Coefficient of colligation

In statistics, 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] and should not be confused with Yule's coefficient for measuring skewness based on quartiles.

Formula

For a 2×2 table for binary variables U and V with frequencies or proportions

	V = 0	V = 1
U = 0	a	b
U = 1	c	d

Yule's Y is given by

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

Yule's Y is closely related to the odds ratio OR = ad/(bc) as is seen in following formula:

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

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. These correspond to the values for the more common Pearson correlation.

Yule's Y is also related to the similar Yule's Q, which can also be expressed in terms of the odds ratio. Q and Y are related by:

${\displaystyle Q={\frac {2Y}{1+Y^{2}}}\ ,}$

${\displaystyle Y={\frac {1-{\sqrt {1-Q^{2}}}}{Q}}\ .}$

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 2×2 table in a crosswise symmetric table wherein b = c = 1 and a = d = √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 perfectly clear way by dividing (a – b) by (a + b). In transformed tables b has to be substituted by 1 and a by √OR. The transformed table has the same degree of association (the same OR) as the original not-crosswise symmetric table. Therefore, the association in asymmetric tables can be measured by Yule's Y, interpreting it in just the same way as with symmetric tables. Of course, Yule's Y and (a − b)/(a + b) give the same result in crosswise symmetric tables, presenting the association as a fraction in both cases.

Yule's Y measures association in a substantial, intuitively understandable way and therefore it is the measure of preference to measure association.^{[citation needed]}

Examples

The following crosswise symmetric table

	V = 0	V = 1
U = 0	40	10
U = 1	10	40

can be split up into two tables:

	V = 0	V = 1
U = 0	10	10
U = 1	10	10

and

	V = 0	V = 1
U = 0	30	0
U = 1	0	30

It is obvious that the degree of association equals 0.6 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).

	V = 0	V = 1
U = 0	3	1
U = 1	3	9

Here follows the transformed table:

	V = 0	V = 1
U = 0	3	1
U = 1	1	3

The odds ratios of both tables are equal to 9. Y = (3 − 1)/(3 + 1) = 0.5 (50%)

References

1. Yule, G. Udny (1912). "On the Methods of Measuring Association Between Two Attributes". Journal of the Royal Statistical Society. 75 (6): 579–652. doi:10.2307/2340126. JSTOR 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