Reduced rank multinomial logistic regression in Markov chains with application to cognitive data

Finite Markov chains are useful tools for studying transitions among health states; these chains can be complex consisting of a mix of transient and absorbing states. The transition probabilities, which are often affected by covariates, can be difficult to estimate due to the presence of many covariates and/or a subset of transitions that are rarely observed. The purpose of this article is to show how to estimate the effect of a subset of covariates of interest after adjusting for the presence of multiple other covariates by applying multidimensional dimension reduction to the latter. The case in which transitions within each row of the one-step transition probability matrix are estimated by multinomial logistic regression is discussed in detail. Dimension reduction for the adjustment covariates involves estimating the effect of the covariates by a product of matrices iteratively; at each iteration one matrix in the product is fixed while the second is estimated using either standard software or nonlinear estimation, depending on which of the matrices in the product is fixed. The algorithm is illustrated by an application where the effect of at least one Apolipoprotein-E (APOE) gene (Formula presented.) allele on transition probability is estimated in a Markov Chain that includes adjustment for eight covariates and focuses on transitions from normal cognition to several forms of mild cognitive impairment, with possible absorption into dementia. Data were drawn from annual cognitive assessments of 649 participants enrolled in the BRAiNS cohort at the University of Kentucky's Alzheimer's Disease Research Center.