## Abstract

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.

Original language | English |
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Pages (from-to) | 2650-2664 |

Number of pages | 15 |

Journal | Statistics in Medicine |

Volume | 40 |

Issue number | 11 |

DOIs | |

State | Published - May 20 2021 |

### Bibliographical note

Publisher Copyright:© 2021 John Wiley & Sons, Ltd.

### Funding

information National Center for Advancing Translational Sciences, Ul11 TR001998; National Institute on Aging, AG028383; AG0386561This research was partially supported by grant UL11 TR001998 from the National Center for Advancing Translational Sciences and grants AG0386561 and AG028383 from the National Institute on Aging.

Funders | Funder number |
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National Institute on Aging | AG028383, AG0386561, P30AG072946 |

National Institute on Aging | |

National Center for Advancing Translational Sciences (NCATS) | Ul11 TR001998 |

National Center for Advancing Translational Sciences (NCATS) |

## Keywords

- Markov chains
- cognitive assessments
- dementia
- multinomial logistic regression
- reduced rank

## ASJC Scopus subject areas

- Epidemiology
- Statistics and Probability