is observed but the number initially treated is unknown. Data in this setting are also often overdispersed, that is the variability within the data exceeds that described by the distribution modelling it. The aim of this thesis is to explore distributions that can accommodate overdispersion in a Wadley’s problem setting. Two methods are essentially considered. The first considers adapting the beta-binomial and multiplicative binomial models that are frequently used for overdispersed binomial-type data to a Wadley’s problem setting. The second strategy entails modelling Wadley’s problem with a distribution that is suitable for modelling overdispersed count data. Some of the distributions introduced can be used for modelling overdispersed count data as well as overdispersed doseresponse data from a Wadley context. These models are compared using goodness of fit tests, deviance and Akaike’s Information Criterion and their properties are explored. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.

Longitudinal data tend to be correlated and hence posing a challenge in the analysis since the correlation has to be accounted for to obtain valid inference. We study various statistical methods for such correlated longitudinal binary responses. These models can be grouped into five model families, namely, marginal, subject-specific, transition, joint and semi-parametric models. Each one of the models has its own strengths and weaknesses. Application of these models is carried out by analyzing data on patient’s adherence status to highly active antiretroviral therapy (HAART). One other complicating issue with the HAART adherence data is missingness. Although some of the models are flexible in handling missing data, they make certain assumptions about missing data mechanisms, the most restrictive being missing completely at random (MCAR). The test for MCAR revealed that dropout did not depend on the previous outcome. A logistic regression model was used to identify predictors for the patients’ first month’s adherence status. A marginal model was then fitted using generalized estimating equations (GEE) to identify predictors of long-term HAART adherence. This provided marginal population-based estimates, which are important for public health perspective. We further explored the subject’s specific effects that are unique to a particular individual by fitting a generalized linear mixed model (GLMM). The GLMM was also used to assess the association structure of the data. To assess whether the current optimal adherence status of a patient depended on the previous adherence measurements (history) in addition to the explanatory variables, a transition model was fitted. Moreover, a joint modeling approach was used to investigate the joint effect of the predictor variables on both HAART adherence status of patients and duration between successive visits. Assessing the association between the two outcomes was also of interest. Furthermore, longitudinal trajectories of observed data may be very complex especially when dealing with practical applications and as such, parametric statistical models may not be flexible enough to capture the main features of the longitudinal profiles, and so a semiparametric approach was adopted. Specifically, generalized additive mixed models were used to model the effect of time as well as interactions associated with time non-parametrically. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

Estimation of current levels of human immunodeficiency virus (HIV) incidence is essential for monitoring the impact of an epidemic, determining public health priorities, assessing the impact of interventions and for planning purposes. However, there is often insufficient data on incidence as compared to prevalence. A direct approach is to estimate incidence from longitudinal cohort studies. Although this approach can provide direct and unbiased measure of incidence for settings where the study is conducted, it is often too expensive and time consuming. An alternative approach is to estimate incidence from cross sectional survey using biomarkers that distinguish between recent and non-recent/longstanding infections. The original biomarker based approach proposes the detection of HIV-1 p24 antigen in the pre-seroconversion period to identify persons with acute infection for estimating HIV incidence. However, this approach requires large sample sizes in order to obtain reliable estimates of HIV incidence because the duration of antigenemia before antibody detection is short, about 22.5 days. Subsequently, another method that involves dual antibody testing system was developed. In stage one, a sensitive test is used to diagnose HIV infection and a less sensitive test such is used in the second stage to distinguish between long standing infections and recent infections among those who tested positive for HIV in stage one. The question is: how do we combine this data with other relevant information, such as the period an individual takes from being undetectable by a less sensitive test to being detectable, to estimate incidence? The main objective of this thesis is therefore to develop likelihood based method that can be used to estimate HIV incidence when data is derived from cross sectional surveys and the disease classification is achieved by combining two biomarker or assay tests. The thesis builds on the dual antibody testing approach and extends the statistical framework that uses the multinomial distribution to derive the maximum likelihood estimators of HIV incidence for different settings. In order to improve incidence estimation, we develop a model for estimating HIV incidence that incorporate information on the previous or past prevalence and derive maximum likelihood estimators of incidence assuming incidence density is constant over a specified period. Later, we extend the method to settings where a proportion of subjects remain non-reactive to a less sensitive test long after seroconversion. Diagnostic tests used to determine recent infections are prone to errors. To address this problem, we considered a method that simultaneously makes adjustment for sensitivity and specificity. In addition, we also showed that sensitivity is similar to the proportion of subjects who eventually transit the “recent infection” state. We also relax the assumption of constant incidence density by proposing linear incidence density to accommodate settings where incidence might be declining or increasing. We extend the standard adjusted model for estimating incidence to settings where some subjects who tested positive for HIV antibodies were not tested by a less sensitive test resulting in missing outcome data. Models for the risk factors (covariates) of HIV incidence are considered in the last but one chapter. We used data from Botswana AIDS Impact (BAIS) III of 2008 to illustrate the proposed methods. The general conclusion and recommendations for future work are provided in the final chapter. / Theses (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

