Coefficient of colligation

Yule’s Y, also known as the coefficient of colligation, is a measure of association between two binary variables. The measure was developed by George Udny Yule in 1912.^[1] .^[2]

Formula

For a 2X2 table for binary variables X and Y with frequencies or proportions

	Y=0	Y=1
X=0	a	b
X=1	c	d

a formula is given by:

$Y = \frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}.$

Yule's Y is closely related to the odds ratio as is seen in following formula:

$Y = \frac{\sqrt{OR}-1}{\sqrt{OR}+1}$

Yule's Y varies from -1 to +1. -1 reflects total negative correlation, +1 reflects perfect positive association while 0 reflects no association at all.

Interpretation

Yule’s Y gives the fraction of perfect association in per unum (multiplied by 100 it represents this fraction in a more familiar percentage). Indeed the formula transforms the original 2x2 table in a crosswise symmetric table wherein b = c = 1 and a = d = SQRT(OR).

For a crosswise symmetric table with frequencies or proportions a = d and b = c it is very easy to see that it can be split up in two tables. In such tables association can be measured in a perfect clear way by dividing (a – b) by (a + b). In transformed tables b has to be substituted by 1 and a by SQRT(OR). The transformed table has the same degree of association (the same OR) as the original not crosswise symmetric table. So the association in not symmetric tables can as well be measured by Yule’s Y interpreting Yule’s Y in the same way as it can be interpreted for symmetric tables. Of course Yule’s Y and (a-b)/(a+b) gives the same result in crosswise symmetric tables. So Yule’s measures association as a fraction for the two kinds of tables.

Yule’s Y measures association in a substantial, intuitively understandable way and therefore it is the measure of preference to measure association.

Examples

The following crosswise symmetric table

	Y=0	Y=1
X=0	40	10
X=1	10	40

can be split up into two tables:

	Y=0	Y=1
X=0	10	10
X=1	10	10

and

	Y=0	Y=1
X=0	30	0
X=1	0	30

It is obvious that de degree of association equals .6 per unum (60%).

The following asymmetric table can be transformed in a table with an equal degree of association (the odds ratios of both tables are equal).

	Y=0	Y=1
X=0	3	1
X=1	3	9

Here follows the transformed table:

	Y=0	Y=1
X=0	3	1
X=1	1	3

The odds ratios of both tables are equal to 9. Y= (3-1)/(3+1) = .5 (50%)

References

1. Yule, G. Udny. "On the Methods of Measuring Association Between Two Attributes". Journal of the Royal Statistical Society 75 (6): 579. doi:10.2307/2340126.
2. Michel G. Soete. A new theory on the measurement of association between two binary variables in medical sciences: association can be expressed in a fraction (per unum, percentage, pro mille....) of perfect association (2013), e-article, BoekBoek.be