Growth mixture modeling (GMM) is a useful tool for examining both between- and within-persons change over time and uncovering unobserved heterogeneity in growth trajectories. Importantly, the correct extraction of latent classes and parameter recovery can be dependent upon the type of covariates used. Time-varying covariates (TVCs) can influence class membership but are scarcely included in GMMs as predictors. Other times, TVCs are incorrectly modeled as time-invariant covariates (TICs). Additionally, problematic results can occur with the use of maximum likelihood (ML) estimation in GMMs, including convergence issues and sub-optimal maxima. In such cases, Bayesian estimation may prove to be a useful solution. The present Monte Carlo simulation study aimed to assess class enumeration accuracy and parameter recovery of GMMs with a TVC, particularly when a TVC has been incorrectly specified as a TIC. Both ML estimation and Bayesian estimation were examined. Results indicated that class enumeration indices perform less favorably in the case of TVC misspecification, particularly absolute class enumeration indices. Additionally, in the case of TVC misspecification, parameter bias was found to be greater than the generally accepted cutoff of 10%, particularly for variance estimates. It is recommended that researchers continue to use a variety of class enumeration indices during class enumeration, particularly relative indices. Additionally, researchers should take caution when interpreting variance parameter estimates when the GMM contains a misspecified TVC.

Survival models are used in analysing time-to-event data. This type of data is very common in medical research. The Cox proportional hazard model is commonly used in analysing time-to-event data. However, this model is based on the proportional hazard (PH) assumption. Violation of this assumption often leads to biased results and inferences. Once non-proportionality is established, there is a need to consider time-varying effects of the covariates. Several models have been developed that relax the proportionality assumption making it possible to analyse data with time-varying effects of both baseline and time-updated covariates. I present various approaches for handling time-varying covariates and time-varying effects in time-to-event models. They include the extended Cox model which handles exogenous time-dependent covariates using the counting process formulation introduced by cite{andersen1982cox}. Andersen and Gill accounts for time varying covariates by each individual having multiple observations with the total-at-risk follow up for each individual being further divided into smaller time intervals. The joint models for the longitudinal and time-to-event processes and its extensions (parametrization and multivariate joint models) were used as it handles endogenous time-varying covariates appropriately. Another is the Aalen model, an additive model which accounts for time-varying effects. However, there are situations where all the covariates of interest do not have time-varying effects. Hence, the semi-parametric additive model can be used. In conclusion, comparisons are made on the results of all the fitted models and it shows that choice of a particular model to fit is influenced by the aim and objectives of fitting the model. In 2002, an AntiRetroviral Treatment (ART) service was established in the Cape Town township of Gugulethu, South Africa. These models will be applied to an HIV/AIDS observational dataset obtained from all patients who initiated ART within the programme between September 2002 and June 2007.

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In this thesis, ties and time-varying covariates in survival analysis are investigated. There are two types of ties: ties between event times (Type 1 ties) and ties between event times and the time that discrete time-varying covariates change or "jump"(Type 2 ties). The Cox proportional hazards model is one of the most important regression models for survival analysis. Methods for including Type 1 ties and time-varying covariates in the Cox proportional hazards model are well established in previous studies, but Type 2 ties have been ignored in the literature. This thesis discusses the effect of Type 2 ties on Cox's partial likelihood, the current default method to treat Type 2 ties in statistical packages SAS and R (called Fail before Jump in this thesis), and proposes alternative methods (Random and Equally Weighted) for Type 2 ties. A simulation study as well as an analysis of data sets from real research both suggest that both Random and Equally Weighted methods perform better than the other two methods. Also the effect of the percentages of Type 1 and Type 2 ties on these methods for handling both types of ties is discussed. / NSERC

Doctor of Philosophy(PhD) / This thesis employs survival analysis framework (Allison, 1984) and the Cox’s hazard model (Cox, 1972) to estimate the probability that a credit rating survives in its current grade at a certain forecast horizon. The Cox’s hazard model resolves some significant drawbacks of the conventional estimation approaches. It allows a rigorous testing of non-Markovian behaviours and time heterogeneity in rating dynamics. It accounts for the changes in risk factors over time, and features the time structure of probability survival estimates. The thesis estimates three stratified Cox’s hazard models, including a proportional hazard model, and two dynamic hazard models which account for the changes in macro-economic conditions, and the passage of survival time over rating durations. The estimation of these stratified Cox’s hazard models for downgrades and upgrades offers improved understanding of the impact of rating history in a static and a dynamic estimation framework. The thesis overcomes the computational challenges involved in forming dynamic probability estimates when the standard proportionality assumption of Cox’s model does not hold and when the data sample includes multiple strata. It is found that the probability of rating migrations is a function of rating history and that rating history is more important than the current rating in determining the probability of a rating change. Switching from a static estimation framework to a dynamic estimation framework does not alter the effect of rating history on the rating migration hazard. It is also found that rating history and the current rating interact with time. As the rating duration extends, the main effects of rating history and current rating variables decay. Accounting for this decay has a substantial impact on the risk of rating transitions. Downgrades are more affected by rating history and time interactions than upgrades. To evaluate the predictive performance of rating history, the Brier score (Brier, 1950) and its covariance decomposition (Yates, 1982) were employed. Tests of forecast accuracy suggest that rating history has some predictive power for future rating changes. The findings suggest that an accurate forecast framework is more likely to be constructed if non-Markovian behaviours and time heterogeneity are incorporated into credit risk models.