A novel sandwich algorithm for empirical Bayes analysis of rank data

Rank data arises frequently in marketing, finance, organizational behavior, and psychology. Most analysis of rank data reported in the literature assumes the presence of one or more variables (sometimes latent) based on whose values the items are ranked. In this paper we analyze rank data using a purely probabilistic model where the observed ranks are assumed to be perturbed versions of the true rank and each perturbation has a specific probability of occurring. We consider the general case when covariate information is present and has an impact on the rankings. An empirical Bayes approach is taken for estimating the model parameters. The Gibbs sampler is shown to converge very slowly to the target posterior distribution and we show that some of the widely used empirical convergence diagnostic tools may fail to detect this lack of convergence. We propose a novel, fast mixing sandwich algorithm for exploring the posterior distribution. An EM algorithm based on Markov chain Monte Carlo (MCMC) sampling is developed for estimating prior hyperparameters. A real life rank data set is analyzed using the methods developed in the paper. The results obtained indicate the usefulness of these methods in analyzing rank data with covariate information.

Keywords

algebraic–probabilistic model, convergence, Gibbs sampler, Markov chains, MCMC, permutation, rank data, sandwich algorithm

2010 Mathematics Subject Classification

Primary 62F07, 62F15. Secondary 60J05.

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Published 30 May 2017