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.
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
Y=
| \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:
Y=
| \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:
Q=
| 2Y | |
| 1+Y2 |
,
Y=
| 1-\sqrt{1-Q2 | |
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 = .
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 . 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.
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%)