The Joint Models for Non-Linear Longitudinal and Time-To-Event Data Using Penalized Splines

Author: Thi Thu Huong Pham

Pham, Thi Thu Huong, 2018 The Joint Models for Non-Linear Longitudinal and Time-To-Event Data Using Penalized Splines , Flinders University, College of Science and Engineering

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Abstract

Joint models for longitudinal and time-to-event data have been applied in many different fields of statistics and clinical studies. My interest is in modelling the relationship between event time outcomes and internal time-dependent covariates. In practice, the longitudinal responses often show non-linear and fluctuated curves. Therefore, the main aim of this thesis is to use penalized splines with a truncated polynomial basis to parameterise the non-linear longitudinal process. Then, the linear mixed effects model is applied to subjectspecific curves and to control the smoothing. The association between the dropout process and longitudinal outcomes is modeled through a proportional hazard model. Two types of baseline risk functions are considered, namely a Gompertz distribution and a piecewise constant model. The resulting models are referred to as penalized spline joint models; an extension of the standard joint models. The expectation conditional maximization (ECM) algorithm is applied to estimate the parameters in the proposed models. To validate the proposed algorithm, extensive simulation studies were implemented followed by a case study. Simulation studies show that the penalized spline joint models improve the existing standard joint models.

The main difficulty that the penalized spline joint models have to face with is the computational problem. The requirement for numerical integration becomes severe when the dimension of random effects increases. In this thesis, a modified two-stage approach has been proposed to estimate the parameters in joint models. This approach not only improves a previous two-stage approach but also allows for the application of extended joint models with a high dimension of random effects in the longitudinal submodel. In particular, in the first stage, the linear mixed effects models (LMEs) and best linear unbiased predictors (BLUPs) are applied to estimate parameters in the longitudinal submodel. Then, in the second stage, an approximation of the fully joint log-likelihood is proposed using the estimated values of these parameters from the longitudinal submodel. The survival parameters are estimated by maximizing the approximation of the fully joint log-likelihood. Simulation studies show that the modified two-stage approach performs well, especially when the dimension of the random effects in the penalized splines joint models increases.

Finally, a Bayesian approach is applied to estimate the parameters in the penalized splines joint models. This approach provides alternative ways to infer the uncertainties of the parameters in the penalized splines joint models. Moreover, this approach can avoid approximations resulting from calculating multiple integrals in the frequentist approach. The Markov chain Monte Carlo (MCMC) algorithm is proposed containing the Gibbs sampler (GS) and Metropolis Hastings (MH) algorithms to sample for the target conditional posterior distributions. Extensive simulation studies were implemented to validate the proposed algorithm. In addition, the prior sensitivity analysis for the baseline hazard rate and association parameters is performed through simulation studies and a case study. The results show that the fully Bayesian approach produces reliable estimates and complete inferences for the parameters in the penalized splines joint models.

Keywords: Survival data, Longitudinal data, Joint models, Time-dependent covariates, Random effects,Two-stage approach, Shared random effects approach

Subject: Statistical Science thesis

Thesis type: Doctor of Philosophy
Completed: 2018
School: College of Science and Engineering
Supervisor: Dr Darfiana Nur