Convert Cohen's d to a point-biserial correlation

Converts a standardized mean difference to the point-biserial correlation implied by a dichotomous criterion with base rate \(p\). The implemented formula is \(r_{pb} = d\sqrt{p(1-p)} / \sqrt{1 + d^2p(1-p)}\). When base_rate = .50, this reduces to the common equal-group conversion \(r = d / \sqrt{d^2 + 4}\).

Usage

d_to_point_biserial(d, base_rate = 0.5)

Arguments

d

Cohen's d. Must be numeric and finite.

base_rate

Proportion in the focal or successful group, usually denoted \(p\). Must be in (0, 1). The default is .50.

Value

Numeric vector of point-biserial correlations.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Erlbaum.

Rice, M. E., & Harris, G. T. (2005). Comparing effect sizes in follow-up studies: ROC area, Cohen's d, and r. Law and Human Behavior, 29(5), 615-620.

Salgado, J. F. (2018). Transforming the area under the normal curve (AUC) into Cohen's d, Pearson's r_pb, odds-ratio, and natural log odds-ratio: Two conversion tables. The European Journal of Psychology Applied to Legal Context, 10(1), 35-47.

Examples

# Minimal example: equal base-rate conversion equals d_to_cor().
d_to_point_biserial(.50, base_rate = .50)
#> [1] 0.2425356
d_to_cor(.50)
#> [1] 0.2425356

# Unequal base rates reduce the attainable point-biserial correlation.
d_to_point_biserial(.50, base_rate = c(.50, .20, .10))
#> [1] 0.2425356 0.1961161 0.1483405