Proportional Reversed Hazard Rate Models with Exponential Baseline

The proportional hazard regression models have been used extensively in survival analysis to understand and exploit the relationship between survival time and covariates. For left censored survival times, reversed hazard rate functions are more appropriate. In this paper, we discuss a parametric proportional reversed hazard rates model using exponential baseline. The estimation for the parameters are discussed. We also assess the performance of the proposed procedure based on a large number of Monte Carlo simulations. Finally, we illustrate the proposed method using a real case example and then we show that it provides a good and better fit than the usual proportional hazards model

Parametric Proportional Hazard Models with Applications in Survival Analysis

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

Optimal Burn-In under Complex Failure Processes: Some New Perspectives

Ph.DDOCTOR OF PHILOSOPH

Investigating the Epidemiology of bovine Tuberculosis in the European Badger

Global health is becoming increasingly reliant on our understanding and management of wildlife disease. An estimated 60% of emerging infectious diseases in humans are zoonotic and with human-wildlife interactions set to increase as populations rise and we expand further into wild habitats there is pressure to seek modelling frameworks that enable a deeper understanding of natural systems. Survival and mortality are fundamental parameters of interest when investigating the impact of disease with far reaching implications for species conservation, management and control. Survival analysis has traditionally been dominated by non- and semi-parametric methods but these can sometimes miss subtle yet important dynamics. Survival and mortality trajectory analysis can alleviate some of these problems by fitting fully parametric functions that describe lifespan patterns of mortality and survival. In the first part of this thesis we investigate the use of survival and mortality trajectories in epidemiology and uncover novel patterns of age-, sex- and infection-specific mortality in a wild population of European badgers (Meles meles) naturally infected with Mycobacterium bovis, the causative agent of bovine tuberculosis (bTB). Limitations of dedicated software packages to conduct such analyses led us to investigate alternative methods to build models from first principles and we found the NIMBLE package to offer an attractive blend of flexibility and speed. We create a novel parameterisation of the Siler model to enable more flexible model specification but encounter the common problem of competing models having comparable fits to the data. Multi-model inference approaches can alleviate some of these issues but require efficient methods to carry out model comparisons; we present an approach based on the estimation of the marginal likelihood through importance sampling and demonstrate its application through a series of simulation- and case-studies. The approach works well for both census and capture-mark-recapture (CMR) data, both of which are common within ecological research, but we uncover challenges in recording and modelling early life mortality dynamics that occur as a result of the CMR sampling process. The final part of the thesis looks at another alternative approach for model comparison that doesn’t require direct estimation of the marginal likelihood, Reversible Jump Markov Chain Monte Carlo (RJMCMC), which is particularly efficient when models to be compared are nested and the problem can reduce to one of variable selection. In the final chapter we carry out an investigation of age-, sex-, infection- and inbreeding-specific variation in survival and mortality in a wild population of European badgers naturally infected with bovine Tuberculosis. Using the methods and knowledge presented through the earlier chapters of this thesis we uncover patterns of mortality consistent with both the mutation accumulation and antagonistic pleiotropy theories of senescence but most interestingly uncover antagonistic pleiotropic effects of inbreeding on age-specific mortality in a wild population for the first time. This thesis provides a number of straightforward approaches to Bayesian survival analysis that are widely applicable to ecological research and can offer greater insight and uncover subtle patterns of survival and mortality that traditional methods can overlook. Our investigation into the epidemiology of bovine Tuberculosis and in particular the effects of inbreeding have far-reaching implications for the control of this disease. This research can also inform future conservation efforts and management strategies as all species are likely to be at increasing risk of inbreeding in an age of dramatic global change, rapid habitat loss and isolation