Joint analysis of longitudinal and time to event data using accelerated failure time models: A Bayesian approach
Date
2020-01-09
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Type
Thesis
Degree Level
Masters
Abstract
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.
Description
Keywords
Joint Modeling, Longitudinal Data, Time-to-event Data, Parametric Accelerated Failure Time, Bayesian Inference, Markov Chain Monte Carlo
Citation
Degree
Master of Science (M.Sc.)
Department
School of Public Health
Program
Biostatistics