Standard errors of correlations adjusted for incidental selection

Abstract

The standard error of correlations that have been adjusted for selection with commonly used formulas developed by Pearson (1903) was investigated. The major purposes of the study were (1) to provide large-sample approximations of the standard error of a correlation adjusted using the Pearson-Lawley three-variable correction formula; (2) to examine the standard errors of adjusted correlations under specific conditions; and (3) to compare various estimates of the standard errors under direct and indirect selection. Two theory-based large-sample estimates of the standard error of a correlation adjusted for indirect selection were developed using the delta method. These two estimates were compared to one another, to a bootstrap estimate, and to an empirical standard deviation of a series of adjusted correlations generated in a simulation study. The simulation study manipulated factors defined by sample size, selection ratio, underlying population distribution, and population correlations in situations that satisfied the basic assumptions of the Pearson-Lawley procedures. The results indicated that the large-sample and bootstrap estimates were very similar when the sample size was 500 and, in most cases, the simpler of the two large-sample approximations appears to offer a reasonable estimate of the standard error of an adjusted correlation without resorting to complex, computer-intensive approaches. Index terms: correlation coefficients, missing data, Pearson-Lawley corrections, selection, standard errors of correlations, validity studies.