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

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

Note too that since for one factor ANOVA

it follows that

Thus

η² = .01 is considered a small effect and η² = .06 and .14 are considered medium and large effects respectively.

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 formulas:

The first version uses the terminology of regression analysis, while the second uses the terminology of ANOVA. Since the numerator can be expressed as

we 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, ω² is a more accurate measure of the effect than η². Once again, ω² = .01 is considered a small effect and ω² = .06 and .14 are considered medium and large effects respectively.

Partial eta-squared and partial omega-squared

Another commonly used measure of effect size is partial eta-squared, which is defined as:

Clearly, for one factor ANOVA, eta-squared and partial eta-squared have the same value.

Similarly there is a partial omega-squared  version of ω², which is defined as

But since the denominator can be expressed as

once again, for a one factor ANOVA, partial omega-squared is equal to omega-squared.

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

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

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