Longitudinal Joint Modelling Utilising the Coxian Phase-type Distribution

  • Conor Donnelly

Student thesis: Doctoral ThesisDoctor of Philosophy


Joint modelling techniques for the analysis of longitudinal and survival data are a relatively recent statistical development, employed to appropriately account for the possible association which can exist between these two processes. Since their development at the end of the 20th century, joint models have become increasingly popular within statistical literature due to their broad applicability across many areas of statistical research. Within a joint modelling framework, each process is represented by its own submodel, where most often a linear mixed effects (LME) model is employed for the longitudinal process and a proportional hazards model for the survival outcome. The parameters of both submodels are estimated simultaneously from a single joint likelihood, therefore overcoming the bias which can occur when the processes are modelled independently.

Within this thesis, the Coxian phase-type regression model is explored as a novel approach to represent the survival process within a joint model. Phase-type distributions, in general, are a diverse family of distributions which describe the absorption times of a continuous time Markov process with a single absorbing state, formulated by a convolution of exponential distributions, either in series or parallel. Employing phase-type distributions to model failure times can potentially uncover latent stages of the process under investigation, and insight can be gained from the estimated parameters regarding the rates of flow through these uncovered phases. Within a medical context, mapping the uncovered states onto distinct stages of a disease's progression, for example, allows predictions to be made regarding the time spent within the different stages of the disease, and inferences can be drawn from these predictions on the individuals' expected quality of life for their remaining survival time.

Whilst previous research has explored the use of phase-type distributions to represent typical survival analysis problems, there are a number of limitations which have hindered their popularity, particularly with regards to the fitting of the models. Consequently, the first portion of this research is concerned with investigating the Coxian phase-type regression model so as to improve its suitability at representing typical survival processes. To this end, a new expectation-maximisation (EM) algorithm approach to fitting phase-type distributions is developed, shown through a simulation study to improve upon alternative algorithm approaches, employed within the current literature, in terms of both the accuracy of the parameter estimates and the rate of convergence. Due to its increased stability, this approach is then extended to allow for the effect of a covariate to vary across the transitions of the model, as opposed to remaining fixed, as is the common assumption within the literature.

Subsequently, a joint modelling framework is developed, within which the longitudinal process is represented by a LME model and the survival process by the Coxian phase-type regression model. This new methodology is shown to be beneficial to both areas of research. For instance, previously the Coxian was limited to modelling time-invariant covariates, greatly limiting its scope, whereas it can now be employed within the new joint modelling framework to model the association between longitudinal biomarkers and survival outcome, extending its applicability, particularly within the medical field.

Further, employing the Coxian to represent the survival process offers a number of advantages over alternative parametric models; the Coxian phase-type distribution can represent any positive distribution to an arbitrary degree of accuracy, overcoming the noted limitations of survival models which assume more restrictive distributions, and greater insight into the survival process can be gained from the uncovered phases of the distribution.
Date of Award2019
Original languageEnglish
Awarding Institution
  • Queen's University Belfast
SupervisorAdele Marshall (Supervisor) & Lisa McFetridge (Supervisor)

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