Conditional Transformation Models

The ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables. This goal is, however, seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not affected by the regressors. The underlying reason for such a restriction is the assumption of additivity of signal and noise. We propose to relax this common assumption in the framework of transformation models. The novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables. These transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts, e.g. the continuous ranked probability score. The corresponding estimated conditional distribution functions are consistent. Conditional transformation models are potentially useful for describing possible heteroscedasticity, comparing spatially varying distributions, identifying extreme events, deriving prediction intervals and selecting variables beyond mean regression effects. An empirical investigation based on a heteroscedastic varying coefficient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more beneficial than kernel-based non-parametric approaches or parametric generalised additive models for location, scale and shape

Joint analysis of longitudinal and time to event data using accelerated failure time models: A Bayesian approach

Joint modeling is a collection of statistical methods to properly handle a longitudinal response while investigating its effects on time to the occurrence of an event. Joint modeling also allows an investigation of the effects of baseline covariates on both the longitudinal response and the event process. In practice, the inspiration of biostatistical research arises from clinical and biomedical studies. The data collected from these studies have always been getting attention due to their particular features that need special consideration when doing an analysis. New statistical methods have developed over time to handle an analysis of such data coming from these sources. A typical clinical study often involves collecting repeated measurements on a biomarker (e.g., lvmi measurements) along with an observation of the time to the occurrence of an event (e.g., death), resulting in a joint modeling setup, a model becomes increasingly popular in clinical studies. Joint models can be formulated with a probability distribution (parametric models) or without assuming a probability distribution (Cox model or semi-parametric Cox PH model) for time-to-event process. In general, parametric models are pivotal in the joint modeling of longitudinal and time-to-event data. A non-parametric or semi-parametric model usually leads to an underestimation of standard errors of the parameter estimates in the joint analysis. However, selection for the joint model framework is quite limited in the literature. The best choice for the selection of longitudinal model can be made based on the observed longitudinal data, and the best survival model can be selected based on the survival data, using standard model selection procedures for these models. In this thesis, we develop and implement a Bayesian joint model framework, consisting of longitudinal process involving continuous longitudinal outcome and two parametric accelerated failure time (AFT) models (Log-logistic (model 1) and Weibull (model 2)) for survival process. We introduce a link between the parametric AFT survival processes and the longitudinal process via one parameter of association corresponding to shared random effects. A linear mixed-effect model approach is used for the analysis of longitudinal process with the normality assumption of longitudinal response along with normal and independent distribution assumption for both random effects and the error term of the longitudinal process. Finally, Bayesian approach using the Markov chain Monte Carlo method with the Gibbs Sampler technique is adopted for the statistical inference. The motivating ideas behind our work on Bayesian joint models using parametric AFT event processes are: (a) although there are well-known techniques to test the proportionality assumption for the Cox PH model, checking this assumption for joint modeling has received less attention in the literature. To our knowledge, no statistical package is available to check the PH assumption under the joint modeling setup. AFT models are particularly useful when the PH assumption is in question, (b) there are two integrals involved in the specification of joint models: (1) a unidimensional integral with respect to time which is relatively straightforward to approximate using numerical techniques, and (2) a multidimensional integral with respect to random effects which is the main computational burden to fit a joint model. It is relatively straightforward to handle (2) under the Bayesian framework, implemented using Markov Chain Monte Carlo (MCMC) techniques, (c) Bayesian approach does not depend on asymptotic approximation for statistical inference and (d) availability of software makes Bayesian implementation for complicated models relatively more straightforward and simpler than frequentist methods. We also develop computational algorithms to fit the proposed Bayesian joint model approach and implemented it in WinBUGS (a Bayesian software) and R software. Analysis are performed with an application to aortic heart valve replacement surgery data (available in joineR package in R software) to illustrate the performance of our two proposed models with the aim of comparing the efficiency of two types of valves based on tissue type (Stentless porcine tissue or Homograft) implanted during surgery and the association between internal covariate (longitudinal response: log.lvmi) and the occurrence of an event (death) after the surgery. Model selection is performed using the deviance information criterion (DIC). Study analysis results for both joint models indicate the statistically significant and strong association between internal covariate (longitudinal response: log.lvmi) and the relative risk of death after aortic valve replacement surgery. Results show that one gm/m^2 increase in the value of log.lvmi after the surgery reduces the relative risk of death by about 62 % (model 1) and 60 % (model 2), respectively, after controlling for other factors. Moreover, age of the patient (age) and preoperative left ventricular ejection fraction (lv) are found statistically significant for the risk of death after surgery. However, we found no significant difference between the efficiency of two types of valves implanted during surgery based on tissue type (Stentless porcine tissue or Homograft) associated with reducing the risk of death in the patients after surgery. Finally, based on DIC, we recommend, Bayesian joint AFT model with Weibull distribution fits the motivated data set more efficiently than Bayesian joint AFT model with Log-Logistic distribution. Developing joint models using AFT event processes, writing the model in a hierarchical framework for Bayesian implementation and developing computational algorithms to fit proposed joint models is the novelty of this thesis

Statistical Degradation Models for Electronics

With increasing presence of electronics in modern systems and in every-day products, their reliability is inextricably dependent on that of their electronics. We develop reliability models for failure-time prediction under small failure-time samples and information on individual degradation history. The development of the model extends the work of Whitmore et al. 1998, to incorporate two new data-structures common to reliability testing. Reliability models traditionally use lifetime information to evaluate the reliability of a device or system. To analyze small failure-time samples within dynamic environments where failure mechanisms are unknown, there is a need for models that make use of auxiliary reliability information. In this thesis we present models suitable for reliability data, where degradation variables are latent and can be tracked by related observable variables we call markers. We provide an engineering justification for our model and develop parametric and predictive inference equations for a data-structure that includes terminal observations of the degradation variable and longitudinal marker measurements. We compare maximum likelihood estimation and prediction results obtained by Whitmore et. al. 1998 and show improvement in inference under small sample sizes. We introduce modeling of variable failure thresholds within the framework of bivariate degradation models and discuss ways of incorporating covariates. In the second part of the thesis we investigate anomaly detection through a Bayesian support vector machine and discuss its place in degradation modeling. We compute posterior class probabilities for time-indexed covariate observations, which we use as measures of degradation. Lastly, we present a multistate model used to model a recurrent event process and failure-times. We compute the expected time to failure using counting process theory and investigate the effect of the event process on the expected failure-time estimates

Nonparametric methods for the estimation of the conditional distribution of an interval-censored lifetime given continuous covariates

Cette thèse contribue au développement de l'estimation non paramétrique de la fonction de survie conditionnelle étant donné une covariable continue avec données censurées. Elle est basée sur trois articles écrits avec mon directeur de thèse, le professeur Thierry Duchesne. Le premier article, intitulé "Une généralisation de l'estimateur de Turnbull pour l'estimation non paramétrique de la fonction de survie conditionnelle avec données censurées par intervalle, " a été publié en 2011 dans Lifetime Data Analysis, vol. 17, pp. 234 - 255. Le deuxième article, intitulé "Sur la performance de certains estimateurs nonparamétriques de la fonction de survie conditionnelle avec données censurées par intervalle, " est parru en 2011 dans la revue Computational Statistics & Data Analysis, vol. 55, pp. 3355-3364. Le troisième article, intitulé "Estimation de la fonction de survie conditionnelle d'un temps de défaillance étant donné une covariable variant dans le temps avec observations censurées par intervalles", sera bientôt soumis à la revue Statistica Sinica