A two-stage approach to the joint analysis of longitudinal and survival data utilising the Coxian phase-type distribution

Abstract

The Coxian phase-type distribution is a special type of Markov model which can be utilised both to uncover underlying stages of a survival process and to make inferences regarding the rates of flow of individuals through these latent stages before an event of interest occurs. Such models can be utilised, for example, to identify individuals who are likely to deteriorate faster through a series of disease states and thus require more aggressive medical intervention. Within this paper, a two-stage approach to the analysis of longitudinal and survival data is presented. In Stage 1, a linear mixed effects model is first used to represent how some longitudinal response of interest changes through time. Within this linear mixed effects model, the individuals' random effects can be considered as a proxy measure for the effect of the individuals' genetic profiles on the response of interest. In Stage 2, the Coxian phase-type distribution is employed to represent the survival process. The individuals' random effects, estimated in Stage 1, are incorporated as covariates within the Coxian phase-type distribution so as to evaluate their effect on the individuals' rates of flow through the system represented by the Coxian. The approach is illustrated using data collected on individuals suffering from chronic kidney disease, where focus is given to an emerging longitudinal biomarker of interest -an individual's haemoglobin level.

Estimating and comparing cancer progression risks under varying surveillance protocols

“…Furthermore, rather than using joint modeling we incorporate PSA via a two-step approach: first estimating random effects and then using them as covariates in our disease progression model. This method may under-represent uncertainty and lead to some bias in projections of underlying upgrade, but has been used by others with latent CTMC models (Donnelly et al, 2017). Another implicit assumption is that that individuals who undergo treatment without upgrading do not have an undetected grade progression before treatment.…”

A Longitudinal Study of the Bladder Cancer Applying a State-Space Model with Non-Exponential Staying Time in States

“…The staying time of patients in hospitals has been studied applying a Coxian phasetype process; it is an increasing Markov process in which all individuals initiate in the first phase, and the only transitions possible are to the next phase or to the absorbent one [30,31]. This process has been used to approach a two-stage model for studying the evolution in time of patients suffering chronic kidney disease; in it, covariates are introduced, allowing to analyze the disease in detail for different groups of patients [32]. An investigating work about the state-space models applied to dynamic systems presents the PH distributions as a good alternative to other models by having a greater applicability and the advantage of the matrix-analytic methods [33].…”