Maximum Likelihood Estimation in Mallows’s Model Using Partially Ranked Data

Consider a sample from a population in which each individual is characterized by a ranking on k items, but only partial information about the ranking is available for the individuals in the sample. The problem is to estimate the population distribution of rankings, given the partially ranked data. This paper proposes use of an EM algorithm to obtain maximum likelihood estimates of the parameters in Mallows’s model for the distribution of rankings. Medical applications are discussed where the items are manifestations of a disease or a developmental process, the ranking is the sequence in which they first appear over time, and the partial ranking results from observation of the subjects cross-sectionally or at a few specified times. The methods are illustrated for a longitudinal study of a community population aged 65 years and older, where the signs are self-reporting of impairment in different physical activities.