Shared random effects analysis of multi-state Markov models: application to a longitudinal study of transitions to dementia

Multi-state models are appealing tools for analysing data about the progression of a disease over time. In this paper, we consider a multi-state Markov chain with two competing absorbing states: dementia and death and three transient non-demented states: cognitively normal, amnestic mild cognitive impairment (amnestic MCI), and non-amnestic mild cognitive impairment (non-amnestic MCI). The likelihood function for the data is derived and estimates for the effects of the covariates on transitions are determined when the process can be viewed as a polytomous logistic regression model with shared random effects. The presence of a shared random effect not only complicates the formulation of the likelihood but also its evaluation and maximization. Three approaches for maximizing the likelihood are compared using a simulation study; the first method is based on the Gauss-quadrature technique, the second method is based on importance sampling ideas, and the third method is based on an expansion by Taylor series. The best approach is illustrated using a longitudinal study on a cohort of cognitively normal subjects, followed annually for conversion to mild cognitive impairment (MCI) and/or dementia, conducted at the Sanders Brown Center on Aging at the University of Kentucky.