Finite Markov chains with absorbing states and mis-specified random effects: application to cognitive data

Finite Markov chains with absorbing states are valuable tools for analyzing longitudinal data with categorical responses. However, defining the one-step transition probabilities in terms of fixed and random effects presents challenges due to the large number of unknown parameters involved. To address this, we employ a marginal model to estimate the fixed effects across various choices of the distribution governing the random effects. Subsequently, we utilize an h-likelihood method to estimate the random effects based on these fixed effect estimates. The estimation approach is applied to analyze longitudinal cognitive data from the Nun Study. Our findings highlight that the fixed effects remain relatively robust across a wide range of assumptions. However, the analysis of random effects utilizing tools such as AIC, Q-Q plots, and gradient plots appears to be sensitive to mis-specifications in the distribution of the random effects. Our proposed approach allows researchers to verify the assumptions of random effects and provides more accurate estimation of these effects. Additionally, the precisely estimated random effects enable researchers to identify individuals at high risk for absorbing states (e.g. incurable diseases) and to determine the progression rates for certain diseases.