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A new method for approximating the asymptotic variance of Spearman’s rank correlation. (English) Zbl 0930.62100
Summary: Epidemiologists use Spearman’s rank correlation, $$\widehat \rho_s$$, and the quantile correlation, $$\widehat\rho_q$$, to measure the agreement between the bivariate ranks and the bivariate quantile-categories of bivariate continuous data, respectively. In this paper we explore the relationship between the finite and asymptotic means and variances of these statistics. We show that the asymptotic means and variances of $$\widehat \rho_q$$ converge to the same limits as those of $$\widehat \rho_s$$, as the number of quantile-categories increases. Also, these point estimates have distributions derived from the “empirical bivariate quantile-partitioned” (EBQP) distribution [C. B. Borkowf, M. H. Gail, R. Carroll and R. D. Gill, Biometrics 53, No. 3, 1054-1069 (1997; Zbl 0896.62114)], so we can use nonparametric EBQP methods to estimate the finite variances of these statistics from data and to compute the asymptotic variance of $$\widehat\rho_q$$ for any underlying bivariate distribution that satisfies certain regularity conditions. These results imply that we can numerically approximate the asymptotic variance of $$\widehat\rho_s$$, for which an explicit formula is not available except in special cases, to a degree of accuracy limited only by computing power.

##### MSC:
 62P10 Applications of statistics to biology and medical sciences; meta analysis 62G05 Nonparametric estimation 62H20 Measures of association (correlation, canonical correlation, etc.)