Antedependence models for nonstationary categorical longitudinal data with ignorable missingness: likelihood‐based inference

Abstract

Time index-ordered random variables are said to be antedependent (AD) of order (p1 ,p2 , … ,pn ) if the kth variable, conditioned on the pk immediately preceding variables, is independent of all further preceding variables. Inferential methods associated with AD models are well developed for continuous (primarily normal) longitudinal data, but not for categorical longitudinal data. In this article, we develop likelihood-based inferential procedures for unstructured AD models for categorical longitudinal data. Specifically, we derive maximum likelihood estimators (MLEs) of model parameters; penalized likelihood criteria and likelihood ratio tests for determining the order of antedependence; and likelihood ratio tests for homogeneity across groups, time invariance of transition probabilities, and strict stationarity. We give closed-form expressions for MLEs and test statistics, which allow for the possibility of empty cells and monotone missing data, for all cases save strict stationarity. For data with an arbitrary missingness pattern, we derive an efficient restricted expectation-maximization algorithm for obtaining MLEs. We evaluate the performance of the tests by simulation. We apply the methods to longitudinal studies of toenail infection severity (measured on a binary scale) and Alzheimer's disease severity (measured on an ordinal scale). The analysis of the toenail infection severity data reveals interesting nonstationary behavior of the transition probabilities and indicates that an unstructured first-order AD model is superior to stationary and other structured first-order AD models that have previously been fit to these data. The analysis of the Alzheimer's severity data indicates that the antedependence is second order with time-invariant transition probabilities, suggesting the use of a second-order autoregressive cumulative logit model.