## Introduction

Categorical variables are common in biomedical and psychosocial studies. For regression analysis of a binary response, logistic regression models may be the most popular. In a logistic model, the coefficients can be easily interpreted in terms of odds ratios (ORs). For an ordinal response where the response levels are ordered, it is common to model it through cumulative probabilities. In other words, the response is dichotomised based on the order using all possible cutpoints, and then regression models are applied to the resulting binary responses. More precisely, suppose the ordinal response levels of an ordinal response Y are labelled as according to their order, then for each , we may dichotomise the outcome into two groups: if and if These dichotomised binaries all together convey the original level. Using the binaries allows us to model the ordinal outcome using models for binaries such as logistic models.

Let be the cumulative probability for the response to take a level up to j , then a cumulative logistic regression model can be specified as

where x is the vector of independent variables. It is commonly assumed that all the in model (1) are the same, resulting in the following proportional odds model:

The aforementioned equation is also called a model with parallel or equal slopes.1

Under model (2), for any two subjects with independent variables and , the OR

is independent of the cut point
j
. This property is called the *proportional odds property*, and model (2) is called a *proportional odds model*. This proportional odds property comes from the assumption that all are the same. The proportional odds model may be the most popular model for ordinal response; however, the proportional odds assumption may be too strong. Thus, it is generally desired to test the proportional odds assumptions.

Based on model (1), the null hypothesis for testing the proportional odds assumption is given by

Score, Wald and likelihood ratio (LR) tests may all be applied to the hypothesis test. However, model (1) may not be estimable; when the coefficients of the covariates are different across different levels, the fitted probabilities for some levels may be negative. To overcome the issue, Brant proposed an approach which first estimates separately based on the dichotomised binary responses and then compares the estimates through a Wald-like statistic based on their joint asymptotic distribution.2 Wolfe and Gould generalised the idea of obtaining an LR test.3

In the section ‘Testing the proportional odds assumption’, we provide a brief description of these tests, as well as their availability in R, SAS, SPSS and Stata. In the section ‘Simulation studies’, simulation studies are carried out to assess the performances of these tests. Finally, a real data example is given in the section ‘Examples’, and the paper concludes with a discussion.