Bootstrap Standard Error and Confidence Intervals for the Correlation Corrected for Range Restriction: A Simulation Study.

Abstract

The standard Pearson correlation coefficient is a biased estimator of the true population correlation, rho, when the predictor and the criterion are range restricted. To correct the bias, the correlation corrected for range restriction, rc, has been recommended, and a standard formula based on asymptotic results for estimating its standard error is also available. In the present study, the bootstrap standard-error estimate is proposed as an alternative. Monte Carlo simulation studies involving both normal and nonnormal data were conducted to examine the empirical performance of the proposed procedure under different levels of rho, selection ratio, sample size, and truncation types. Results indicated that, with normal data, the bootstrap standard-error estimate is more accurate than the traditional estimate, particularly with small sample size. With nonnormal data, performance of both estimates depends critically on the distribution type. Furthermore, the bootstrap bias-corrected and accelerated interval consistently provided the most accurate coverage probability for rho.

Standard errors and confidence intervals for correlations corrected for indirect range restriction: A simulation study comparing analytic and bootstrap methods

“…The number of bootstrap replications (needed for the computation of SE B , BSI, and BPI) was B = 2,000 (Chan & Chan, 2004;Li et al, 2011).…”

“…We determined the acceptability of the different evaluation criteria as follows (Chan & Chan, 2004;Li et al, 2011): For Bias R , Bias SE B , and Bias SE F , the parameter estimate was considered excellent if Bias was within AE5% and reasonable if Bias was within AE10%. For the coverage probability of the CIs, a coverage probability falling within [.92, .97], which allows for sampling error, may be considered acceptable (Li et al, 2011).…”

“…Consistent with this standpoint, the present study adds to and refines the existing body of knowledge regarding the best way to estimate standard errors (SEs) and confidence intervals (CIs) for correlations corrected for IRR. Chan and Chan (2004) developed a bootstrap procedure for estimating SEs and CIs for correlations corrected for DRR. They compared its empirical performance with the performance of two formulas for the SEs (Bobko & Rieck, 1980;Fisher, 1954) and the respective CIs of the correlations in simulation studies involving both normal and nonnormal data.…”

“…A bootstrap procedure for estimating the standard error and confidence intervals On the basis of studies conducted by Chan and Chan (2004) and Li et al (2011), we used a bootstrap procedure for estimating the SE and two types of CIs. A detailed description of the bootstrap procedure can be found in Chan and Chan (2004) and Li et al (2011).…”

“…A bootstrap procedure for estimating the standard error and confidence intervals On the basis of studies conducted by Chan and Chan (2004) and Li et al (2011), we used a bootstrap procedure for estimating the SE and two types of CIs. A detailed description of the bootstrap procedure can be found in Chan and Chan (2004) and Li et al (2011). Generally, the data observed under IRR are composed of two subsamples: a selected (restricted) subsample of size n r with scores on all three variables (Z r , X r , Y r ) and an unselected subsample of size n u with scores on the selection variable (Z u ) only.…”