Checking normality of multivariate data

Multivariate Probability Distributions in R

Surajit Ray

Professor, University of Glasgow

Why check normality?

Classical statistical techniques that assume univariate/multivariate normality:
- Multivariate regression
- Discriminant analysis
- Model-based clustering
- Principal component analysis (PCA)
- Multivariate analysis of variance (MANOVA)

Review: univariate normality tests

qqnorm(iris_raw[, 1])
qqline(iris_raw[, 1])

If the values lie along the reference line the distribution is close to normal

Review: univariate normality tests

qqnorm(iris_raw[, 1])
qqline(iris_raw[, 1])

If the values lie along the reference line the distribution is close to normal
Deviation from the line might indicate
- heavier tails
- skewness
- outliers
- clustered data

qqnorm of all variables

mvn(iris_raw[, 1:4], univariatePlot = "qqplot")

MVN library multivariate normality test functions

MVN version 5.9

Multivariate normality tests by
- Mardia
- Henze-Zirkler
- Royston
Graphical appoaches
- chi-square Q-Q
- perspective
- contour plots

MVN library multivariate normality test functions

Multivariate normality tests by
- Mardia $\checkmark$
- Henze-Zirkler $\checkmark$
- ~~Royston~~
Graphical appoaches
- chi-square Q-Q $\checkmark$
- ~~perspective~~
- ~~contour plots~~

Using Mardia Test to check multivariate normality

mvn(iris_raw[, 1:4],mvnTest = "mardia")

$multivariateNormality 
             Test          Statistic              p value Result
1 Mardia Skewness   67.4305087780629 4.75799820400705e-07     NO
2 Mardia Kurtosis -0.230112114480775    0.818004651478188    YES
3             MVN               <NA>                 <NA>     NO

Using qqplot from Mardia Test to check multivariate normality

 mvn(iris_raw[, 1:4],mvnTest = "mardia", multivariatePlot = "qq")

Using Henze-Zirkler's test to check multivariate normality

mvn(iris_raw[, 1:4],mvnTest = "hz")

$multivariateNormality
           Test  HZ p value MVN
1 Henze-Zirkler 2.3       0  NO

Testing multivariate normality by species

mvn(iris[iris$Species  == "setosa", 1:4], 
     mvnTest = "mardia",
     multivariatePlot = "qq")

$multivariateNormality
             Test        Statistic           p value Result
1 Mardia Skewness 25.6643445196298 0.177185884467652    YES
2 Mardia Kurtosis 1.29499223711605 0.195322907441935    YES
3             MVN             <NA>              <NA>    YES

Let's make use of the tests for multivariate normality!

Multivariate Probability Distributions in R