Three-Way ANOVA: Definition & Example

A three-way ANOVA is used to determine how three different factors affect some response variable.

Three-way ANOVAs are less common than a one-way ANOVA (with only one factor) or two-way ANOVA (with only two factors) but they are still used in a variety of fields.

Whenever we perform a three-way ANOVA, we’re interested in whether there is a statistically significant relationship between each factor and the response variable along with whether there are any interaction effects between the factors.

This tutorial shares several scenarios in which you might use a three-way ANOVA along with an example of how to perform one.

When to Use a Three-Way ANOVA

Here are a few scenarios where you may use a three-way ANOVA:

Scenario 1: Botany

A botanist may want to determine how (1) sunlight exposure, (2) watering frequency, and (3) fertilizer type affect plant growth.

In this scenario, she could perform a three-way ANOVA since there are three factors and one response variable.

Scenario 2: Retail

An executive at a retail store may want to determine how (1) day of the week, (2) store location, and (3) advertising campaigns affect total sales.

In this scenario, he could perform a three-way ANOVA since there are three factors and one response variable.

Scenario 3: Medical

A doctor may want to determine how (1) gender, (2) diet, and (3) exercise habits affect weight.

In this scenario, she could perform a three-way ANOVA since there are three factors and one response variable.

Three-Way ANOVA: Example

Suppose a researcher wants to determine if training program, gender, and athletic division affect jumping height.

To test this, he can perform a three-way ANOVA with the following factors:

  • 1. Training program (program 1 vs. program 2)
  • 2. Gender (male vs. female)
  • 3. Athletic Division (division I vs. division II)

The one response variable will be jumping height.

Suppose he collects this data on 40 individuals:

He then uses statistical software to perform a three-way ANOVA and receives the following output:

three way ANOVA table

The P-value column shows the p-value for each individual factor and the interactions between the factors.

From the output we can see that none of the interactions between the three factors were statistically significant.

We can also see that each of the three factors (Program, Gender, and Division) were statistically significant.

In conclusion, we would state that training program, gender, and division are all significant predictors of the jumping height increase among players.

We would also state that there are no significant interaction effects between these three factors.

Note: In practice, we would also calculate the mean jumping height for each program, gender, and division so that we could determine which levels of each factor are associated with increased jumping height.

Additional Resources

The following tutorials explain how to perform a three-way ANOVA in R and Python:

How to Perform a Three-Way ANOVA in R
How to Perform a Three-Way ANOVA in Python

2 Replies to “Three-Way ANOVA: Definition & Example”

  1. Hey Zach, how would you write the results of this 3-way ANOVA in a paper? Would it be like a 2-way, (F(df interaction, df within) = [F-value], p = [p-value])? Using the last table above as an example, F(1,1) = 0.182, p = 0.673. Or might you use the residuals value of 32 somewhere? Thanks in advance!

  2. Hi sir
    My name is bonsa, is am from Ethiopia. I have one question for you, how can can I test a comparison between five banks on for factors on one dependent variable. I have five selected banks, four independent variable and one dependent variable, so I want to compete those five banks based the response that I got from the sample. So can run it on spas. I hope I will get my answer soon. Thank you in advance.

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