Abstract 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. Copyright © 2005 John Wiley & Sons, Ltd.