Computing short‐interval transition matrices of a discrete‐time Markov chain from partially observed data

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Abstract
Markov chains constitute a common way of modelling the progression of a chronic disease through various severity states. For these models, a transition matrix with the probabilities of moving from one state to another for a specific time interval is usually estimated from cohort data. Quite often, however, the cohort is observed at specific times with intervals that may be greater than the interval of interest. The transition matrix computed then needs to be decomposed in order to estimate the desired interval transition matrix suited to the model. Although simple to implement, this method of matrix decomposition can yet result in an invalid short‐interval transition matrix with negative or complex entries. In this paper, we present a method for computing short‐interval transition matrices that is based on regularization techniques. Our method operates separately on each row of the invalid short‐interval transition matrix aiming to minimize an appropriate distance measure. We test our method on various matrix structures and sizes, and evaluate its performance on a real‐life transition model for HIV‐infected individuals. Copyright © 2007 John Wiley & Sons, Ltd.