Biases of success rate differences shown in binomial effect size displays

The intent of a binomial effect size display (BESD) is to show "the [real-world] importance of [an] effect indexed by a correlation [r]" (R. Rosenthal, 1994, p. 242) by reexpressing this correlation as a success rate difference (SRD) (e.g., treatment group success rate-control group success rate). However, SRDs displayed in BESDs generally overestimate real-world SRDs implied by correlations of (a) dichotomous X and Y variables (phi coefficients), (b) dichotomous X and continuous Y variables (point-biserial coefficients [r(pb)S]), and (c) continuous X and Y variables (r(xy)s). Furthermore, overestimation biases are larger for r(xy)s than for r(pb)s. Differences in the sizes of biases linked to different correlations suggest that BESD SRDs reported for different correlations are not comparable. The stochastic difference index (N. Cliff, 1993; A. Vargha & H. D. Delaney, 2000) is recommended as an alternative to the BESD.