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

School of Mathematics and Statistics Research Seminar

Speaker: A/Prof Bryan Shepherd
Time: Tuesday 2nd February 2016 at 11:00 AM - 12:00 PM
Location: Cotton Club, Cotton 350
Groups: "Mathematics" "Statistics and Operations Research"

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Abstract

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; the PSR can be written as E{sign(y,Y)} whereas the popular observed-minus-expected residual can be thought of as E(y-Y). Therefore, the PSR is useful in settings where differences are not meaningful (e.g., ordered categorical data) or where the expectation of the fitted distribution cannot be calculated (e.g., censored data). 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 is illustrated in two application examples: one looking at the association between workers' educational attainments and their wages after controlling for potential confounding variables, and a second application estimating pairwise correlations between responses to all questions in a large survey after adjusting for relevant demographic and community-level factors. This is joint work with Qi Liu and Chun Li.

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