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

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.

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.

shapiro.test(residuals(result))

### 4.2 Homogeneity of Variances

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

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

### 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:

summary(result)

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.

TukeyHSD(result)