"Rank Biserial Correlation" in jamovi

[reason180]

Cureton (1956) wrote:

"A formula is developed for the correlation between a ranking (possibly including ties and a dichotomy, with limits which are always ± 1. This formula is shown to be equivalent both to Kendall's tau and Spearman's ρ."
[ Cureton, E. E. (1956). Rank-biserial correlation. Psychometrika, 3, 287–290. https://doi.org/10.1007/BF02289138 ]

But this appears not to be the case! Jamovi reports the effect-size for the Mann-Whitney U test as a "Rank Biserial Correlation." However, the result seems incorrect to me. I've used the Spearman's rho routine, and alternately have rank-transformed the data and then computed Pearson's r. They confirm, for example, that the rank biserial correlation between y = {3, 9, 6, 5, 7, 2} and x = {0, 1, 0, 1, 1, 0} is 0.683. Not 0.778, which is the value reported as the rank biserial correlation accompanying the Mann-Whitney U. See the attached jamovi file (Version 2.2.2).

Glass (1965) on the other hand seems to write that the rank biserial correlation is merely an estimate of what the Spearman rho would be if the dichotomous variable (in the rank biserial correlation) were the result of the forced dichotomization of a rank variable.

This seems to be a bit of a mess, doesn't it?

[reason180]

What I now see is that whereas Cureton (1956) claimed the the "rank biserial" correlation is equivalent to a Spearman's rho (with one of the two variables being a numerically coded, binary variable), Glass (1965) offered it as only an estimate of rho.

So I think I'll avoid the "rank biserial correlation" since I want to calculate the correlation between a set of ranks and a binary variable, and I want the actual rho (i.e., the actual Pearson's r for ranked values) rather than an estimate.

[nevestv]

I suspect Jamovi uses an ambiguous name for this Effect Size. When you apply it, the software refers a paper by Kerby (2014) that presents three different formulas to be used when you use nonparametric tests for comparing two groups:One by Wendt, which uses the U score from Mann-Whitney, described as: 1-(2*U)/(n1*n2), where n1 is the group 1 e n2 is the group 2 (before and after, case and control, etc.).
Another one which uses z score, from both mann-whitney and Wilcoxon's test: z divided by the root square of N, being N the total number of observations.
And the Simple Difference Formula, which could substitute both the above, based on ranks sum and subtractions (I'm too lazy to describe it here right now...The paper is much more instructive XP).
I've made some comparisons of effect size manualy and with the software and I believe the software applies the Wendt's formula.

[reason180]

In my own investigation it seems to me that jamovi's rank biserial output perfectly matches the result of applying the Mann-Whitney U method.

[MAgojam]

Hi there are some old posts here ...
https://forum.jamovi.org/viewtopic.php?f=5&t=1688&p=6123&hilit=dama#p6073
https://forum.jamovi.org/viewtopic.php?f=5&t=1688&p=6123&hilit=dama#p6117