rankbiserial

Rankbiserial, or rank-biserial correlation, is a nonparametric measure of effect size used to quantify the association between a binary variable and a continuous or ordinal variable. It is most commonly applied in conjunction with the Wilcoxon rank-sum test (Mann-Whitney U test) to describe the magnitude of group differences.

Calculation typically involves two groups with sizes n1 and n2. Let U be the Mann-Whitney statistic; using the smaller of the two U values, the rankbiserial correlation rb is defined as rb = 1 − 2U/(n1 n2). The statistic ranges from -1 to 1: rb = 1 indicates complete separation with all observations in group 1 exceeding those in group 2, rb = -1 the reverse, and rb = 0 no systematic difference.

An equivalent formulation uses sums of ranks. If R1 is the sum of ranks for group 1 and R2 for group 2, rb can also be expressed as rb = (R1 − R2)/(n1 n2). Both expressions yield the same value under appropriate tie handling.

Interpretation and usage: Rankbiserial provides a standardized measure of effect size for nonparametric comparisons. Values closer to -1 or 1 indicate stronger group separation, while values near 0 suggest little to no difference. The interpretation is context-dependent and often compared with other effect sizes.

Limitations and notes: The calculation assumes independent observations and two groups. Ties and unequal sample sizes require standard tie corrections. While informative about magnitude, rankbiserial does not replace hypothesis testing and should be reported alongside p-values from the Wilcoxon test.