Test of Association Between Two Ordinal Variables While Adjusting for Covariates

Abstract

We propose a new set of test statistics to examine the association between two ordinal categorical variables X and Y after adjusting for continuous and/or categorical covariates Z . Our approach first fits multinomial (e.g., proportional odds) models of X and Y , separately, on Z . For each subject, we then compute the conditional distributions of X and Y given Z . If there is no relationship between X and Y after adjusting for Z , then these conditional distributions will be independent, and the observed value of ( X , Y ) for a subject is expected to follow the product distribution of these conditional distributions. We consider two simple ways of testing the null of conditional independence, both of which treat X and Y equally, in the sense that they do not require specifying an outcome and a predictor variable. The first approach adds these product distributions across all subjects to obtain the expected distribution of ( X , Y ) under the null and then contrasts it with the observed unconditional distribution of ( X , Y ). Our second approach computes “residuals” from the two multinomial models and then tests for correlation between these residuals; we define a new individual-level residual for models with ordinal outcomes. We present methods for computing p -values using either the empirical or asymptotic distributions of our test statistics. Through simulations, we demonstrate that our test statistics perform well in terms of power and Type I error rate when compared to proportional odds models which treat X as either a continuous or categorical predictor. We apply our methods to data from a study of visual impairment in children and to a study of cervical abnormalities in human immunodeficiency virus (HIV)-infected women. Supplemental materials for the article are available online.