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. Multiple R is the correlation coefficient 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 R^2, 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

Links

↑ Multiple regression

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. 
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