Estimating a Population Distribution of Sequences of K Items from Cross‐Sectional Data

Consider a population in which each individual is characterized by a specific ordering of k items; this ordering might be the sequence in which k selected manifestations of impaired function or disease would appear over time. The order in which the signs appear may vary from person to person. In cross‐sectional data, however, all that is known is the specific signs present at the time of observation, with order of appearance unknown. The problem is to estimate the population distribution of the orderings on the set Sk of permutations of k integers, given only the observable partial information. This paper proposes use of an EM algorithm to estimate parameters for Mallows's model on Sk. A test of goodness of fit is proposed, and residual analyses are described which can identify patterns of model failure. The general approach is also applicable when there is more than one epoch at which observations are made. The methods are illustrated for a cross‐sectional study of a community population aged 65 years and over, where the signs are self‐reporting of impaired physical function.