Bayesian Semiparametric Quantile Regression for Clustered Data

Traditional frequentist quantile regression makes few assumptions on the form of the error distribution and thus is able to accommodate non-normal errors. However, inference on the quantile regression models could be challenging for the unknown error distribution, though asymptotic or resampling methods were developed. Bayesian literature on quantile regression with random effects is relatively limited. The quantile regression approach proposed in this dissertation is founded on Bayesian probabilistic modeling for the underlying unknown distributions. By adopting the error density with a nonparametric scale mixture models, we developed Bayesian semiparametric models to make an inference on any quantile of interest and to allow for flexible shapes of the error densities. In this dissertation, we aimed to develop Bayesian semiparametric quantile regressions for both longitudinal data and clustered interval-censored data. We first proposed a semiparametric quantile mixed effect regression for clustered data, which relaxed normality assumption for both random effects and the error term. We then developed a semiparametric accelerated failure time quantile regression for the clustered interval-censored data. Both of the methods allow for estimates for the subgroup specific parameters and the detection of heterogeneity in the random effects population under nonparametric settings. Markov chain Monte Carlo (MCMC) methods provide computationally feasible implementations of Bayesian inference and learning. However, the speed of convergence can be challenging for highly complex and nonconjugate models. Specifically, Gibbs sampling algorithm that employs the addition of auxiliary parameters was used to speed up posterior sampling in our study. Several variations of the proposed model were considered and compared via the deviance information criterion. The performance of the proposed methods was evaluated by extensive simulation studies, and examples using data from Orthodontic clinics and lymphatic filariasis drug studies were presented as illustration

Bayesian Semi- and Non-parametric Analysis for Spatially Correlated Survival Data

Flexible incorporation of both geographical patterning and risk effects in cancer survival models is becoming increasingly important, due in part to the recent availability of large cancer registries. The analysis of spatial survival data is challenged by the presence of spatial dependence and censoring for survival times. Accurately modeling the risk factors and geographical pattern that explain the differences in survival is particularly of interest. Within this dissertation, the first chapter reviews commonlyused baseline priors, semiparametric and nonparametric Bayesian survival models and recent approaches for accommodating spatial dependence, both conditional and marginal. The last three chapters contribute three flexible survival models: (1) a proportional hazards model with areal-level covariate-adjusted frailties with application to county-level breast cancer survival data, (2) a marginal Bayesian nonparametric model for time to disease arrival of threatened amphibian populations, and (3) a generalized accelerated failure time model with spatial intrinsic conditionally autoregressive frailties with application to county-level prostate cancer data. An R package spBayesSurv is developed to examine all the proposed models along with some traditional spatial survival models

Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues

In the past two decades, joint models of longitudinal and survival data have received much attention in the literature. These models are often desirable in the following situations: (i) survival models with measurement errors or missing data in time-dependent covariates, (ii) longitudinal models with informative dropouts, and (iii) a survival process and a longitudinal process are associated via latent variables. In these cases, separate inferences based on the longitudinal model and the survival model may lead to biased or inefficient results. In this paper, we provide a brief overview of joint models for longitudinal and survival data and commonly used methods, including the likelihood method and two-stage methods

Novel Computational Methods for Censored Data and Regression

This dissertation can be divided into three topics. In the first topic, we derived a recursive algorithm for the constrained Kaplan-Meier estimator, which promotes the computation speed up to fifty times compared to the current method that uses EM algorithm. We also showed how this leads to the vast improvement of empirical likelihood analysis with right censored data. After a brief review of regularized regressions, we investigated the computational problems in the parametric/non-parametric hybrid accelerated failure time models and its regularization in a high dimensional setting. We also illustrated that, when the number of pieces increases, the discussed models are close to a nonparametric one. In the last topic, we discussed a semi-parametric approach of hypothesis testing problem in the binary choice model. The major tools used are Buckley-James like algorithm and empirical likelihood. The essential idea, which is similar to the first topic, is iteratively computing linear constrained empirical likelihood using optimization algorithms including EM, and iterative convex minorant algorithm

Independent increments in group sequential tests : a review

In order to apply group sequential methods for interim analysis for early stopping in clinical trials, the joint distribution of test statistics over time has to be known. Often the distribution is multivariate normal or asymptotically so, and an application of group sequential methods requires multivariate integration to determine the group sequential boundaries. However, if the increments between successive test statistics are independent, the multivariate integration reduces to a univariate integration involving simple recursion based on convolution. This allows application of standard group sequential methods. In this paper we review group sequential methods and the development that established independent increments in test statistics for the primary outcomes of longitudinal or failure time data

Analysis of Interval Censored Data Using a Longitudinal Biomarker

In many medical studies, interest focuses on studying the effects of potential risk factors on some disease events, where the occurrence time of disease events may be defined in terms of the behavior of a biomarker. For example, in diabetic studies, diabetes is defined in terms of fasting plasma glucose being 126 mg/dl or higher. In practice, several issues complicate determining the exact time-to-disease occurrence. First, due to discrete study follow-up times, the exact time when a biomarker crosses a given threshold is unobservable, yielding so-called interval censored events. Second, most biomarker values are subject to measurement error due to imperfect technologies, so the observed biomarker values may not reflect the actual underlying biomarker levels. Third, using a common threshold for defining a disease event may not be appropriate due to patient heterogeneity. Finally, informative diagnosis and subsequent treatment outside of observational studies may alter observations after the diagnosis. It is well known that the complete case analysis excluding the externally diagnosed subjects can be biased when diagnosis does not occur completely at random. To resolve these four issues, we consider a semiparametric model for analyzing threshold-dependent time-to-event defined by extreme-value-distributed biomarkers. First, we propose a semiparametric marginal model based on a generalized extreme value distribution. By assuming the latent error-free biomarkers to be non-decreasing, the proposed model implies a class of proportional hazards models for the time-to-event defined for any given threshold value. Second, we extend the marginal likelihood to a pseudo-likelihood by multiplying the likelihoods over all observation times. Finally, to adjust for externally diagnosed cases, we consider a weighted pseudo-likelihood estimator by incorporating inverse probability weights into the pseudo-likelihood by assuming that external diagnosis depends on observed data rather than unobserved data. We estimate the three model parameters using the nonparametric EM, pseudo-EM and weighted-pseudo-EM algorithm, respectively. Herein, we theoretically investigate the models and estimation methods. We provide a series of simulations, to test each model and estimation method, comparing them against alternatives. Consistency, convergence rates, and asymptotic distributions of estimators are investigated using empirical process techniques. To show a practical implementation, we use each model to investigate data from the ARIC study and the diabetes ancillary study of the ARIC study.Doctor of Philosoph

Accelerated failure tima models for multivariate interval-censored data with flexible distributional assumptions

Department of Probability and Mathematical StatisticsKatedra pravděpodobnosti a matematické statistikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult