Partially Ordered Mixed Hidden Markov Model for the Disablement Process of Older Adults

Abstract

At both the individual and societal levels, the health and economic burden of disability in older adults is enormous in developed countries, including the U.S. Recent studies have revealed that the disablement process in older adults often comprises episodic periods of impaired functioning and periods that are relatively free of disability, amid a secular and natural trend of decline in functioning. Rather than an irreversible, progressive event that is analogous to a chronic disease, disability is better conceptualized and mathematically modeled as states that do not necessarily follow a strict linear order of good to bad. Statistical tools, including Markov models, which allow bidirectional transition between states, and random effects models, which allow individual-specific rate of secular decline, are pertinent. In this article, we propose a mixed effects, multivariate, hidden Markov model to handle partially ordered disability states. The model generalizes the continuation ratio model for ordinal data in the generalized linear model literature and provides a formal framework for testing the effects of risk factors and/or an intervention on the transitions between different disability states. Under a generalization of the proportional odds ratio assumption, the proposed model circumvents the problem of a potentially large number of parameters when the number of states and the number of covariates are substantial. We describe a maximum likelihood method for estimating the partially ordered, mixed effects model and show how the model can be applied to a longitudinal dataset that consists of N = 2903 older adults followed for 10 years in the Health Aging and Body Composition Study. We further statistically test the effects of various risk factors upon the probabilities of transition into various severe disability states. The result can be used to inform geriatric and public health science researchers who study the disablement process. Supplementary materials for this article are available online.

KEY WORDS:

• Continuation ratio model
• EM algorithm
• Generalized linear model
• Health ABC study
• Latent Markov model

Acknowledgments

The research was supported by the following grants from the National Institutes of Health: R01AG031827A, U01HL101066-01, and P30 AG21332.

Notes

^a130 ⩽ SBP < 140 and DBP < 90 or SBP < 140 and 85 ⩽ DBP < 90, where SBP stands for the systolic blood pressure and DBP stands for the diastolic blood pressure.

^bSBP ⩾ 140 or DBP ⩾ 90.

^a value indicates the transition probability of row to column.

^aThe mixed model is based on Equation (11) such that the log conditional odds where the random slope θ _i ∼ N(0, σ²), and φ _rs and x_it are, respectively, the intercept and the fixed effects. Mean imputation was used for missing values in the predictor variables.