Time-varying rankings with the Bayesian Mallows model

Abstract

We present new statistical methodology for analysing rank data, where the rankings are allowed to vary in time. Such data arise, for example, when the assessments are based on a performance measure of the items, which varies in time, or if the criteria, according to which the items are ranked, change in time. Items can also be absent when the assessments are made, because of delayed entry or early departure, or purely randomly. In such situations, also the dimension of the rank vectors varies in time. Rank data in a time-dependent setting thus lead to challenging statistical problems. These problems are further complicated, from the perspective of computation, by the large dimension of the sample space consisting of all permutations of the items. Here, we focus on introducing and developing a Bayesian version of the Mallows rank model, suitable for situations in which the ranks vary in time and the assessments can be incomplete. The consequent missing data problems are handled by applying Bayesian data augmentation within Markov chain Monte Carlo. Our method is also adapted to the task of future rank prediction. The method is illustrated by analysing some aspects of a data set describing the academic performance, measured by a series of tests, of a class of high school students over a period of 4 years. Copyright © 2016 John Wiley & Sons, Ltd.