Likelihood for generally coarsened observations from multi-state or counting process models

TL;DR

This paper rigorously derives the likelihood for multistate models with mixed discrete and continuous observations, extending to general coarsening schemes, with applications in epidemiology.

Contribution

It provides a formal proof and general framework for likelihood derivation in non-Markov multistate models with coarsened data.

Findings

Rigorous likelihood derivation using Jacod's formula

Extension to general coarsening observation schemes

Application to models of dementia, institutionalization, and death

Abstract

We consider first the mixed discrete-continuous scheme of observation in multistate models; this is a classical pattern in epidemiology because very often clinical status is assessed at discrete visit times while times of death or other events are observed exactly. A heuristic likelihood can be written for such models, at least for Markov models; however, a formal proof is not easy and has not been given yet. We present a general class of possibly non-Markov multistate models which can be represented naturally as multivariate counting processes. We give a rigorous derivation of the likelihood based on applying Jacod's formula for the full likelihood and taking conditional expectation for the observed likelihood. A local description of the likelihood allows us to extend the result to a more general coarsening observation scheme proposed by Commenges & G\'egout-Petit. The approach is illustrated by considering models for dementia, institutionalization and death.