Estimating life expectancy in health and ill health by using a hidden Markov model

Van den Hout, Jagger and Matthews have a paper to appear in JRSS C. The paper applies the misclassification hidden Markov model, developed by Satten and Longini and Jackson and Sharples, to modelling of data on cognitive impairment in the elderly and its effect on mortality. Patients with a cognition score (MMSE) below 22 were considered impaired. However, cognitive decline is considered to be progressive so backwards transitions in the dataset are explained through misclassification.

The main aim of the paper is to estimate life expectancies in the non-impaired and impaired states. As mortality will be highly dependent on age, non-homogeneous transition intensities are required. Rather than employ the standard approach of piecewise constant intensities, the authors instead include age as a log-linear time dependent covariate and assume that an individual observed at ages t and u, for t < u, has constant intensity Q(t) for the interval (t,u). This will clearly result in some degree of bias, particularly if observation times are widely spaced. Life expectancy is then calculated by assuming intensities are constant in 1 year intervals. As this is different from how the data were estimated, the bias may be further compounded.

Rudimentary goodness-of-fit is carried out by comparing estimated survival curves from the HMM with a Cox-regression performed directly on the survival data. It is worth noting that this approach could be problematic in certain circumstances because the HMM is not nested within the Cox-regression model, so there might be discrepancies between the curves even if the HMM is correctly specified.