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Biometrics. 2018 Jun;74(2):595-605. doi: 10.1111/biom.12812. Epub 2017 Nov 13.

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

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Merck, Rahway, New Jersey, U.S.A.
Department of Population and Quantitative Health Sciences, Case Western Reserve University, Cleveland, Ohio, U.S.A.
Departments of Epidemiology and Global Health, University of Washington, Seattle, Washington, U.S.A.
Department of Biostatistics, Vanderbilt University School of Medicine, Nashville, Tennessee, U.S.A.


It is desirable to adjust Spearman's rank correlation for covariates, yet existing approaches have limitations. For example, the traditionally defined partial Spearman's correlation does not have a sensible population parameter, and the conditional Spearman's correlation defined with copulas cannot be easily generalized to discrete variables. We define population parameters for both partial and conditional Spearman's correlation through concordance-discordance probabilities. The definitions are natural extensions of Spearman's rank correlation in the presence of covariates and are general for any orderable random variables. We show that they can be neatly expressed using probability-scale residuals (PSRs). This connection allows us to derive simple estimators. Our partial estimator for Spearman's correlation between X and Y adjusted for Z is the correlation of PSRs from models of X on Z and of Y on Z, which is analogous to the partial Pearson's correlation derived as the correlation of observed-minus-expected residuals. Our conditional estimator is the conditional correlation of PSRs. We describe estimation and inference, and highlight the use of semiparametric cumulative probability models, which allow preservation of the rank-based nature of Spearman's correlation. We conduct simulations to evaluate the performance of our estimators and compare them with other popular measures of association, demonstrating their robustness and efficiency. We illustrate our method in two applications, a biomarker study and a large survey.


Conditional correlation; Cumulative probability model; Partial correlation; Rank-based statistic; Semiparametric transformation model

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