Correcting for measurement error under meta-analysis of z-transformed correlations

Abstract

This study mainly concerns correction for measurement error using the meta-analysis of Fisher's z-transformed correlations. The disattenuation formula of Spearman (American Journal of Psychology, 15, 1904, 72) is used to correct for individual raw correlations in primary studies. The corrected raw correlations are then used to obtain the corrected z-transformed correlations. What remains little studied, however, is how to best correct for within-study sampling error variances of corrected z-transformed correlations. We focused on three within-study sampling error variance estimators corrected for measurement error that were proposed in earlier studies and is proposed in the current study: (1) the formula given by Hedges (Test validity, Lawrence Erlbaum, 1988) assuming a linear relationship between corrected and uncorrected z-transformed correlations (linear correction), (2) one derived by the first-order delta method based on the average of corrected z-transformed correlations (stabilized first-order correction), and (3) one derived by the second-order delta method based on the average of corrected z-transformed correlations (stabilized second-order correction). Via a simulation study, we compared performance of these estimators and the sampling error variance estimator uncorrected for measurement error in terms of estimation and inference accuracy of the mean correlation as well as the homogeneity test of effect sizes. In obtaining the corrected z-transformed correlations and within-study sampling error variances, coefficient alpha was used as a common reliability coefficient estimate. The results showed that in terms of the estimated mean correlation, sampling error variances with linear correction, the stabilized first-order and second-order corrections, and no correction performed similarly in general. Furthermore, in terms of the homogeneity test, given a relatively large average sample size and normal true scores, the stabilized first-order and second-order corrections had type I error rates that were generally controlled as well as or better than the other estimators. Overall, stabilized first-order and second-order corrections are recommended when true scores are normal, reliabilities are acceptable, the number of items per psychological scale is relatively large, and the average sample size is relatively large.