For example, many studies investigate pairwise associations between large numbers of biomarkers to better understand biological processes. When dealing with ordered categorical data, nonlinear relationships, skewed distributions, and extreme values, rank correlation coefficients such as Spearman’s rho or Kendall’s tau are preferred. For symmetrically distributed continuous variables, a common choice is Pearson’s correlation coefficient. Although correlation coefficients have limitations (e.g., an inability to accurately describe non-monotonic relationships), their continued popularity is due in part to their simplicity and interpretability. To this end, associations are frequently described using correlation coefficients, which, well over a century after their introduction, remain popular in practice. It is often of interest to summarize the degree of association between two variables using a single number. We illustrate our method in two applications, a biomarker study and a large survey. 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 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. Our conditional estimator is the conditional correlation of PSRs. 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. This connection allows us to derive simple estimators. We show that they can be neatly expressed using probability-scale residuals (PSRs). The definitions are natural extensions of Spearman’s rank correlation in the presence of covariates and are general for any orderable random variables. We define population parameters for both partial and conditional Spearman’s correlation through concordance-discordance probabilities. 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. It is desirable to adjust Spearman’s rank correlation for covariates, yet existing approaches have limitations.
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