Ordinal outcomes: A cumulative probability model with the log link and an assumption of proportionality

We present here a study of ordinal outcomes with a cumulative probability model. In particular, we consider the log link with the assumption of proportionality. The logit link is currently the most widely used link with ordinal outcomes in the health research literature. With the logit link, one obtains regression coefficients that are functions of odds. When the log link is used, one obtains regression coefficients that are functions of probabilities. While odds might be preferred with certain studies that are retrospective, many health researchers may prefer to have direct statements about probabilities. There are two classes of models with the log link analogous to those with the logit link. We will call these two classes the Proportional Probability Model (PPM) and the Log Cumulative Probability Model (LCPM). These models introduce a challenge not seen with the logit link models. The log link models have constraints on the parameter space. We must insist that the maximum likelihood estimate (MLE) satisfy these constraints. We present conditions for the uniqueness of the MLE and we present other features of the MLE. Several examples and several closed form expressions for the MLE are presented. While this paper is primarily about the PPM, our R package lcpm contains functions to fit both the PPM and the LCPM.