Modelling of repeated ordered measurements by isotonic sequential regression

Abstract

This article introduces a simple model for repeated observations of an ordered categorical response variable which is isotonic over time. It is assumed that the measurements represent an irreversible process such that the response at time t is never lower than the response observed at the previous time point t − 1. Observations of this type occur, for example, in treatment studies when improvement is measured on an ordinal scale. As the response at time t depends on the previous outcome, the number of ordered response categories depends on the previous outcome leading to severe problems when simple threshold models for ordered data are used. To avoid these problems, the isotonic sequential model is introduced. It accounts for the irreversible process by considering the binary transitions to higher scores and allows a parsimonious parameterization. It is shown how the model may easily be estimated using existing software. Moreover, the model is extended to a random effects version which explicitly takes heterogeneity of individuals and potential correlations into account.