Eta-squared
Because of the relationship between ANOVA and multiple regression, we can use the correlation coefficient as a measure of effect size in ANOVA. The value of the correlation coefficient is given by Multiple R in the Regression data analysis tool. E.g., for Example 1 of ANOVA using Regression, r = .285 (see Figure 2 of ANOVA using Regression), which indicates a medium effect.
A more commonly used measure of effect size is the coefficient of determination R2 which in the context of ANOVA is called eta squared, labeled η2. Thus
For the above example, η2 = .0812, which means that 8.12% of the variance is explained by the model.
Note too that since
it follows that
Thus
Omega-squared
Unfortunately, eta squared is a biased estimate of the population’s coefficient of determination. A less biased estimate, called omega squared, is a better measure of effect size. Omega squared is given by the following formula:
The first version uses the terminology of regression analysis, while the second uses the terminology of ANOVA. We also have the following alternative form:
For one-factor ANOVA in Example 3 of Basic Concepts for ANOVA, ω2 = 0.14 (as can be seen in Figure 1 of Confidence Interval for ANOVA).
In general, omega is a more accurate measure of the effect, where ω2 = .01 is considered a small effect and ω2 = .06 and .14 are considered medium and large effects respectively.
Cohen’s f
Another measure of effect size is Cohen’s f effect size that can be calculated by
Note that
In Effect Size for ANOVA, we described a version of Cohen’s f effect size by
These two versions are almost equal. In fact
Note too that if f is known, then eta-square can be expressed as
η2 = f2/(f2+1)
Effect size for post-hoc tests
Effect sizes for the omnibus ANOVA results, however, are not really that interesting. More useful are effect sizes for the follow-up tests. As explained in Linear Regression for Comparing Means, a useful measure of effect size here is
References
Lakens, D. (2015) Why you should omega-squared instead of eta-squared. The 20% Statistician
http://daniellakens.blogspot.com/2015/06/why-you-should-use-omega-squared.html
Carroll, R. M., Nordholm, L. A. (1975). Sampling characteristics of Kelley’s ε and Hays’ ω. Educational and Psychological Measurement, 35(3), 541–554.
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