We present mathematical models to study the mechanism of interaction of tumour infiltrating cytotoxic lymphocytes (TICLs) with tumour cells. We focus on the phase spaces of the systems and the nature of the solutions for the cell densities in the short and long term. The first model describes the production of offspring through cell proliferation, death and local kinetic interactions. The second model characterises the spatial distribution dynamics of the cell densities through reaction diffusion, which describes the random movement of the cells, and chemotaxis, which describes the immune cell movements towards the tumour cells. We then extend these models further to incorporate the effects of immunotherapy by developing two new models. In both situations, we analyse the phase spaces of the homogeneous models, investigate the presence of travelling wave solutions in our systems, and provide numerical simulations. Our analysis shows that cancer dormancy can be attributed to TICLs. Our study also shows that TICLs reduce the tumour cell density to a cancer dormant state but even with immunotherapy do not completely eliminate tumour cells from body tissue. Travelling wave solutions were confirmed to exist in the heterogeneous model, a linear stability analysis of the homogeneous models and numerical simulations show the existence of a stable tumour dormant state and a phase space analysis confirms that there are no limit cycles. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

In sequential testing an observer must choose when to observe additional data points and when to stop observation and make a decision. This stopping rule is traditionally based upon probability of error as well as certain cost parameters. The proposed stopping rule will instruct the observer to cease observation once the expected cost of the next observation increases. There is often a great deal of information about what the observer should see. This information will be used to develop a prior distribution for the parameters. The proposed stopping rule will be analyzed and compared to other stopping rules. Analysis of simulated data shows under which conditions the cost of the proposed stopping rule will approximate the minimum expected cost. / Department of Mathematical Sciences

Acute Myocardial Infarction (AMI), commonly known as heart attack, is a leading cause of death for adult men and women in the world. Studying mortality after AMI is therefore an important problem in epidemiology. This thesis develops statistical methodology for examining geographic patterns in mortality following AMI. Specifically, we develop parametric Accelerated Failure Time (AFT) models for censored survival data, where space-varying regression is used to investigate spatial patterns of mortality after AMI. In addition to important covariates such as age and gender, the regression models proposed here also incorporate spatial random e ects that describe the residual heterogeneity associated with di erent local health geographical units. We conduct model inference under a hierarchical Bayesian modeling framework using Markov Chain Monte Carlo algorithms for implementation. We compare an array of models and address the goodness-of- t of the parametric AFT model through simulation studies and an application to a longitudinal AMI study in Quebec. The application of our AFT model to the Quebec AMI data yields interesting ndings concerning aspects of AMI, including spatial variability. This example serves as a strong case for considering the parametric AFT model developed here as a useful tool for the analysis of spatially correlated survival data.