When you’re working with Analysis of Variance (ANOVA), you might encounter three different “flavors”: Type I, Type II, and Type III. These types differ in how they handle variability in your data, especially when your groups aren’t perfectly balanced. If you have even group sizes, all three types will give you the same results. But when group sizes differ, it’s crucial to understand the distinctions.
In R, the basic aov() function gives you Type I ANOVA, which works well for balanced designs. For unbalanced designs, you might want to use the Anova() function from the car package, where you can choose between Type II and Type III.
At the heart of ANOVA is the concept of Sums of Squares (SS). This represents the variability or spread in your data. The goal is to divide this variability into different components, like the effects of different factors in your model.
Let’s break down how each type of ANOVA deals with variability:
Type I ANOVA calculates the SS for each factor in the order they appear in your model. The first factor gets credit for all the variability it can explain. The second factor then gets credit for any additional variability it explains, and so on. This means that changing the order of factors in your model can change your results.
Type II ANOVA looks at each factor after accounting for all other factors, except for interactions involving that factor. It’s not affected by factor order, making it a good choice when you’re not particularly interested in interactions.
Type III ANOVA examines each factor’s unique contribution after accounting for all other factors and interactions. It’s particularly useful when you care about interactions and have an unbalanced design.
In a perfectly balanced experiment, all your groups have the same number of observations. But in the real world, things aren’t always so neat. Unbalanced designs, where group sizes differ, can complicate how we divide up the variance. Different types of ANOVA handle this imbalance in different ways, which is why understanding these types becomes crucial.
Let’s look at a small dataset to see these differences in action:
df <- data.frame(
A = factor(c("a", "a", "a", "a", "b", "b", "b", "b", "b", "b", "b", "b")),
B = factor(c("x", "y", "x", "y", "x", "y", "x", "y", "x", "x", "x", "x")),
Y = c( 14, 30, 15, 35, 50, 51, 30, 32, 51, 55, 53, 55)
)
# Displaying the first few rows of the simulated data
head(df) A B Y
1 a x 14
2 a y 30
3 a x 15
4 a y 35
5 b x 50
6 b y 51Let’s check if our data is balanced:
As we can see, our groups aren’t balanced. Factor A has different numbers of observations for its levels, and the same is true for factor B. This unbalanced nature means our choice of ANOVA type could affect our results.
Now, let’s run all three types of ANOVA on our data:
library(car)
# Step 1: Calculate regression object with lm()
type.lm <- lm(formula = Y ~ A*B,
data = df)
# Type I ANOVA - aov()
type.I.aov <- aov(type.lm)
# Type II ANOVA - Anova(type = 2)
type.II.aov <- car::Anova(type.lm, type = 2)
# Type III ANOVA - Anova(type = 3)
type.III.aov <- car::Anova(type.lm, type = 3)Let’s look at the outputs from each type of ANOVA and interpret what they’re telling us about our data:
summary(type.I.aov) Df Sum Sq Mean Sq F value Pr(>F)
A 1 1488.4 1488.4 18.389 0.00266 **
B 1 18.2 18.2 0.225 0.64781
A:B 1 390.1 390.1 4.820 0.05941 .
Residuals 8 647.5 80.9
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In the Type I ANOVA, we see that:
Factor A is highly significant (p = 0.00266)
Factor B is not significant (p = 0.64781)
The interaction A:B is marginally significant (p = 0.05941)
Remember, Type I ANOVA calculates these sequentially. So A gets “credit” for all the variance it can explain, then B for what’s left, and finally the interaction.
type.II.aovAnova Table (Type II tests)
Response: Y
Sum Sq Df F value Pr(>F)
A 1476.22 1 18.2391 0.002722 **
B 18.22 1 0.2252 0.647809
A:B 390.15 1 4.8204 0.059406 .
Residuals 647.50 8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In the Type II ANOVA:
Factor A remains highly significant (p = 0.002722)
Factor B is still not significant (p = 0.647809)
The interaction A:B is still marginally significant (p = 0.059406)
The results are similar to Type I, but notice the slight changes in Sum Sq and F values. This is because Type II adjusts for other main effects but not interactions.
type.III.aovAnova Table (Type III tests)
Response: Y
Sum Sq Df F value Pr(>F)
(Intercept) 420.50 1 5.1954 0.052125 .
A 1785.37 1 22.0587 0.001548 **
B 324.00 1 4.0031 0.080420 .
A:B 390.15 1 4.8204 0.059406 .
Residuals 647.50 8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In the Type III ANOVA:
Factor A is still highly significant (p = 0.001548)
Factor B now appears marginally significant (p = 0.080420)
The interaction A:B remains marginally significant (p = 0.059406)
Here’s where things get interesting. In Type III, each effect is calculated as if it were entered last into the model. This can reveal effects that might be masked in Type I or II, especially in unbalanced designs.
Factor A is consistently significant across all ANOVA types. This suggests a robust main effect of A on our dependent variable Y.
Factor B shows different results across ANOVA types. It’s non-significant in Types I and II, but marginally significant in Type III. This suggests that B might have some effect, but it’s being masked by the unbalanced design and its relationship with A.
The interaction A:B is consistently marginally significant. This hints at a potential interplay between factors A and B that might be worth investigating further.
The change in B’s significance in Type III ANOVA underscores the importance of choosing the appropriate ANOVA type for unbalanced designs. If we only looked at Type I or II, we might have missed the potential effect of B entirely.
In conclusion, while all three ANOVA types agree on the significance of A and the marginal significance of the A:B interaction, they differ in their assessment of B. This demonstrates how crucial it is to consider the structure of your data and choose the appropriate statistical approach, especially when dealing with unbalanced designs.
Choosing between Type I, II, or III ANOVA depends on your specific experimental design and research questions. Understanding how each type handles variance is crucial for interpreting your results correctly, especially with unbalanced data. The differences in SS across these types show the varying approaches to capturing and representing the underlying variability in your data.
This is also a very brief introduction, for more information, check out another digestible, but slightly more detailed summary in S. Mangiafico’s R companion