What is a Factorial ANOVA? (Definition & Example)


A factorial ANOVA is any ANOVA (“analysis of variance”) that uses two or more independent factors and a single response variable. 

This type of ANOVA should be used whenever you’d like to understand how two or more factors affect a response variable and whether or not there is an interaction effect between the factors on the response variable.

Factorial ANOVA

This tutorial provides several examples of situations where a factorial ANOVA may be used along with a step-by-step example of how to perform a factorial ANOVA.

Note: A two-way ANOVA is a type of factorial ANOVA.

Examples of Using a Factorial ANOVA

A factorial ANOVA could be used in each of the following situations.

Example 1: Plant Growth

A botanist wants to understand how sunlight exposure and watering frequency affect plant growth. She plants 100 seeds and lets them grow for three months under different conditions for sunlight exposure and watering frequency. After three months, she records the height of each plant.

In this case, she has the following variables:

  • Response variable: plant growth
  • Factors: sunlight exposure, watering frequency

And she would like to answer the following questions:

  • Does sunlight exposure affect plant growth?
  • Does watering frequency affect plant growth?
  • Is there an interaction effect between sunlight exposure and watering frequency?

She could use a factorial ANOVA for this analysis because she wants to understand how two factors affect a single response variable.

Example 2: Exam Scores

A professor wants to understand how class time and teaching method affect exam scores. He uses two different teaching methods and two different teaching times (early morning and early afternoon) and records the average exam scores of each student at the end of the semester.

In this case, he has the following variables:

  • Response variable: exam score
  • Factors: teaching method, teaching time

And he would like to answer the following questions:

  • Does teaching method affect exam scores?
  • Does teaching time affect exam scores?
  • Is there an interaction effect between teaching method and teach time?

He could use a factorial ANOVA for this analysis because he wants to understand how two factors affect a single response variable.

Example 3: Annual Income

An economist collects data to understand how education level (high school diploma, college degree, graduate degree), marital status (single, divorced, married), and region (North, East, South, West) affect annual income.

In this case, he has the following variables:

  • Response variable: annual income
  • Factors: education level, marital status, region

And he would like to answer the following questions:

  • Does education level affect income?
  • Does marital status affect income?
  • Does region affect income?
  • Is there an interaction effect between these three independent factors?

He could use a factorial ANOVA for this analysis because he wants to understand how three factors affect a single response variable.

Step-by-Step Example of a Factorial ANOVA

A botanist wants to know if sunlight exposure and watering frequency affect plant growth. She plants 40 seeds and lets them grow for two months under different conditions for sunlight exposure and watering frequency. After two months, she records the height of each plant.

The results are shown below:

We can see that five plants were grown under each combination of conditions.

For example, there were five plants grown with daily watering and no sunlight and their heights after two months were 4.8 inches, 4.4 inches, 3.2 inches, 3.9 inches, and 4.4 inches:

The botanist uses this data to perform a factorial ANOVA in Excel and ends up with the following output:

The last table shows the result of the factorial ANOVA:

  • The p-value for the interaction between watering  frequency and sunlight exposure was 0.310898. This is not statistically significant at alpha level 0.05.
  • The p-value for watering frequency was 0.975975. This is not statistically significant at alpha level 0.05.
  • The p-value for sunlight exposure was 3.9E-8 (0.000000039). This is statistically significant at alpha level 0.05.

We can conclude that sunlight exposure is the only factor that has a statistically significant effect on plant growth.

We can also conclude that there is no interaction effect between sunlight exposure and watering frequency and that watering frequency does not have a statistically significant effect on plant growth.

Additional Resources

An Introduction to the One-Way ANOVA
An Introduction to the Two-Way ANOVA
An Introduction to the Repeated Measures ANOVA
The Differences Between ANOVA, ANCOVA, MANOVA, and MANCOVA

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