### Department of Biostatistics Seminar/Workshop Series

# Test of Association between Two Ordinal Variables while Adjusting for Covariates

## Chun Li, PhD

### Assistant Professor, Department of Biostatistics

Vanderbilt University School of Medicine

### Wednesday, April 8, 1:30-2:30pm, MRBIII Conference Room 1220

### Intended Audience: Persons interested in applied statistics, statistical theory, epidemiology, health services research, clinical trials methodology, statistical computing, statistical graphics, R users or potential users

We consider the situation where there are two ordinal categorical random variables, X and Y, and we want to examine the association between X and Y after adjusting for continuous and/or categorical covariates Z. Although this is a common scenario in medical research and social sciences, there does not appear to be a good solution for this problem. Proportional odds model, or ordinal logistic regression, is commonly used to examine the association between an ordinal response and continuous or categorical predictors. However, when one of the predictors is ordered categorical, all traditional regression approaches including ordinal logistic regression have to treat the ordinal predictor as either numerical or categorial. The former enforces a linearity assumption and the latter ignores the order information. Existing solutions, either regression-based approaches such as splines and isotonic regression or nonparametric approaches such as partial tau and partial gamma, have undesirable properties.

We propose a new approach to this problem. We first fit proportional odds models of Y and X, separately, on Z. Then we construct test statistics using the conditional distributions Y|Z and X|Z. The significance of our test statistics can be evaluated either empirically using parametric bootstrap or asymptotically using estimating equations. We also define residuals for proportional odds models.

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