Rank-Biserial Correlation

Statistics Definitions > Rank-Biserial Correlation

What is Rank-Biserial Correlation?

Rank-biserial correlation (rank biserial r or r_rb) is used when you want to find a correlation between dichotomous (binary) nominal data and ordinal (ranked) data. It is a special case of Somers’ D.

Formula

The formula (assuming there are no tied ranks) is:
r_rb = 2 * (Y₁ – Y₀) / n.
Where:

n = number of data pairs in the sample,
Y₀ = Y score means for data pairs with x = 0,
Y₁ = Y score means for data pairs with x = 1.

For example, let’s say you had the following data:
Dichotomous variable: 1,1,1,0,1
Ordinal variable: 3,1,5,4,2

Y₀ = 4 (only one ordinal variable is paired with 0).
Y₁ = 3+1+5+2/4 = 11/4 = 2.75
n = 5

Giving a rank-biserial correlation coefficient of: 2 * (2.75 – 4)/6 = -0.21.

Running the Test

In SAS: Run the %BISERIAL macro.
In SPSS: Click Analyze → Correlate → Bivariate. Add your variables, deselect Pearson (the default) and click Spearman.* Click OK.

*Glass(1966) noted that the rank biserial correlation is appropriate to estimate Spearman correlation, so I’m assuming this works both ways. You should note though, that it is an estimate.

Matched Pairs Rank Biserial

Kerby (2014) suggested the following for calculating matched pairs rank biserial correlation:

Run a Wilcoxon signed ranks test.
Add the sum of negative ranks and the sum of positive ranks (from the output). This equals the total sum of ranks.
Divide the sum of negative ranks by the total sum of ranks to get a proportion.
Divide the sum of positive ranks by the total sum of ranks to get a proportion.
Find the difference between the two proportions. This is the matched pairs rank biserial.

References:
Glass, G. V. (1966). Note on rank biserial correlation. Educational and Psychological Measurement, 26, 623-631. Retrieved Jan 1, 2017 from here: http://journals.sagepub.com/doi/pdf/10.1177/001316446602600307
Kerby, D. S. (2014). The simple difference formula: an approach to teaching nonparametric correlation. Innovative Teaching, 3(1), Article-1.
Willson, V. L. (1976). Critical values of the rank-biserial correlation coefficient. Educational and Psychological Measurement, Vol. 36, pp. 297-300.