Interval estimation for rank correlation coefficients based on the probit transformation with extension to measurement error correction of correlated ranked data

Publisher Website
Google Scholar
Abstract
The Spearman (ρs) and Kendall (τ) rank correlation coefficient are routinely used as measures of association between non‐normally distributed random variables. However, confidence limits for ρs are only available under the assumption of bivariate normality and for τ under the assumption of asymptotic normality of . In this paper, we introduce another approach for obtaining confidence limits for ρs or τ based on the arcsin transformation of sample probit score correlations. This approach is shown to be applicable for an arbitrary bivariate distribution. The arcsin‐based estimators for ρs and τ (denoted by s,a, a) are shown to have asymptotic relative efficiency (ARE) of 9/π2 compared with the usual estimators s and when ρs and τ are, respectively, 0. In some nutritional applications, the Spearman rank correlation between nutrient intake as assessed by a reference instrument versus nutrient intake as assessed by a surrogate instrument is used as a measure of validity of the surrogate instrument. However, if only a single replicate (or a few replicates) are available for the reference instrument, then the estimated Spearman rank correlation will be downwardly biased due to measurement error. In this paper, we use the probit transformation as a tool for specifying an ANOVA‐type model for replicate ranked data resulting in a point and interval estimate of a measurement error corrected rank correlation. This extends previous work by Rosner and Willett for obtaining point and interval estimates of measurement error corrected Pearson correlations. Copyright © 2006 John Wiley & Sons, Ltd.