How to Conduct a Two-Way Analysis of Variance (ANOVA) in R

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Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. While one-way ANOVA is used to test differences between several groups based on one single factor, the two-way ANOVA is an extension that allows us to evaluate the influence of two different categorical independent variables at the same time.

Two-way ANOVA provides insights into the interactions between two factors and their impact on a dependent variable, offering a more in-depth view of complex relationships.

This extensive article covers how to conduct a two-way ANOVA in R.

Table of Contents

  1. Overview of Two-Way ANOVA
  2. Data Preparation
  3. Conducting Two-Way ANOVA in R
  4. Checking Assumptions
  5. Interpretation of Results
  6. Post-hoc Tests
  7. Reporting the Results
  8. Conclusion

1. Overview of Two-Way ANOVA

Two-way ANOVA evaluates how two factors impact a dependent variable, and it also looks at the interaction between the two factors. We can categorize it into:

  1. Two-Way ANOVA with Replication: Multiple observations for each combination of factors.
  2. Two-Way ANOVA without Replication: Only one observation for each combination of factors.

2. Data Preparation

Your data should be organized with one column for each factor and one for the dependent variable. For example:

  • Factor 1: Different diets (Vegan, Mediterranean, etc.)
  • Factor 2: Age groups (Young, Middle-Aged, etc.)
  • Dependent Variable: Cholesterol level

Here’s how sample data might look in R:

data <- data.frame(
  Cholesterol = c(200, 220, 185, 210, 190, 235, 180, 225),
  Diet = c("Vegan", "Vegan", "Mediterranean", "Mediterranean", "Vegan", "Vegan", "Mediterranean", "Mediterranean"),
  Age_Group = c("Young", "Old", "Young", "Old", "Young", "Old", "Young", "Old")

3. Conducting Two-Way ANOVA in R

The aov() function in R can perform a two-way ANOVA when you specify both factors.

Here’s the general syntax:

result <- aov(DependentVariable ~ Factor1 * Factor2, data=YourDataFrame)

For our sample data:

result <- aov(Cholesterol ~ Diet * Age_Group, data=data)

4. Checking Assumptions

4.1 Normality

For each group combination, the residuals should be approximately normally distributed. Use the Shapiro-Wilk test or QQ plots to verify this.


4.2 Homogeneity of Variances

The variances for each combination of the groups should be equal. Use Levene’s test to check this:


4.3 Independence

This is usually guaranteed by the study design.

5. Interpretation of Results

To interpret the two-way ANOVA, use the summary() function:


You’ll get an output showing the main effects of each factor and their interaction. Pay attention to the F-values and p-values to determine significance.

6. Post-hoc Tests

If you find significant interactions or main effects, post-hoc tests like Tukey’s HSD can help identify which groups differ significantly.


7. Reporting the Results

In your report, include:

  1. The main effects of each factor.
  2. Interaction effects.
  3. F-values, degrees of freedom, and p-values.
  4. Results of any post-hoc tests.

8. Conclusion

Two-way ANOVA in R allows for a more complex analysis than one-way ANOVA. It provides insights into how two factors impact a dependent variable individually and interactively. This article should provide a solid foundation for conducting and interpreting two-way ANOVA in R for your own data. By following these steps, you can confidently explore the complex relationships in your data and present your findings effectively.

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