Other measures of effect size for ANOVA

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

Eta-square

For the above example, η2 = .0812, which means that 8.12% of the variance is explained by the model.

Note too that since

F-statistic

it follows that

SS_Bet

Thus

Eta-square formula

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:

Omega square

The first version uses the terminology of regression analysis, while the second uses the terminology of ANOVA. We also have the following alternative form:

Omega squared

For one-factor ANOVA in Example 3 of Basic Concepts for ANOVAω2 = 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

Cohen's f

Note that

f-square

In Effect Size for ANOVA, we described a version of Cohen’s f effect size by

f prime

These two versions are almost equal. In fact

f' and f

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

Effect size contrast r

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. 
Available through Sage Journals

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