Proportional hazard (PH) models can be formulated with or without assuming a probability distribution for survival times. The former assumption leads to parametric models, whereas the latter leads to the semi-parametric Cox model which is by far the most popular in survival analysis. However, a parametric model may lead to more efficient estimates than the Cox’s model under certain conditions. Only a few parametric models are closed under PH assumption, the most common of which is the Weibull that accommodates only monotone hazard functions.
The main objective of this thesis is to develop flexible and parsimonious parametric models which are capable of adequately describing different shapes of hazard function. In particular, we propose a generalization of the log-logistic distribution that belongs to the PH family. It has properties similar to those of log-logistic, and approaches the Weibull in the limit. These features enable it to handle both monotone and unimodal (inverse U-shape) hazard functions. Applications to four data sets and a simulation study revealed that the model could potentially be very useful in adequately describing different types of time-to-event data.
The generalized log-logistic PH model naturally accommodates monotone decreasing and unimodal hazard functions, and has the ability to model increasing hazard shapes satisfactorily. However, it is not flexible enough to deal with bathtub-shaped hazard functions. This type of shape is widely used to describe data in medical research and reliability engineering. Motivated by this, we propose a more general parametric proportional hazards model by modifying the Kumaraswamy Weibull (MKumW) distribution. The model is parsimonious and flexible in the sense that it accommodates all four standard shapes of the hazard function at the small cost of estimating only three distributional parameters.
We also consider two commonly encountered problems in survival analysis which require further extension of the standard PH models. More specifically, we propose methods for recurrent event data analysis and joint modeling as described below.
In biomedical studies and clinical trials, the individuals under study may experience multiple events over time. Such processes are called recurrent event processes, and the data generated by such processes are called recurrent event data. We propose a parametric recurrent event model, formulated using our MKumW distribution. Specifically, we consider the Poisson process formulation, with the baseline intensity function modeled parametrically.
Another problem considered in this study is joint modeling. In many longitudinal studies, a longitudinal response is observed along with an observation of the time to the occurrence of an event; the event can be timed from the beginning of an observation period, resulting in survival or time-to-event data. A typical goal in such studies is to investigate the effects of the longitudinal response (internal covariate for the event process) on the development of the event. The motivating idea behind the joint modeling techniques is to couple the time-to-event model with the longitudinal model through shared random effects. Although the Cox PH model is appealing to analyze standard survival data mainly due to its robustness property, the use of the Cox PH in joint modeling usually leads to an underestimation of the standard errors of the parameter estimates. Therefore, most methods for joint modeling are based on parametric response distributions. We propose a joint modeling framework based on our MKumW distribution. The novelty lies in formulating a hierarchical model based on the MKumW distribution, proposing a Bayesian approach for statistical inference, and computationally intensive Bayesian implementation of the methodology in the statistical software WinBUGS.
In this thesis, we propose two parametric PH models for time-to-event data, and develop theory for statistical inference. As demonstrated, the proposed models are fairly flexible and parsimonious, and can be valuable in survival analysis theory and applications. Perhaps the most important contribution of this thesis involves further extension of one of the proposed models to recurrent event data analysis and joint modeling of longitudinal and time-to-event data.
We have published one article based on the generalized log-logistic PH model in the Journal of Statistical Distributions and Applications. We intend to publish at least two more articles out of this thesis: the focus of one article will be on statistical methodologies for recurrent event and joint modeling based on the MKumW distribution, and another article could be on the software implementation of the proposed models, possibly in a journal in computational statistics
