### Department of Biostatistics Seminar/Workshop Series

# A New Residual for Ordinal Outcomes

## Chun Li, PhD

### Associate Professor, Department of Biostatistics, Vanderbilt University School of Medicine

### Wednesday, April 14, 1:30-2:30pm, MRBIII Conference Room 1220

Ordinal categorical variables are common in medical and social research. Ordinal regression models allow us to study the effect of covariates on ordinal outcomes. We propose a new residual for ordinal outcomes. Unlike some existing definitions of residuals, the new residual is individual based, and its sign reflects the direction of an observed outcome compared to its fitted distribution. In addition, each individual has a single residual irrespective of the number of categories of the ordinal outcome. We study the properties of this new residual and describe its connections with other residuals. The new residual is a useful tool for constructing test statistics and performing model diagnostics. When testing for association between two ordinal variables Y and X while adjusting for covariates Z, we can fit separate ordinal regression models of Y on Z and X on Z and construct test statistics using the residuals of these models. We describe some properties of these conditional ordinal-by-ordinal tests and their connections with Kendall's tau, Goodman and Kruskal's gamma, and Spearman's rank correlation. For ordinal regression models, the new residual allows us to carry out model diagnostics analogously to those for linear models. We demonstrate the use of these diagnostic tools using simulated and real data sets.