One-way ANOVA (analysis of variance) is an effective tool for comparing three or more groups for differences. ANOVA is a statistical method for testing differences between two or more means. One-way ANOVA works when there’s one independent variable such as a treatment group and one dependent variable such as effectiveness measurement.

## Selecting a Package

For this tutorial, main package we will be using is called “car”, which is short for “Companion to Applied Regression”. This package has functions for performing various statistical analyses, including ANOVA.

To install the “car” package, we use the following code:

```
install.packages("car")
library(car)
```

Now that we have the “car” package installed and loaded, we can begin setting up our data for the one-way ANOVA analysis. We will need a data frame with at least two columns: one for the grouping variable (the factor) and one for the continuous variable (the response).

For example, let’s say we have a dataset of exam scores for students in three different classes:

Class | Score |
---|---|

1 | 85 |

2 | 78 |

1 | 92 |

3 | 80 |

2 | 79 |

3 | 87 |

We can create a data frame from this data using the following code:

```
set.seed(1233423423)
class <- sample(1:3, 30, replace = TRUE)
score <- round(runif(30, 70, 100), 0)
exam_scores <- data.frame(class, score)
```

This code uses the sample() function to randomly select numbers between 1 and 3, replacing them 30 times in order to create the class column. The score column is created with runif(), which generates 30 random numbers between 70 and 100 that are then rounded to the nearest integer using round(). Finally, data.frame() is employed in order to combine both columns into a data.frame called exam_scores, with seed parameter locked into specific starting point generation for consistent results.

We’re going to do a quick check on our data, looking at the means of each group:

```
> tapply(exam_scores$score, exam_scores$class, mean)
1 2 3
81.87500 84.93750 88.83333
```

Looks um… pretty random… as we would have expected….

Now we are ready to perform the one-way ANOVA analysis on our data using the “aov()” function from the “car” package. We will specify the response variable as the first argument and the grouping variable as the second argument:

`anova_results <- aov(score ~ class, data = exam_scores)`

With these few steps, we have installed the necessary package and set up our data for one-way ANOVA analysis. We are now ready to move on to the next step of the analysis, which is interpreting the results.

The results, by the way, are…

```
aov(formula = score ~ class, data = exam_scores)
Call:
aov(formula = score ~ class, data = exam_scores)
Terms:
class Residuals
Sum of Squares 164.7721 2235.9279
Deg. of Freedom 1 28
Residual standard error: 8.936138
Estimated effects may be unbalanced
```

More on how to interpret this in a moment… but first, a word about getting your data in clean shape…

## Data Preparation

Before performing a one-way ANOVA in R, it is essential to prepare the data. This section discusses the necessary steps to prepare the data for ANOVA analysis.

### Checking Data Assumptions

The first step in preparing data for one-way ANOVA is to check data assumptions. The assumptions of ANOVA include normality, homogeneity of variance, and independence. Violation of any of these assumptions can lead to inaccurate results.

To check for normality, we can use graphical methods such as histograms, Q-Q plots, and box plots. We can also perform statistical tests such as Shapiro-Wilk test or Anderson-Darling test. For homogeneity of variance, we can use graphical methods such as box plots or statistical tests such as Levene’s test or Bartlett’s test. To check for independence, we can use time series plots or autocorrelation plots.

### Data Cleaning and Transformation

The next step is to clean and transform the data. This involves identifying and handling missing values, outliers, and extreme values. We can use visual methods like box plots or statistical tests like Grubbs’ test or Dixon’s Q test to detect outliers. Missing values can be imputed with either mean, median, or mode values while extreme values should be handled through winsorizing or trimming off excess information from the dataset.

After cleaning the data, it may be necessary to transform it in order to meet the assumptions of ANOVA. For instance, if the distribution is irregular, we can transform it using logarithmic, square root or inverse transformations; however it should not alter how we interpret the results.

Here is an example of checking the normality assumption with a Q-Q plot and performing a logarithmic transformation in R:

# Checking normality assumption qqnorm(data) qqline(data) # Logarithmic transformation log_data <- log(data)

## One-Way ANOVA

One-Way ANOVA (One Way Analysis of Variance) is a statistical method used to test for any significant difference in the means between three or more groups. In this section, we’ll review the assumptions, run the One-Way ANOVA, and interpret its results using R.

### ANOVA Assumptions

Before running the One-Way ANOVA, it is important to check the following assumptions:

**Normality:**The dependent variable should be normally distributed in each group.**Homogeneity of variance:**The variance of the dependent variable should be equal across all groups.**Independence:**The observations in each group should be independent of each other.

You can verify the normality assumption using a normal probability plot or Shapiro-Wilk test. Similarly, homogeneity of variance can be checked using Levene’s test. If these conditions aren’t fulfilled, then you may need to transform the data or perform non-parametric testing.

### Running the One-Way ANOVA

In R, you can use the `aov()`

function to run the One-Way ANOVA. The syntax is as follows:

`model <- aov(dependent_variable ~ group_variable, data = your_data)`

For example, let’s say we want to test if there is any significant difference in the mean weight of scores across three different classes:

`model <- aov(score ~ class, data = exam_scores)`

### Interpreting ANOVA Results

After running the One-Way ANOVA, you can use the `summary()`

function to get the ANOVA table:

`summary(model)`

The ANOVA table presents the sum of squares, degrees of freedom, mean square, F-statistic, and p-value. The latter indicates whether there is a significant difference in means between groups. If the p-value falls below 0.05, we reject the null hypothesis and conclude there is indeed an effect between means.

For example, the ANOVA table for our apple weight example might look like this:

```
Df Sum Sq Mean Sq F value Pr(>F)
class 1 164.8 164.77 2.063 0.162
Residuals 28 2235.9 79.85
```

In this example, the p-value is greater than 0.05, so we accept the null hypothesis and conclude that there is no significant difference in the mean scores across the three classes. Which, as you can see from the orignial criteira we used to randomly generate the data, is as you would expect.

## Section 5: Post Hoc Analysis

### Multiple Comparison Tests

After performing the one-way ANOVA and detecting a significant difference between at least one pair of groups, multiple comparison tests must be conducted to establish which groups differ significantly from each other. There are various methods for conducting these analyses such as Tukey’s HSD, Bonferroni, and Scheffe’s tests.

Tukey’s HSD (honestly significant difference) test is a widely-used post hoc analysis method. It compares all possible pairs of means and determines which ones differ significantly from one another while controlling for the family-wise error rate. Bonferroni and Scheffe’s tests are other commonly employed techniques to control type I error rates.