Joint modelling of longitudinal and time-to-event data with functional components under a copula approach

Abstract

There has been growing interests in joint modelling of longitudinal measurements and time-to-event data in recent decades. In a longitudinal study, subjects are followed up over a period and some information, such as biomarkers, are recorded at the follow-up time points. This process is eventually terminated because of the occurrence of an event of interest, such as death of the subjects, the quitting of individuals from the study due to some reasons irrelevant to the longitudinal or event processes, or the limit of maximum study time. The intra-subject outcomes of longitudinal measurements and the event time data are naturally correlated and separately modelling of them may lead to biased results as indicated by previous research. Joint modelling of the two processes through unobservable latent random variables, where the latent classes, shared or correlated random effects introduce dependency between the two sub-models while also accounting for heterogeneity among the population, is the most popular approach. However, the key assumption of conditional independence between the two processes given the latent random variables is rarely discussed. The aim of this thesis is to discuss this issue and propose approaches to cope with it. In Chapter 2, joint modelling of the marginals for longitudinal measurements and survival outcome by multivariate copulas, in which the dependency of the two sub-models are determined by the parameters in the copulas instead of latent random variables, is discussed. The model is flexible in the sense that the copula, which introduces association, and the marginal distributions can be modelled independently. Compared with the regular approach of the latent random variable joint model, computation of this proposed model is reduced as numerical integration with respect to the latent random variables is not required and this model still provides predictions on survival probabilities at a subject-specific level, but the longitudinal trajectories can only be fitted at a population level. Unlike the latent random variables joint model, the interpretation of the association between the two processes is not as straightforward and the modelling of the correlation matrix in the copula becomes difficult when its dimension is high (i.e., the number of repeated longitudinal measurements in each subject is large). In Chapter 3, we propose a joint model with one more layer of dependency, compared with the one in Chapter 2, between the two sub-models, where the joint distribution of the two processes given the random effects are specified by a multivariate Gaussian copula, thus relaxing the conditional independence assumption of the two processes given the random effects assumed in the regular joint model. Our model includes the regular joint model as a special case when the correlation matrix in the multivariate Gaussian copula is an identity matrix. A simulation study indicates that misspecification of the correlation matrix of the multivariate Gaussian copula results in biased parameter estimations even if the two sub-models are correctly specified, i.e., assuming conditional independence when it is not causes biased results. Further study indicates the dynamic predictions of survival probabilities are also affected by misspecification of the correlation matrix in the multivariate Gaussian copula. The correctly specified joint model has better prediction performance compared with the regular joint model given the correct sub-models. As in Chapter 2, the modelling of the correlation matrix is still troublesome when the number of the longitudinal measurements per subject is large. In Chapter 4, a functional bivariate copula joint model, which allows a nonparametric way of modelling residual dependency of the two processes by B-spline basis functions, is applied to specify the joint distribution of the time-to-event data and a single longitudinal measurement after conditioning on the random effects and the subject b

Date of Award	1 Aug 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Foster (Supervisor) & Christiana Charalambous (Supervisor)