Normality Test for Multi-Grouped Data in R

When analyzing multi-grouped data in R, it's crucial to assess whether the data within each group follows a normal distribution. The assumption of normality is vital for many statistical tests like ANOVA and t-tests. This article provides a detailed explanation of how to perform normality tests for multi-grouped data in R, using common methods such as the Shapiro-Wilk Test, Q-Q plots, and Kolmogorov-Smirnov Test.

Why Test for Normality?

In statistics, many tests assume that the data follows a normal distribution. For example:

• ANOVA (Analysis of Variance) assumes that the residuals are normally distributed within each group.
• t-tests require the normality assumption for the data in each group.
• If data is non-normal, other methods such as non-parametric tests (Kruskal-Wallis) may be used.

By conducting normality tests, we ensure that the data meets the assumptions of these tests. now we will discuss different Methods for Testing Normality using R Programming Language.

1: Shapiro-Wilk Test

The Shapiro-Wilk test is one of the most commonly used tests for checking normality. It tests the null hypothesis that the data is normally distributed.

shapiro.test(x)

2: Kolmogorov-Smirnov Test

Kolmogorov-Smirnov Test compares the sample distribution to a reference normal distribution. However, it has limitations when testing normality, particularly with small samples.

ks.test(x, "pnorm", mean(x), sd(x))

3: Q-Q Plot (Quantile-Quantile Plot)

A Q-Q plot helps visually assess the normality of data. If the data points fall approximately on the reference line, the data is considered normally distributed.

qqnorm(x)
qqline(x)

4: Anderson-Darling Test

The Anderson-Darling test is a more powerful version of the Kolmogorov-Smirnov test, specifically designed for detecting deviations from normality in the tails of the distribution.

library(nortest)
ad.test(x)

Now we Performing Normality Tests for Multi-Grouped Data. Suppose we have data from multiple groups and want to check whether each group follows a normal distribution.

Step 1: Load and Explore the Data

We’ll use the built-in iris dataset, which contains data for three species of flowers.

# Load dataset
data(iris)

# View the first few rows
head(iris)

Output:

  Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1          5.1         3.5          1.4         0.2  setosa
2          4.9         3.0          1.4         0.2  setosa
3          4.7         3.2          1.3         0.2  setosa
4          4.6         3.1          1.5         0.2  setosa
5          5.0         3.6          1.4         0.2  setosa
6          5.4         3.9          1.7         0.4  setosa

The iris dataset has five columns: Sepal.Length, Sepal.Width, Petal.Length, Petal.Width, and Species.

Step 2: Split Data by Group

We can split the data based on the Species column to perform the normality test for each species.

# Split data by species
iris_split <- split(iris$Sepal.Length, iris$Species)

Step 3: Perform Shapiro-Wilk Normality Test

We can apply the Shapiro-Wilk test to each group to assess normality.

# Apply Shapiro-Wilk test to each group
lapply(iris_split, shapiro.test)

Output:

$setosa

	Shapiro-Wilk normality test

data:  X[[i]]
W = 0.9777, p-value = 0.4595


$versicolor

	Shapiro-Wilk normality test

data:  X[[i]]
W = 0.97784, p-value = 0.4647


$virginica

	Shapiro-Wilk normality test

data:  X[[i]]
W = 0.97118, p-value = 0.2583

The output will return the W statistic and p-value for each species. If the p-value is greater than 0.05, the data is considered normally distributed.

Step 4: Visualize with Q-Q Plots

We can generate Q-Q plots for each group to visually assess normality.

# Q-Q plot for each species
par(mfrow = c(1, 3))  # Set layout for 3 plots
for (species in names(iris_split)) {
  qqnorm(iris_split[[species]], main = paste("Q-Q Plot for", species))
  qqline(iris_split[[species]])
}
par(mfrow = c(1, 1))  # Reset layout

Output:

• Shapiro-Wilk Test: The null hypothesis is that the data is normally distributed. If the p-value is greater than 0.05, we fail to reject the null hypothesis, indicating the data is normally distributed.
• Q-Q Plot: If the data points align closely to the reference line, the data is considered to be normally distributed.

Conclusion

Testing for normality is an essential step in ensuring that assumptions for parametric tests, like ANOVA or t-tests, are met. In R, there are multiple ways to test for normality in multi-grouped data, including the Shapiro-Wilk test, Q-Q plots, and the Kolmogorov-Smirnov test. By performing these tests, you can make informed decisions about whether to proceed with parametric tests or opt for non-parametric alternatives.