What is Eta Squared? (Definition & Example)


Eta squared is a measure of effect size that is commonly used in ANOVA models.

It measures the proportion of variance associated with each main effect and interaction effect in an ANOVA model.

How to Calculate Eta Squared

The formula to calculate Eta squared is straightforward:

Eta squared = SSeffect / SStotal

where:

  • SSeffect: The sum of squares of an effect for one variable.
  • SStotal: The total sum of squares in the ANOVA model.

The value for Eta squared ranges from 0 to 1, where values closer to 1 indicate a higher proportion of variance that can be explained by a given variable in the model.

The following rules of thumb are used to interpret values for Eta squared:

  • .01: Small effect size
  • .06: Medium effect size
  • .14 or higher: Large effect size

Example: Calculating Eta Squared

Suppose we want to determine if exercise intensity and gender impact weight loss.

To test this, we recruit 30 men and 30 women to participate in an experiment in which we randomly assign 10 of each to follow a program of either no exercise, light exercise, or intense exercise for one month.

The following table shows the results of a two-way ANOVA using exercise and gender as factors and weight loss as the response variable:

            Df Sum Sq Mean Sq F value p value    
gender       1   15.8   15.80   9.916 0.00263
exercise     2  505.6  252.78 158.610 < 2e-16
Residuals   56   89.2    1.59  

We can calculate SStotal, the total sum of squares, as follows: 15.8 + 505.6 + 89.2 = 610.6.

We can then calculate Eta squared for gender and exercise as follows:

  • Eta squared for gender: 15.8 / 610.6 = .02588
  • Eta squared for exercise: 505.6 / 610.6 = .828

We would conclude that the effect size for exercise is very large while the effect size for gender is quite small.

These results match the p-values shown in the output of the ANOVA table. The p-value for exercise ( <.000) is much smaller than the p-value for gender (.00263), which indicates that exercise is much more significant at predicting weight loss.

This example also illustrates why Eta squared is useful: Although gender is statistically significant (p = .00263), the effect size associated with it is actually quite small.

A p-value can only tell us whether or not there is some significant association between two variables, but a measure of effect size like Eta squared can tell us the strength of association between the variables.

Additional Resources

An Introduction to the One-Way ANOVA
An Introduction to the Two-Way ANOVA
A Guide to Using Post Hoc Tests with ANOVA

Leave a Reply

Your email address will not be published. Required fields are marked *