*74*

When weâ€™d like to test whether or not a single variable is normally distributed, we can create a Q-Q plot to visualize the distribution or we can perform a formal statistical test like an Anderson Darling Test or a Jarque-Bera Test.

However, when weâ€™d like to test whether or notÂ *severalÂ *variables are normally distributed as a group we must perform aÂ **multivariate normality test**.

This tutorial explains how to perform the following multivariate normality tests for a given dataset in R:

- Mardiaâ€™s Test
- Energy Test
- Multivariate Kurtosis and Skew Tests

**Related:Â **If weâ€™d like to identify outliers in a multivariate setting, we can use the Mahalanobis distance.

**Example: Mardiaâ€™s Test in R**

**Mardiaâ€™s Test** determines whether or not a group of variables follows a multivariate normal distribution. The null and alternative hypotheses for the test are as follows:

H_{0} (null): The variables follow a multivariate normal distribution.

H_{a} (alternative): The variables *do not *follow a multivariate normal distribution.

The following code shows how to perform this test in R using theÂ **QuantPsyc** package:

library(QuantPsyc) #create dataset set.seed(0) data #perform Multivariate normality test mult.norm(data)$mult.test Beta-hat kappa p-val Skewness 1.630474 13.5872843 0.1926626 Kurtosis 13.895364 -0.7130395 0.4758213

TheÂ **mult.norm()Â **function tests for multivariate normality in both the skewness and kurtosis of the dataset. Since both p-values are not less than .05, we fail to reject the null hypothesis of the test. We donâ€™t have evidence to say that the three variables in our dataset do not follow a multivariate distribution.

**Example: Energy Test in R**

An **Energy****Â Test** is another statistical test that determines whether or not a group of variables follows a multivariate normal distribution. The null and alternative hypotheses for the test are as follows:

H_{0} (null): The variables follow a multivariate normal distribution.

H_{a} (alternative): The variables *do not *follow a multivariate normal distribution.

The following code shows how to perform this test in R using the **energy **package:

library(energy) #create dataset set.seed(0) data #perform Multivariate normality test mvnorm.etest(data, R=100) Energy test of multivariate normality: estimated parameters data: x, sample size 50, dimension 3, replicates 100 E-statistic = 0.90923, p-value = 0.31

The p-value of the test isÂ **0.31**. Since this is not less than .05, we fail to reject the null hypothesis of the test. We donâ€™t have evidence to say that the three variables in our dataset do not follow a multivariate distribution.

**Note:Â **The argument R=100 specifies 100 boostrapped replicates to be used when performing the test. For datasets with smaller sample sizes, you may increase this number to produce a more reliable estimate of the test statistic.

**Additional Resources**

How to Create & Interpret a Q-Q Plot in R

How to Conduct an Anderson-Darling Test in R

How to Conduct a Jarque-Bera Test in R

How to Perform a Shapiro-Wilk Test in R