An Empirical Bayes Model for Markov-Dependent Binary Sequences with Randomly Missing Observations

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.We develop an improved empirical Bayes estimation methodology for the analysis of two-state Markov chains observed from heterogeneous individuals. First, the two transition probabilities corresponding to each chain are assumed to be drawn from a common, bivariate distribution that has beta marginals. Second, randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations. A likelihood ratio test is used to test for dependence between the transition probabilities. Posterior distributions for the transition probabilities are also derived, as is an approximation for the equilibrium probabilities. The proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data. Management, McGill University, for preparing the display data set for our use. The authors express their appreciation to an associate editor and two referees for their helpful comments.Their procedure uses a prior distribution that assumes that the rows of the transition matrix are stochastically inde- pendent. Bayes decision methods for finite-state Markov chains with complete observations have been discussed by Martin (1967) and Basawa and Prakasa Rao (1980). Maximum likelihood estimation with randomly missing data was discussed by Muenz and Rubinstein (1985) and Lee (1993).In this article we consider the case where some observations in each chain are missing at random (in the terminology of Rubin 1976) and the transition probabilities are drawn from a new family of bivariate beta prior distributions. That is, we develop extensions that allow for randomly missing data and for the rows of the transition matrix to be correlated. In Section 2 we develop an empirical Bayes procedure for the complete data case, and in Section 3 we extend the results to the missing-at-random case. We demonstrate the proposed procedures in Section 4 with a numerical example and an analysis of longitudinal data for store displays promoting a food product. Finally, in Section 5 we make comments on the usefulness of the proposed procedures and possible extensions. THE COMPLETE-DATA CASEWe begin by presenting the likelihood for the transition probabilities of a single, stationary Markov chain. We then introduce the Bayesian framework using a bivariate beta prior distribution and develop the empirical Bayes estimation procedure for the case where an ensemble of chains is observed. Likelihood for the Transition ProbabilitiesLet X -(X1,...,Xm+i) be a s...

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