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

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.