This correction is mathematically equivalent to correcting the correlation for direct range restriction in the split variable.
correct_r_split(r, pi, pa = 0.5, n = NULL)Vector of correlations affected by an uneven or unrepresentative split of a dichotomous variable.
Vector of proportions of incumbent/sample cases in one of the categories of the dichotomous variable.
Vector of proportions of applicant/population cases in one of the categories of the dichotomous variable.
Optional vector of sample sizes.
Vector of correlations corrected for unrepresentative splits (if n is supplied, corrected error variance and adjusted sample size is also reported).
$$r_{c}=\frac{r_{obs}}{u\sqrt{\left(\frac{1}{u^{2}}-1\right)r_{obs}^{2}+1}}$$ where \(u=\sqrt{\frac{p_{i}(1-p_{i})}{p_{a}(1-p_{a})}}\), the ratio of the dichotomous variance in the sample (\(p_{i}\) is the incumbent/sample proportion in one of the two groups) to the dichotomous variance in the population (\(p_{a}\) is the applicant/population proportion in one of the two groups). This correction is identical to the correction for univariate direct range restriction, applied to a dichotomous variable.
Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: SAGE. https://doi.org/10/b6mg. pp. 287-288.
# NOT RUN {
correct_r_split(r = 0.3, pi = .9, pa = .5, n = 100)
# }
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