How to Perform a Three-Way Analysis of Variance (ANOVA) in R

As researchers or data analysts dig deeper into the complexities of their data, they often find that considering more factors allows for a richer understanding of the variables at play. A Three-Way Analysis of Variance (ANOVA) is a powerful statistical test that allows you to examine the influence of three different factors on a dependent variable.

While Two-Way ANOVA examines the effects of two factors, a Three-Way ANOVA allows you to explore the interplay among three factors, adding a layer of complexity and richness to your analysis.

In this extensive guide, we will walk you through the process of conducting a Three-Way ANOVA in R, from data preparation to interpretation and reporting.

1. Basics of Three-Way ANOVA
2. Data Preparation
3. Running a Three-Way ANOVA in R
4. Checking Assumptions
5. Interpreting Results
6. Conducting Post-hoc Tests
8. Conclusion

1. Basics of Three-Way ANOVA

Three-Way ANOVA allows you to evaluate how three factors affect a dependent variable. You can also examine the interaction effects among these three factors. The test extends the concept of the Two-Way ANOVA by adding another factor to the model.

2. Data Preparation

Your data should be structured in a way that each column represents either a dependent variable or an independent factor. Here’s a hypothetical example:

• Factor 1: Diet (Vegan, Omnivore)
• Factor 2: Exercise (Yes, No)
• Factor 3: Age Group (Young, Middle-Aged, Elderly)
• Dependent Variable: Cholesterol Level
data <- data.frame(
Cholesterol = c(180, 190, 200, 210, 185, 175, 205, 220, 170, 210),
Diet = c("Vegan", "Omnivore", "Vegan", "Omnivore", "Vegan", "Omnivore", "Vegan", "Omnivore", "Vegan", "Omnivore"),
Exercise = c("Yes", "Yes", "No", "No", "Yes", "Yes", "No", "No", "Yes", "No"),
Age_Group = c("Young", "Young", "Middle-Aged", "Middle-Aged", "Elderly", "Elderly", "Young", "Young", "Middle-Aged", "Elderly")
)

3. Running a Three-Way ANOVA in R

In R, you can perform a Three-Way ANOVA using the aov() function, much like you would for One-Way or Two-Way ANOVA.

Here is the general format:

result <- aov(DependentVariable ~ Factor1 * Factor2 * Factor3, data=YourDataFrame)

For our hypothetical example:

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

4. Checking Assumptions

Before interpreting the results, you should check the assumptions underlying ANOVA:

4.1 Normality

The residuals should be approximately normally distributed for each combination of the groups:

shapiro.test(residuals(result))

4.2 Homogeneity of Variances

Variances should be equal across groups. Levene’s test can be used for this:

install.packages("car")
library(car)
leveneTest(result)

4.3 Independence

Ensure that the samples are independent, which is usually a feature of the study design.

5. Interpreting Results

After running the test and ensuring the assumptions are met, use the summary() function to interpret the results:

summary(result)

Look at the F-values and p-values to understand the main and interaction effects.

6. Conducting Post-hoc Tests

If your Three-Way ANOVA shows significant effects, you may wish to follow up with post-hoc tests to examine pairwise comparisons. The TukeyHSD() function is commonly used for this purpose:

TukeyHSD(result)