Bootstrap standard error and confidence intervals for the correlations corrected for indirect range restriction

Abstract

The standard Pearson correlation coefficient, r, is a biased estimator of the population correlation coefficient, ρ(XY) , when predictor X and criterion Y are indirectly range-restricted by a third variable Z (or S). Two correction algorithms, Thorndike's (1949) Case III, and Schmidt, Oh, and Le's (2006) Case IV, have been proposed to correct for the bias. However, to our knowledge, the two algorithms did not provide a procedure to estimate the associated standard error and confidence intervals. This paper suggests using the bootstrap procedure as an alternative. Two Monte Carlo simulations were conducted to systematically evaluate the empirical performance of the proposed bootstrap procedure. The results indicated that the bootstrap standard error and confidence intervals were generally accurate across simulation conditions (e.g., selection ratio, sample size). The proposed bootstrap procedure can provide a useful alternative for the estimation of the standard error and confidence intervals for the correlation corrected for indirect range restriction.