Learn how to perform an Analysis Of VAriance (ANOVA) in R to compare 3 groups or more. See also how to interpret the results and test the assumptions

ANOVA (ANalysis Of VAriance) is a statistical test to determine whether two or more population means are different. In other words, it is used to **compare two or more groups** to see if they are significantly **different**.

In practice, however, the:

**Student t-test**is used to compare**2 groups**;**ANOVA**generalizes the t-test beyond 2 groups, so it is used to compare**3 or more groups**.

Note that there are several versions of the ANOVA (e.g., one-way ANOVA, two-way ANOVA, mixed ANOVA, repeated measures ANOVA, etc.). In this article, we present the simplest form only — the **one-way ANOVA**1 — and we refer to it as ANOVA in the remaining of the article.

Although ANOVA is used to make inference about means of different groups, the method is called “analysis of *variance*”. It is called this because it compares the “between” variance (the variance between the different groups) and the variance “within” (the variance within each group). If the between variance is significantly larger than the within variance, the group means are declared to be different. Otherwise, we cannot conclude one way or the other. The two variances are compared to each other by taking the ratio (between variance/within variance) and then by comparing this ratio to a threshold from the Fisher probability distribution (a threshold based on a specific significance level, usually 5%).

This is enough theory regarding the ANOVA method for now. In the remaining of this article, we discuss it from a more practical point of view, and in particular, we will cover the following points:

- the aim of the ANOVA, when it should be used and the null/alternative hypothesis
- the underlying assumptions of the ANOVA and how to check them
- how to perform the ANOVA in R
- how to interpret results of the ANOVA
- understand the notion of post-hoc test and interpret the results
- how to visualize results of ANOVA and post-hoc tests

Data for the present article is the `penguins`

dataset (an alternative to the well-known `iris`

dataset), accessible via the {palmerpenguins} package:

```
## install.packages("palmerpenguins")
library(palmerpenguins)
```

The dataset contains data for 344 penguins of 3 different species (Adelie, Chinstrap and Gentoo). The dataset contains 8 variables, but we focus only on the flipper length and the species for this article, so we keep only those 2 variables:

```
library(tidyverse)
dat <- penguins %>%
select(species, flipper_length_mm)
```

(If you are unfamiliar with the pipe operator (`%>%`

), you can also select variables with `penguins[, c("species", "flipper_length_mm")]`

. Learn more ways to select variables in the article about data manipulation.)

Below some basic descriptive statistics and a plot (made with the {ggplot2} package) of our dataset before we proceed to the goal of the ANOVA:

```
summary(dat)
### species flipper_length_mm
### Adelie :152 Min. :172.0
### Chinstrap: 68 1st Qu.:190.0
### Gentoo :124 Median :197.0
### Mean :200.9
### 3rd Qu.:213.0
### Max. :231.0
### NA's :2
```

Flipper length varies from 172 to 231 mm, with a mean of 200.9 mm. There are respectively 152, 68 and 124 penguins of the species Adelie, Chinstrap and Gentoo.

```
library(ggplot2)
ggplot(dat) +
aes(x = species, y = flipper_length_mm, color = species) +
geom_jitter() +
theme(legend.position = "none")
```

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