| In a clinical trial of a treatment for alcoholism, the usual response variable of interest is the number of alcoholic drinks consumed by each subject each day. Subjects in these trials are typically volunteers who are alcoholics, and thus are prone to erratic drinking behaviors, often characterized by alternating periods of heavy drinking and abstinence. Simple statistical models often fail to capture the complicated nature of drinking behavior as it changes through time. We propose to describe subjects' drinking behavior using a Hidden Markov model (HMM) for ordinal data, where the counts of drinks per day are summarized as an ordinal variable with three levels, as is the convention in alcohol research. The hidden states correspond to underlying states of health/drinking behavior. We first fit a number of HMMs to data without incorporating covariates, using the Baum-Welch algorithm. We compare predictions made by the HMMs to those made by other models, and we also discuss the clinical interpretations of parameter estimates from the HMMs. The HMMs describe the most common patterns of drinking among the subjects, and can be used to address two relevant questions from the alcoholism literature having to do with the danger of moderate drinking and the definition of a relapse, respectively. Next, we use Bayesian methods to fit an HMM that incorporates covariates---most importantly, the treatment effect. We use the Forward-Backward Gibbs Sampler to do this, using a multinomial logit model for hidden state transitions, and an ordinal probit model for observations. The effects of treatments on the hidden state transition matrix and stationary distribution can be measured. Lastly, we discuss the problem of choosing starting points when fitting an HMM using the Baum-Welch algorithm. When the parameter space is large, the grid of starting points can be sparse, allowing for the global maximum to go undiscovered. We use the entropy of the marginal state distributions to devise a scheme for choosing starting points that may reach the global maximum somewhat reliably. |
