Rank Biserial Correlation

Explanation

Two variations on this coefficient are in circulation. One is used with a one-sample (OS) or paired samples (PS) Wilcoxon signed rank test, the other with the Wilcoxon rank sum test for independent samples (IS) (equal to the Mann-Whitney U test).The later has two variations that you might come across.

These are, in my opinion, an effect size measure and not really a correlation. The OS and PS expresses the difference between positive and negative ranks, as a proportion of the maximum possible rank. The version of the IS is twice the difference in average ranks, divided by the total sample size. The measure is derived from a similar approach as the Spearman correlation.

For the IS version there are two variations. Cureton (1956) was perhaps the first to mention this term and provided a formula. His formula actually yields the same result as Goodman-Kruskal gamma (Goodman & Kruskal, 1954). Glass (1965; 1966) also developed a formula, but only for cases when there are no ties between the two categories. His formula will yield the same result as Somers'd (1962) and Cliff delta (1993). Cureton (1968) responded to Glass and gave his formula in an alternative form. Willson (1976) showed the link with Cureton formula and the Mann-Whitney U statistic. For more details on this see the article from Rubia (2022).

Obtaining the Coefficient

Click here for one-sample

with Excel

Excel file from video: ES - Rank Biserial (One-Sample) (E).xlsm

with stikpetE

without stikpetE

with Flowgorithm

Flowgorithm file: FL-ESrankbisOS.fprg

with Python

Notebook from video: ES - Rank Biserial (One-Sample) (P).ipynb

with stikpetP

without stikpetP, with scipy or pandas

without libraries

with R

Notebook from video: ES - Rank Biserial (One-Sample) (R).ipynb

with stikpetR

without stikpetR

with SPSS

Formulas

The Rank Biserial Correlation, can be calculated using (Cureton, 1956, p. 288; King & Minium, 2008, p. 403):

$r_{rb} = \frac{4\times\left|R_{min} - \frac{R_{pos} + R_{neg}}{2}\right|}{n\times\left(n+1\right)} =\frac{\left|R_{pos} - R_{neg}\right|}{R} $

Where $n$ is the sample size, $R$ the sum of all ranks, $R_{neg}$ the sum of ranks of scores below the hypothesized median, $R_{pos}$ of the scores above, $R_{min}$ the minimum of those two, and $R$ the sum of all ranks.

Click here for independent-samples

with Excel

Excel file from video: ES - Rank-Biserial (ind samp) (E).xlsm

with stikpetE

without stikpetE

with Python

Notebook from video: ES - Rank-Biserial (ind samp) (P).ipynb

with stikpetP

without stikpetP

with R

Notebook from video: ES - Rank-Biserial (ind samp) (R).ipynb

with stikpetR

without stikpetR

with SPSS

Formulas

The formula from Cureton's version is (Cureton, 1956, p. 289):

$$r_{rb} = \frac{P - Q}{n_1 \times n_2 - \sum_{i=1}^c t_{i,1} \times t_{i,2}}$$

Symbols:

$n_i$, the number of scores in category i
If one category has two scores of 3 and the other has three scores of 3, then $t_1,1 = 2, t_1,2 = 3$, if the first category has also one score of 4 and the second has two scores of 4, then $t_2,1 = 1,t_2,2 = 2$, etc.
for each score in the first category, determine how many scores in the second are less, $P$ is then the sum of these.
for each score in the second category, determine how many scores in the first are less, $Q$ is then the sum of these.

The formula was also rewritten to (Cureton, 1968, p. 78):

$$r_{rb} = \frac{\bar{R}_1 - \left(n + 1\right)/2}{n_2/2 - B/n_1}$$

with:

$$B = \frac{\sum_{i=1}^c t_{i,1} \times t_{i,2}}{2}$$

The $\bar{R}_1$ is the average of the ranks of the first category.

Willson (1976, p. 298) adapted this to be calculated using the Mann-Whitney U value:

$$r_{rb} = \left(\frac{n_1 n_2}{n_1 n_2 - 2B}\right)\left(1 - \frac{2U}{n_1 n_2}\right)$$

Glass's version is (1965, p. 91; 1966, p. 626):

$$r_{rb} = \frac{2}{n}\left(\bar{R}_1 - \bar{R}_2\right)$$

Which is the same as Cliff delta, and can also be expressed using the Mann-Whitney U value (Cliff, 1993, p. 495):

$$r_{rb} = \frac{2U}{n_1 n_2} - 1$$

Interpretation

Since the Cureton version is the same as Goodman-Kruskal gamma, we could use rules-of-thumb from Goodman-Kruskal gamma for this version as shown in Table 1.

Table 1
Rule of thumb for Goodman-Kruskal gamma
\|r_rb\|	Blaikie (2003, p. 100)	Rea and Parker (2014, p. 229)	Metsämuuronen (2023, p. 17)
0.00 < 0.10	negligible	negligible	negligible
0.10 < 0.14	weak	low	negligible
0.14 < 0.30	weak	low	small
0.30 < 0.31	moderate	moderate	small
0.31 < 0.45			medium
0.45 < 0.60			large
0.60 < 0.62	strong	strong	large
0.62 < 0.75	strong	strong	very large
0.75 < 0.84	very strong	very strong	very large
84 or more	very strong	very strong	huge

The Glass version is the same as Cliff delta, so we could use rules-of-thumb for that one, as shown in Table 2

Table 2
Rule of thumb for Cliff delta
\|r_rb\|	Romano et al. (2006, p. 14)	Metsämuuronen (2023, p. 17)
0.000 < 0.110	negligible	negligible
0.110 < 0.147	negligible	small
0.147 < 0.280	small	small
0.280 < 0.330	small	medium
0.330 < 0.430	medium	medium
0.430 < 0.474	medium	large
0.474 or more	large	large

or as Somers d, as shown in Table 3

Table 3
Rule of thumb for Somers d
\|r_rb\|	Metsämuuronen (2023, p. 17)
0.000 < 0.13	negligible
0.13 < 0.29	small
0.29 < 0.43	medium
0.43 < 0.59	large
0.59 < 0.81	very large
0.81 or more	huge

Alternative, the Glass version can be converted to Cohen's d (Marfo & Okyere, 2019, p. 4), so rules of thumb of Cohen d can then also be used, or to Vargha-Delaney A (= CLES) and rules of thumb for that could be used.

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