Bryan Shepherd, Ph.D., Associate Professor, Department of Biostatistics, Vanderbilt University

"Covariate-adjusted Spearman's rank correlation with probability-scale residuals"

It is often of interest to summarize the degree of association between two variables using a single number, the correlation coefficient.  When dealing with ordered categorical data, nonlinear correlation, skewed distributions, and/or outliers, rank correlation coefficients, such as Spearman's correlation are preferred. In many applications, it is desirable to adjust Spearman's correlation for covariates, yet existing approaches are ad hoc or problematic with discrete data.  We propose two new estimators for covariate-adjusted Spearman's rank correlations, partial and conditional, using probability-scale residuals (PSRs).  The PSR is defined as P(Y<y)-P(Y>y), where y is the observed outcome and Y is a random variable from the fitted distribution; we will briefly describe the use of PSRs for diagnostics.  Our partial Spearman's rank correlation is the correlation of PSRs from models of X on Z and Y on Z, which is analogous to the partial Pearson's correlation derived as the correlation of observed-minus-expected residuals. With PSRs obtained from semiparametric ordinal models, our estimators preserve the rank-based nature of Spearman's rank correlations.  We derive properties of our estimators, conduct simulations to evaluate their performance, and compare them with other popular measures of association, demonstrating their robustness and efficiency.  Our method will be illustrated using several relevant datasets.  This is joint work with Qi Liu and Chun Li.

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