# Overview of Type I, Type II, and Type III ANOVA in R (with Examples)

This tutorial explains what Type I, Type II, and Type III ANOVA are. Further, it provides examples showing how you can do Type I, Type II, and Type III ANOVA in R.

## 1. Introduction

Type I, Type II, and Type III ANOVA are 3 different ways of calculating sum of squares in ANOVA.

Type I ANOVA:

• SS(A) for factor A
• SS(B | A) for factor B
• SS(AB | A, B) for interaction AB

Type II ANOVA:

• SS(A | B) for factor A
• SS(B | A) for factor B
• SS(AB | A, B) for interaction AB

Type III ANOVA:

• SS(A | B, AB) for factor A
• SS(B | A, AB) for factor B
• SS(AB | A, B) for interaction AB

You can calculate them using R and the following shows the examples. First, we will generate the data being use later. The data has two categorical IVs (cities and stores) and one DV (sales).

```x_1 = rep(c('City1','City2'),each=5)
x_2 = rep(c('store1','store2'), 5)
sales=c(10,20,20,50,30,10,5,4,12,4)

df <- data.frame (cities  = x_1,
stores = x_2,
sales=sales)
print(df)```

The following is the output:

```   cities stores sales
1   City1 store1    10
2   City1 store2    20
3   City1 store1    20
4   City1 store2    50
5   City1 store1    30
6   City2 store2    10
7   City2 store1     5
8   City2 store2     4
9   City2 store1    12
10  City2 store2     4```

## 2. Type I ANOVA in R

You can use either anova() or aov() to do Type I ANOVA in R.

```#Use anova() to Calculate Type I ANOVA in R
sales_ANOVA <- lm(sales ~ cities*stores, data = df)

ANOVA_result<-anova(sales_ANOVA)
print(ANOVA_result)```

Output:

```Analysis of Variance Table

Response: sales
Df Sum Sq Mean Sq F value  Pr(>F)
cities         1 902.50  902.50  7.7523 0.03182 *
stores         1  93.75   93.75  0.8053 0.40408
cities:stores  1 183.75  183.75  1.5784 0.25569
Residuals      6 698.50  116.42
```
```#Use aov() to Calculate Type I ANOVA in R
aov_result<-aov(sales ~ cities*stores, data = df)
summary(aov_result)```

Output:

```              Df Sum Sq Mean Sq F value Pr(>F)
cities         1  902.5   902.5   7.752 0.0318 *
stores         1   93.8    93.8   0.805 0.4041
cities:stores  1  183.7   183.7   1.578 0.2557
Residuals      6  698.5   116.4

```

As we can see that, the p-values from anova() and aov() are exactly the same. You can read an independent tutorial about Type 1 ANOVA in R here.

## 3. Type II ANOVA in R

We can use the Anova() function within the package of Companion to Applied Regression (CAR) to calculate Type II ANOVA in R.

Note that, the Anova() function only provides options of Type II and Type III ANOVA. Thus, you can not use it to calculate Type I ANOVA.

```#Using Anova() for Type II ANOVA
result4<-car::Anova(lm(sales ~ cities*stores, data = df),type=2)
print(result4)```

Output:

```Anova Table (Type II tests)

Response: sales
Sum Sq Df F value  Pr(>F)
cities        984.15  1  8.4537 0.02707 *
stores         93.75  1  0.8053 0.40408
cities:stores 183.75  1  1.5784 0.25569
Residuals     698.50  6
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```

We can see that the SS (cities) is 984.15, which is different from SS (cities) in Type I ANOVA. You can read an independent tutorial about Type 2 ANOVA in R here.

## 4. Type III ANOVA in R

We use Anova() function from CAR to calculate Type III ANOVA. The following is the R code.

```# Using Anova() for Type III ANOVA
result5<-car::Anova(lm(sales ~ cities*stores, data = df),type=3)
print(result5)```

Output:

```Anova Table (Type III tests)

Response: sales
Sum Sq Df F value  Pr(>F)
(Intercept)   1200.00  1 10.3078 0.01835 *
cities         158.70  1  1.3632 0.28728
stores         270.00  1  2.3193 0.17861
cities:stores  183.75  1  1.5784 0.25569
Residuals      698.50  6
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1```

As we can see, the results are different from what are in Type I and Type II ANOVA. For more detailed discussions about Type 3 ANOVA in R, please refer to this tutorial.