A nonstationary Markov transition model for computing the relative risk of dementia before death

This paper investigates the long-term behavior of the k-step transition probability matrix for a nonstationary discrete-time Markov chain in the context of modeling transitions from intact cognition to dementia with mild cognitive impairment and global impairment as intervening cognitive states. The authors derive formulas for the following absorption statistics: (1) the relative risk of absorption between competing absorbing states and (2) the mean and variance of the number of visits among the transient states before absorption. As absorption is not guaranteed, sufficient conditions are discussed to ensure that the substochastic matrix associated with transitions among transient states converges to zero in limit. Results are illustrated with an application to the Nun Study, a cohort of 678 participants, 75-107 years of age, followed longitudinally with up to 10 cognitive assessments over a 15-year period.