A three-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups that have been split on three factors.
The following example shows how to perform a three-way ANOVA in R.
Example: Three-Way ANOVA in R
Suppose a researcher wants to determine if two training programs lead to different mean improvements in jumping height among college basketball players.
The researcher suspects that gender and division (Division I or II) may also affect jumping height so he collects data for these factors as well.
His goal is to perform a three-way ANOVA to determine how training program, gender, and division affect jumping height.
Use the following steps to perform this three-way ANOVA in R:
Step 1: Create the Data
First, let’s create a data frame to hold the data:
#create dataset df <- data.frame(program=rep(c(1, 2), each=20), gender=rep(c('M', 'F'), each=10, times=2), division=rep(c(1, 2), each=5, times=4), height=c(7, 7, 8, 8, 7, 6, 6, 5, 6, 5, 5, 5, 4, 5, 4, 3, 3, 4, 3, 3, 6, 6, 5, 4, 5, 4, 5, 4, 4, 3, 2, 2, 1, 4, 4, 2, 1, 1, 2, 1)) #view first six rows of dataset head(df) program gender division height 1 1 M 1 7 2 1 M 1 7 3 1 M 1 8 4 1 M 1 8 5 1 M 1 7 6 1 M 2 6
Step 2: View Descriptive Statistics
Before performing the three-way ANOVA, we can use dplyr to quickly summarize the mean jumping height increase grouped by training program, gender, and divison:
library(dplyr) #calculate mean jumping height increase grouped by program, gender, and division df %>% group_by(program, gender, division) %>% summarize(mean_height = mean(height)) # A tibble: 8 x 4 # Groups: program, gender [4] program gender division mean_height 1 1 F 1 4.6 2 1 F 2 3.2 3 1 M 1 7.4 4 1 M 2 5.6 5 2 F 1 2.6 6 2 F 2 1.4 7 2 M 1 5.2 8 2 M 2 4
Here’s how to interpret the output:
- The mean jumping height increase among division I females who used training program 1 was 4.6 inches.
- The mean jumping height increase among division II females who used training program 1 was 3.2 inches.
- The mean jumping height increase among division I males who used training program 1 was 7.4 inches.
And so on.
Step 3: Perform the Three-Way ANOVA
Next, we can use the aov() function to perform the three-way ANOVA:
#perform three-way ANOVA model <- aov(height ~ program * gender * division, data=df) #view summary of three-way ANOVA summary(model) Df Sum Sq Mean Sq F value Pr(>F) program 1 36.1 36.10 65.636 2.98e-09 *** gender 1 67.6 67.60 122.909 1.71e-12 *** division 1 19.6 19.60 35.636 1.19e-06 *** program:gender 1 0.0 0.00 0.000 1.000 program:division 1 0.4 0.40 0.727 0.400 gender:division 1 0.1 0.10 0.182 0.673 program:gender:division 1 0.1 0.10 0.182 0.673 Residuals 32 17.6 0.55 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The Pr(>F) column shows the p-value for each individual factor and the interactions between the factors.
From the output we can see that none of the interactions between the three factors were statistically significant.
We can also see that each of the three factors – program, gender, and division – were statistically significant.
We can now use dplyr once again to find the mean jumping height increase for program, gender, and division separately:
library(dplyr) #find mean jumping increase by program df %>% group_by(program) %>% summarize(mean_height = mean(height)) # A tibble: 2 x 2 program mean_height 1 1 5.2 2 2 3.3 #find mean jumping increase by gender df %>% group_by(gender) %>% summarize(mean_height = mean(height)) # A tibble: 2 x 2 gender mean_height 1 F 2.95 2 M 5.55 #find mean jumping increase by division df %>% group_by(division) %>% summarize(mean_height = mean(height)) # A tibble: 2 x 2 division mean_height 1 1 4.95 2 2 3.55
From the output we can observe the following:
- The mean jumping height increase among individuals who used training program 1 (5.2 inches) was higher than the mean increase among individuals who used training program 2 (3.3 inches).
- The mean jumping height increase among males (5.55 inches) was higher than the mean increase among females (2.95 inches).
- The mean jumping height increase among division 1 players (4.95 inches) was higher than the mean increase among division 2 players (3.55 inches).
In conclusion, we would state that training program, gender, and division are all significant predictors of the jumping height increase among players.
We would also state that there are no significant interaction effects between these three factors.
Additional Resources
The following tutorials explain how to fit other ANOVA models in R:
How to Perform a One-Way ANOVA in R
How to Perform a Two-Way ANOVA in R
what do you do when there is a significant finding in one of the interactions?
Excellent question — this is *exactly* where the real value of ANOVA shows up:
A significant interaction means *the effect of one variable depends on the level of another variable*.
### If your Three-Way ANOVA has a significant interaction (let’s say A:B interaction is significant), here’s what you should do next:
—
## 1. *Do Not* Interpret Main Effects in Isolation
> Why?
Significant interaction means the main effects can’t be generalized — because the effect changes based on the other factor(s).
Example:
> “Factor A increases the outcome” — might only be true when Factor B = High.
But for Factor B = Low, Factor A might decrease the outcome.
—
## 2. Perform *Simple Effects* Analysis
Break down the interaction to examine the effect of one factor at specific levels of the other factor(s).
### In R:
Use `emmeans` (Estimated Marginal Means) or `lsmeans` to explore this.
“`R
library(emmeans)
# Let’s say your model is:
model <- aov(Y ~ A * B * C, data=mydata) # Explore A at each level of B emmeans(model, pairwise ~ A | B) # Or explore interaction plots interaction.plot(mydata$A, mydata$B, mydata$Y) ``` --- ## 3. Visualize the Interaction Plots make complex interactions easier to interpret. In R: ```R library(ggplot2) ggplot(mydata, aes(x=A, y=Y, color=B)) + stat_summary(fun=mean, geom="line") + facet_wrap(~C) ``` --- ## 4. Report Findings Like This: > “There was a significant interaction between A and B (p < 0.01), indicating the effect of A differed depending on the level of B. Specifically, A increased Y when B was High (M=XX), but had little effect when B was Low (M=YY)." --- ## Final Step: If your *three-way* interaction (A:B:C) is significant: → Perform Simple Effects for combinations of 2 factors within levels of the third factor. ```R emmeans(model, pairwise ~ A | B * C) ``` --- ## Summary Flow: | Step | Action | R Tools | |------|--------|---------| | 1 | Don't interpret main effects alone | - | | 2 | Perform simple effects analysis | `emmeans()` | | 3 | Visualize interaction | `ggplot2` or `interaction.plot()` | | 4 | Report findings | Contextual interpretation | | 5 | If 3-way interaction significant | Simple effects within combinations of 2 factors | ---