Rank-Biserial Correlation Calculator

Convert the Mann-Whitney U statistic into the rank-biserial correlation, provide approximate confidence intervals, and interpret the effect size alongside the probability of superiority.

1. Enter Mann-Whitney Summary

Requires U statistic and sample sizes

Use the U value associated with group 1. If you have both, supply the smaller value and note the direction.

Direction affects the sign of the correlation.

Confidence interval uses the large-sample variance of the rank-biserial correlation.

Formula Reference

Rank-biserial correlation

\\[ r_{\text{rb}} = 1 - \frac{2U}{n_1 n_2} \\]

Positive values indicate a tendency for group 1 to have larger ranks (if U corresponds to group 1).

Variance approximation

\\[ \text{Var}(r_{\text{rb}}) \approx \frac{n_1 + n_2 + 1}{3 n_1 n_2} \\]

Derived from the variance of the Mann-Whitney U statistic under the null hypothesis.

Probability of superiority

\\[ P = \frac{U}{n_1 n_2} = \frac{r_{\text{rb}} + 1}{2} \\]

Represents the probability that a random observation from group 1 exceeds a random observation from group 2.

Step-by-Step Guide

  1. Obtain the Mann-Whitney U statistic and sample sizes for the two groups.
  2. Convert U to the rank-biserial correlation using the linear transformation.
  3. Estimate variance and a z-based confidence interval for large samples.
  4. Express the effect in terms of probability of superiority and qualitative magnitude.
  5. Note whether ties or small samples necessitate exact methods or bootstrap intervals.

References

  • Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11.IT.3.1.
  • McGrath, R. E., & Meyer, G. J. (2006). When effect sizes disagree: The case of r and d. Psychological Methods, 11(4), 386-401.
  • Fritz, C. O., Morris, P. E., & Richler, J. J. (2012). Effect size estimates: Current use, calculations, and interpretation. Journal of Experimental Psychology: General, 141(1), 2-18.