## 1. Introduction

What are the meanings of different types of ANOVA? In other words, what are Type 1 (Type I), Type 2 (Type *II*), and Type 3 (*Type III*) ANOVA? The following uses the model of factors A and B, and its interaction A*B as an example to explain the difference of Type 1 (Type I), Type 2 (Type

*II*), and Type 3 (

*Type*) ANOVA.

*III*## 2. **Type 1 ANOVA (Type I ANOVA)**

### 2.1 **Definition of Type 1 ANOVA**

`SS(A) for factor A`

`SS(B | A) for factor B`

`SS(AB | A, B) for interaction AB`

The following includes the explanations for each statement in Type 1 ANOVA.

- SS(A) is SS attributable to A, even including those that could also have been attributed to other factors like B or A*B. Thus, SS(A) is equal to the model of only putting A as the single predictor.
- SS(B | A) = SS (A, B) – SS(A)
- SS(AB|A,B) = SS (AB, A, B) – SS(A, B)

### 2.2 **Other Attributes**

- Type 1 ANOVA is
, since it considers the order of entering factors into the model. The order of you write factors A and B in the model impacts the SS calculated for the factors.**sequential sum of squares** - If you add all different components of SS and do some simple math, you will get the following.

SS_{A} + SS_{B | A} +SS_{AB | A, B} + SSE = SST

## 3. **Type 2 ANOVA (Type II ANOVA)**

### 3.1 **Definition of Type 2 ANOVA**

`SS(A | B) for factor A`

`SS(B | A) for factor B`

`SS(AB | A, B) for interaction AB`

The following includes the explanations for each statement in Type 2 ANOVA.

- SS(A|B) = SS (A, B) – SS(B)
- SS(B|A) = SS (A, B) – SS(A)
- SS(AB|A,B) = SS (AB, A, B) – SS(A, B).

### 3.2 **Other Attributes and Comments**

`SS(A|B)`

and`SS(B|A)`

do not control the impact of AB. Thus, if the interaction AB is significant, you need to be cautious interpretating`SS(A | B)`

and`SS(B | A)`

.- When AB is insignificant, it seems to me that you do not really have to add AB into the model either, unless you really have a good reason.
- Type 2 ANOVA does not include the shared SS between A and B (i.e., the shared area of A and B circles, see below). At first, it is a bit counter-intuitive, as you would kind of guess that
`SS(AB | A, B)`

should be from the shared area of A and B circles. However,`SS(AB | A, B)`

means that it excludes SS from the combination of A and B. Thus,`SS(AB | A, B)`

is SS totally outside of A and B, and at the same time attributable to AB. That is, the shared SS of A and B (i.e.,`the overlap in two circles`

) is NOT part of`SS(AB | A, B)`

at all. (For more, see a discussion on Stackexchange.)

- Note that,
`SS`

and_{AB|A,B}`SSE`

are the same as the counterparts in Type I ANOVA. - We can do some simple manth as follows. If A and B have no overlap, namely
`SS(A,B)`

=`SS(A)`

+`SS(B)`

, we can get SS_{A|B}+SS_{B|A}+SS_{AB|A,B}+SSE = SST. However, if there is overlap (i.e., there is shared area between A and B circles), SS_{A|B}+SS_{B|A}+SS_{AB|A,B}+SSE ≠ SST

SS_{A|B}+SS_{B|A}+SS_{AB|A,B}

=SS(A,B)-SS(B)+SS(A,B)-SS(A)+SS(AB,A,B)-SS(A, B)

=SS(AB,A,B)+SS(A,B)-SS(A)-SS(B)

## 4. **Type **3 **ANOVA** **(Type III ANOVA)**

### 4.1 **Definition of Type 3 ANOVA**

`SS(A | B, AB) for factor A`

`SS(B | A, AB) for factor B`

`SS(AB | A, B) for interaction AB`

The following includes the explanations for each statement in Type 3 ANOVA.

- SS(A|B,AB) = SS (AB, A, B) – SS(AB, B)
- SS(B|A,AB
**)**= SS (AB, A, B) – SS(AB, A) - SS(AB|A,B)= SS (AB, A, B) – SS(A, B)

### 4.2 **Other Attributes and Comments**

- SS that could be attributed to 2 or more combination (e.g.,
**B and AB**,**A and AB**,**A and B**) are not used. - I agree with the post on R-bloggers that, when the interaction AB is significant, you need to use Type 3 ANOVA. Further, when the interaction effect is significant, typically you need to focus on the simple main effects, rather than the main effects of
**A | B, AB**or**B | A, AB**, which are difficult to explain, in my opinions. - When the interaction AB is not significant, you can just drop the AB in the model, unless you have a good reason to keep it. If you drop AB in the model, Type 2 and Type 3 ANOVA are the same.
- When using Type 3 ANOVA (or even Type 1 and 2), it is better to specify which type of ANOVA you are using when reporting the results. Then, readers can make their judgment.
- If A and B are not related (orthogonal), we can get SS
_{A|B,AB}+SS_{B|A,AB}+SS_{AB|A,B}+SSE=SST. Otherwise, we likely to get SS_{A|B,AB}+SS_{B|A,AB}+SS_{AB|A,B}+SSE≠SST.

## Reference

- How to interpret type I, type II, and type III ANOVA and MANOVA?
- TYPE I vs. TYPE III Sum of Squares
- Anova – Type I/II/III SS explained