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
George Udny Yule FRS (18 February 1871 – 26 June 1951), usually known as Udny Yule, was a British statistician, particularly known for the Yule distribution.
Personal life
Yule was born at Beech Hill, a house in Morham near Haddington, ...
in 1912,
[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] and should not be confused with
Yule's coefficient for measuring
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
based on
quartiles
In statistics, a quartile is a type of quantile which divides the number of data points into four parts, or ''quarters'', of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are a ...
.
Formula
For a 2×2 table for
binary variables ''U'' and ''V'' with frequencies or proportions
:
Yule's ''Y'' is given by
:
Yule's ''Y'' is closely related to the
odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
''OR'' = ''ad''/(''bc'') as is seen in following formula:
:
Yule's ''Y'' varies from −1 to +1. −1 reflects total negative
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
, +1 reflects perfect positive association while 0 reflects no association at all. These correspond to the values for the more common
Pearson correlation
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
.
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:
:
:
Interpretation
Yule's ''Y'' gives the fraction of perfect association in per
unum
Unum Group is an American insurance company headquartered in Chattanooga, Tennessee. Founded as Union Mutual in 1848 and known as UnumProvident from 1999-2007, the company is part of the Fortune 500. Unum Group was created by the 1999 merger of ...
(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.
Examples
The following crosswise symmetric table
:
can be split up into two tables:
:
and
:
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).
:
Here follows the transformed table:
:
The odds ratios of both tables are equal to 9. ''Y'' = (3 − 1)/(3 + 1) = 0.5 (50%)
References
{{Reflist
Summary statistics for contingency tables