How to Calculate Eta Squared in R

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 and is calculated as follows:

Eta squared = SS_effect / SS_total

where:

• SS_effect: The sum of squares of an effect for one variable.
• SS_total: 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

This tutorial provides a step-by-step example of how to calculate Eta squared for variables in an ANOVA model in R.

Step 1: Create the Data

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 code shows how to create a data frame to hold the data we’re working with:

#make this example reproducible
set.seed(10)

#create data frame
data rep(c("Male", "Female"), each = 30),
                   exercise=rep(c("None", "Light", "Intense"), each = 10, times=2),
                   weight_loss=c(runif(10, -3, 3), runif(10, 0, 5), runif(10, 5, 9),
                                 runif(10, -4, 2), runif(10, 0, 3), runif(10, 3, 8)))

#view first six rows of data frame
head(data)

#  gender exercise weight_loss
#1   Male     None  0.04486922
#2   Male     None -1.15938896
#3   Male     None -0.43855400
#4   Male     None  1.15861249
#5   Male     None -2.48918419
#6   Male     None -1.64738030

#see how many participants are in each group
table(data$gender, data$exercise)

#         Intense Light None
#  Female      10    10   10
#  Male        10    10   10

Step 2: Fit the ANOVA Model

The following code shows how to fit a two-way ANOVA using exercise and gender as factors and weight loss as the response variable:

#fit the two-way ANOVA model
model #view the model output
summary(model)

            Df Sum Sq Mean Sq F value  Pr(>F)    
gender       1   15.8   15.80   9.916 0.00263 ** 
exercise     2  505.6  252.78 158.610 

Step 3: Calculate Eta Squared

We can calculate the effect size Eta squared for each variable in our model by using the etaSquared() function from the lsr package:

#load lsr package
library(lsr)

#calculate Eta Squared
etaSquared(model)

            eta.sq eta.sq.part
gender   0.0258824   0.1504401
exercise 0.8279555   0.8499543

The Eta squared for gender and exercise are as follows:

• Eta squared for gender: 0.0258824
• Eta squared for exercise: 0.8279555

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 which indicates that exercise more significant at predicting weight loss.>

Additional Resources

The following tutorials explain how to fit various ANOVA models in R:

How to Conduct a One-Way ANOVA in R
How to Conduct a Two-Way ANOVA in R
How to Conduct a Repeated Measures ANOVA in R