Statistics How To

Coefficient of Association: Definition, Types, Examples

Statistics Definitions >

What is a Coefficient of Association?

A Coefficient of Association measures the strength of a relationship. “Association” means that the variables have shared or common elements or some degree of agreement.

A large number of different association coefficients is available. Which you choose is dependent on many factors, including the data type (e.g. Kendall’s Tau for ranked nominal variables or Yule’s Y for binary variables). That said, a coefficient of association is independent of its measurement scale.

These coefficients typically range between 0 and 1, where 0 is no relationship and 1 is a perfect relationship. However, some measures of association range from -1 to 1, where -1 indicates a perfect inverse relationship.

Coefficient of Association for Nominal Variables

Kendall’s Tau (Kendall Rank Correlation Coefficient) measures relationships between columns of ranked data.

  • Tau-A and Tau-B are usually used for square tables (with equal columns and rows).
  • Tau-B will adjust for tied ranks.
  • Tau-C is usually used for rectangular tables. For square tables, Tau-B and Tau-C are essentially the same.

Binary Variables

1. Coefficient of Colligation (Yule’s Y)

Yules Y (Coefficient of Colligation) or, more simply, Y, can be used to approximate tetrachoric correlation (Warren’s, 2008); Tetrachoric correlation is used to measure rater agreement for binary data. Yule’s Y, a transformation of the odds ratio, is not used very often. One reason is that its used is generally restricted to 2×2 tables; In addition Digby’s (1983) coefficient H, is generally considered to be a better approximation.

Yule’s Q, Yule’s Y, and Digby’s H coefficients are part of a general family of coefficients which raise the odds ratio to a power (c) (Bonnett & Price, 2007).

  • Yule’s Q: c = 1
  • Yule’s Y: c = .5 (i.e. the square root of the OR)
  • Digby’s H = .75

2. Phi Coefficient of Association

The Phi Coefficient of association is used for contingency tables when:

  • At least one variable is a nominal variable.
  • Both variables are dichotomous variables.

Cramer’s V is a similar measure, used when tables are 3×3 or larger.

Related Measures

See also: Measures of Association.


Bonett, D.G. and Price, R.M, (2007) Statistical Inference for Generalized Yule Coefficients in 2 x 2 Contingency Tables. Sociological Methods and Research, 35, 429-446.
Digby, P.G.N. (1983). Approximating the tetrachoric correlation coefficient. Biometrics, 39, 753–757.
Warrens, M. (2008). On Association Coefficients for 2×2 Tables and Properties That Do Not Depend on the Marginal Distributions. Psychometrika. 2008 Dec; 73(4): 777–789. Published online 2008 Jul 23. doi: 10.1007/s11336-008-9070-3.
Yule, G.U. (1912). On the methods of measuring the association between two attributes. Journal of the Royal Statistical Society, 75, 579–652.


Confused and have questions? Head over to Chegg and use code “CS5OFFBTS18” (exp. 11/30/2018) to get $5 off your first month of Chegg Study, so you can understand any concept by asking a subject expert and getting an in-depth explanation online 24/7.

Comments? Need to post a correction? Please post a comment on our Facebook page.

Check out our updated Privacy policy and Cookie Policy