Applications and Advances of Regime Switching Models

Multi-state models offer a versatile framework for modelling complex event history data, including panel observed data subject to misclassification error and left truncation. The state occupied by a patient is governed by a continuous-time discrete-state stochastic process, defined by transition intensities. The main computational challenge to fitting the models to panel observed data is the evaluation of the transition probabilities. Under a Markov assumption, subject-specific transition probabilities can be computed by directly solving the Kolmogorov Forward equations as a system of ordinary differential equations. However, the presence of continuous covariates makes this approach computationally unattractive for large datasets since a separate system must be solved for each unique covariate pattern in the data. To address this issue, the transition probabilities can be considered as a function of both time (for instance age) and continuous covariates (for instance calendar date of birth), where the function can be defined as the solution to a system of partial differential equations (PDEs). Standard approaches for obtaining approximate solutions to PDEs can then be applied to obtain solutions.

The method is illustrated through modelling of cognitive decline in the elderly population using the Health and Retirement Study (HRS). A four-state progressive process is assumed, consisting of normal cognition, mild cognitive impairment, dementia and death, where backward transitions in the observed trajectories are accounted for by allowing misclassification to adjacent cognition states. HRS aims to have a representative sample of the older population leading to subjects entering the study at various ages over 50. To account for this late entry, subjects can be assumed to be representative of living subjects of the same age and covariate pattern, with their initial state distribution inferred from the state distribution at age 50 and the transition probabilities between age 50 and their age at entry. However, HRS has run for over 30 years meaning cohort effects are present which affects the validity of standard left-truncation assumptions. To accommodate the cohort effect, calendar time is introduced as a second timescale which affects the magnitude of transition intensities and interacts with key covariates. The resulting model can be used to explore trends in prevalence and incidence of cognitive impairment.
 

Osteoarthritis (OA) is a prevalent degenerative joint disease, with the knee being the most commonly affected joint. Modern studies of knee joint injury and OA often measure biomechanical variables, particularly forces exerted during walking. However, the relationship among gait patterns, clinical profiles, and OA disease remains poorly understood. These biomechanical forces are typically represented as curves over time, but until recently, studies have relied on discrete values (or landmarks) to summarize these curves. This work aims to demonstrate an approach for analyzing full movement curves that capture key features of the gait cycle. Using data from the Intensive Diet and Exercise for Arthritis (IDEA) study, we developed a shape-based representation of variation in the full biomechanical curves and leveraged data integration methods to understand the shared variation in movement curves patterns with clinical traits. This talk will cover details of the nonparametric methods used in our approach and summarize key clinical discoveries. 

Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations -- a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs).