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Standard Errors of Correlations Adjusted for Incidental Selection

Allen, Nancy L.; Dunbar, Stephen B.
Publication Year:
Report Number:
ETS Research Report
Document Type:
Page Count:
Subject/Key Words:
Correlation, Error of Measurement, Pearson-Lawley Formula, Statistical Analysis, Validity Studies


This investigation examined the standard error of correlations which have been adjusted for selection with commonly used formulas developed by Pearson (1903). Specifically, it had three major purposes: (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 boot-strap 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 (1) sample size, (2) selection ratio, (3) underlying population distribution, and (4) 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, when the sample size was 100. On the basis of results of the simulation study, 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. (32pp.)

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