Understanding One-Way and Two-Way ANOVA [With Examples]

This article provides a comprehensive overview of the two main types of Analysis of Variance (ANOVA) tests: one-way ANOVA vs two-way ANOVA. It begins by explaining the general concept of ANOVA and its widespread use in various fields to compare means across groups. The article details when each type of ANOVA is appropriate, illustrating this with examples. It then contrasts the two tests by highlighting their primary differences, such as the number of independent variables involved. Additionally, it offers a summary of these differences to help readers understand the distinct applications of one-way and two-way ANOVA in statistical analysis.

Analysis of Variance (ANOVA): An Overview

Generally, Analysis of Variance (ANOVA) is among the most widely used statistical tests in research. It is widely used in various fields such as economics, psychology, marketing, biology, and business to analyze datasets. ANOVA is mainly used when the researcher aims to compare the means of three or more independent groups.

There are two main types of ANOVA, which are widely used in research. These include one-way Anova and two-way Anova. This article will summarize each of these ANOVA tests, outlining when each of the ANOVA tests is appropriate. The article then presents the main difference between one-way and two-way ANOVA and provides a summary of these differences.

What is a One-Way ANOVA and What is it Used For?

A one-way ANOVA is a statistical test that is mainly used when the researcher wishes to compare the variation in group means between three or more independent groups. In fact, it is an extension of the two-sample independent t-test, when the number of independent groups is more than two. In other words, a one-way ANOVA aims to determine whether there exists a significant mean difference in the dependent variable between the various independent groups.

Note. The term “Independent groups” is different from “independent variables”.  In one-way ANOVA, there is one dependent variable and one independent variable/factor. The independent variable is the one that contains more than three or more independent groups.

You can check out our comprehensive article on one-way analysis of variance for more details.

One-Way ANOVA Example

Consider this University Scenario:

A university is conducting a study to determine the effectiveness of different teaching methods on students’ exam scores in an introductory psychology course. The course instructors have been using three different teaching styles: traditional lectures, interactive workshops, and online modules. The university wants to know if these teaching methods lead to significant differences in students’ exam performance. The university collects exam scores from students who have experienced each teaching method to compare their performance.

Based on the above example:

The one-way ANOVA will be the most appropriate statistical test to address the university’s research problem. This is because there is one dependent variable and one independent variable (with 3 independent groups/categories).

Let’s identify the dependent and independent variables.

Dependent Variable: Students’ exam scores in the introductory psychology course.

Independent Variable: Teaching Methods

Independent Groups/Categories. The 3 teaching methods, which included; Traditional lectures, Interactive workshops, and online modules

Thus, performing a one-way ANOVA would help the university determine if there is a statistically significant difference in exam scores across the three teaching methods. This would consequently help the university make informed decisions about the most effective teaching approach for the introductory psychology course.

What is a Two-Way ANOVA and What is it Used For?

Similar to the one-way ANOVA, a two-way ANOVA is an inferential statistical test that is used to determine whether there is a significant mean difference in the dependent variable based on two independent variables (two factors, each with two or more categories). Simply stated, a two-way ANOVA is used to investigate how two independent variables (combined), influence the dependent variable.

Two-Way ANOVA Example

Consider a scenario of a pharmaceutical company, that aims to conduct a study to investigate the effects of two independent variables, dosage, and gender, on the effectiveness of a new medication in treating a specific medical condition. The company wants to determine if there are significant interactions between dosage levels, gender, and the medication’s effectiveness. Thus, the company recruits both male and female participants and assigns them randomly to receive one of the three dosage levels of the medication (Low, Medium, and High). Participants undergo treatment for a specified period, and their response to the medication is measured in terms of effectiveness.

Let’s identify the independent and dependent variables;

Dependent Variable: Effectiveness of the new medication in treating the medical condition.

Independent Variables: The independent variables are Dosage levels (Low, Medium, and High) and gender (Male, Female)

Independent Groups (Categories)

Dosage levels

• Low dosage
• Medium dosage
• High dosage

Gender

• Male
• Female

Now, based on the above scenario, a two-way ANOVA would help the company examine the main effects of dosage and gender on medication effectiveness. It would also help the company determine whether there is an interaction effect between dosage and gender. The findings would provide insights into how different dosage levels and gender influence the effectiveness of the medication. This would consequently guide the pharmaceutical company in optimizing treatment strategies for the medical condition.

One-Way vs Two-Way ANOVA: What is the difference?

The main difference between a one-way ANOVA and a Two-Way ANOVA is in the number of independent variables. While a one-way ANOVA has one independent variable (with 3 or more categories/independent groups), a two-way ANOVA has two independent variables (each with 2 or more levels/categories).

The similarity between a one-way and a two-way ANOVA is that they are both designed to test for differences in means when the independent groups are three or more.

NOTE: If you’re only interested in determining whether there is a significant mean difference between two independent groups, the most appropriate statistical test is the independent sample t-test. You can check out our tutorial on how to conduct an independent sample t-test in SPSS for more details.

Summary of the Differences between One-way and Two-way ANOVA

There are other notable differences between one-way and two-way ANOVA. These are summarized below;

• While a one-way ANOVA is mainly used for testing the equality in means between three or more independent groups, a two-way ANOVA is mainly used for determining the effects of two independent variables (each with two or more levels) on a single dependent variable.
• While a one-way ANOVA has one independent variable (Single Factor), a two-way ANOVA on the other hand, has two independent variables.
• The single factor in a one-way ANOVA should have three or more categories. On the other hand, each of the independent variables in a two-way ANOVA can have 2 or more categories/levels.