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Ezekiel's adjusted R 2 is widely used in linear regression analysis. The present study examined the statistical properties of Ezekiel's measure through a series of Monte Carlo simulations. Specifically, we examined the bias and root mean squared error (RMSE) of Ezekiel's adjusted R statistic, and (b) the sample 2 relative to (a) the sample R 2 statistic, and (b) the sample R 2 minus the expected value of R 2 . Simulation design factors consisted of sample sizes ( N  = 50, 100, 200, 400), number of predictors (2, 3, 4, 5, 6), and population squared multiple correlations ( ρ 2 = 0, .10, .25, .40, .60). Factorially crossing these design factors resulted in 100 simulation conditions. All populations were normal/Gaussian, and for each condition, we drew 10,000 Monte Carlo samples. Regarding systematic variation (bias), results indicated that with few exceptions, Ezekiel's adjusted R 2 demonstrated the lowest bias. Regarding unsystematic variation (RMSE), the performance of Ezekiel's measure was comparable to the other statistics, suggesting that the bias-variance tradeoff is minimal for Ezekiel's adjusted R 2 . Additional findings indicated that sample size-to-predictor ratios of 66.67 and greater were associated with low bias and that ratios of this magnitude were accompanied by large sample sizes ( N  = 200 and 400), thus suggesting that researchers using Ezekiel's adjusted R 2 should aim for sample sizes of 200 or greater in order to minimize bias when estimating the population squared multiple correlation coefficient. Overall, these findings indicate that Ezekiel's adjusted R 2 has desirable properties and, in addition, these findings bring needed clarity to the statistical literature on Ezekiel's classic estimator.

With nonnormal data, the typical confidence interval of the correlation (Fisher z') may be inaccurate. The literature has been unclear as to which of several alternative methods should be used instead, and how extreme a violation of normality is needed to justify an alternative. Through Monte Carlo simulation, 11 confidence interval methods were compared, including Fisher z', two Spearman rank-order methods, the Box-Cox transformation, rank-based inverse normal (RIN) transformation, and various bootstrap methods. Nonnormality often distorted the Fisher z' confidence interval-for example, leading to a 95 % confidence interval that had actual coverage as low as 68 %. Increasing the sample size sometimes worsened this problem. Inaccurate Fisher z' intervals could be predicted by a sample kurtosis of at least 2, an absolute sample skewness of at least 1, or significant violations of normality hypothesis tests. Only the Spearman rank-order and RIN transformation methods were universally robust to nonnormality. Among the bootstrap methods, an observed imposed bootstrap came closest to accurate coverage, though it often resulted in an overly long interval. The results suggest that sample nonnormality can justify avoidance of the Fisher z' interval in favor of a more robust alternative. R code for the relevant methods is provided in supplementary materials.

The authors conducted a Monte Carlo simulation of 8 statistical tests for comparing dependent zero-order correlations. In particular, they evaluated the Type I error rates and power of a number of test statistics for sample sizes (Ns) of 20, 50, 100, and 300 under 3 different population distributions (normal, uniform, and exponential). For the Type I error rate analyses, the authors evaluated 3 different magnitudes of the predictor-criterion correlations (rho(y,x1) = rho(y,x2) = .1, .4, and .7). For the power analyses, they examined 3 different effect sizes or magnitudes of discrepancy between rho(y,x1) and rho(y,x2) (values of .1, .3, and .6). They conducted all of the simulations at 3 different levels of predictor intercorrelation (rho(x1,x2) = .1, .3, and .6). The results indicated that both Type I error rate and power depend not only on sample size and population distribution, but also on (a) the predictor intercorrelation and (b) the effect size (for power) or the magnitude of the predictor-criterion correlations (for Type I error rate). When the authors considered Type I error rate and power simultaneously, the findings suggested that O. J. Dunn and V. A. Clark's (1969) z and E. J. Williams's (1959) t have the best overall statistical properties. The findings extend and refine previous simulation research and as such, should have greater utility for applied researchers.

The authors conducted a Monte Carlo simulation of 4 test statistics for comparing dependent correlations with no variables in common. Empirical Type I error rates and power estimates were determined for K. Pearson and L. N. G. Filon's (1898) z, O. J. Dunn and V. A. Clark's (1969) z, J. H. Steiger's (1980) original modification of Dunn and Clark's z, and Steiger's modification of Dunn and Clark's z using a backtransformed average z procedure for sample sizes of 10, 20, 50, and 100 under 3 different population distributions. For the Type I error rate analyses, the authors evaluated 3 different magnitudes of the predictor-criterion correlations (ρ12 = ρ34 = .10, .30, and .70). Likewise, for the power analyses, 3 different magnitudes of discrepancy or effect sizes between correlations with no variables in common (ρ12 and ρ14) were examined (values of .10, .40, and .60). All of the analyses were conducted at 3 different levels of predictor intercorrelation. Results indicated that the choice as to which test statistic is optimal, in terms of power and Type I error rate, depends not only on sample size and population distribution but also on (a) the predictor intercorrelations and (b) the effect size (for power) or the magnitude of the predictor-criterion correlations (for Type I error rate). For the conditions examined in the present study, Pearson and Filon's z had inflated Type I error rates when the predictor-criterion correlations were low to moderate. Dunn and Clark's z and the 2 Steiger procedures had similar, but conservative. Type I error rates when the predictor- criterion correlations were low. Moreover, the power estimates were similar among the 3 procedures. [ABSTRACT FROM AUTHOR]

Presents a Monte Carlo evaluation of four test statistics for comparing dependent zero-order correlations. Use of three different magnitudes of the predictor-criterion correlations in type I error; Factors that affect optimality of test statistic; Effect of size or magnitude on the predictor-criterion correlations.