# Assessing Classical Test Assumptions

In classical parametric procedures we often assume normality and constant variance for the model error term. Methods of exploring these assumptions in an ANOVA/ANCOVA/MANOVA framework are discussed here. Regression diagnostics are covered under multiple linear regression.

## Outliers

Since outliers can severly affect normality and homogeneity of variance, methods for detecting disparate observerations are described first.

The **aq.plot( )** function in the

**mvoutlier**package allows you to identfy multivariate outliers by plotting the ordered squared robust Mahalanobis distances of the observations against the empirical distribution function of the MD

^{2}

_{i}. Input consists of a matrix or data frame. The function produces 4 graphs and returns a boolean vector identifying the outliers.

`# Detect Outliers in the MTCARS Data`

library(mvoutlier)

outliers <-

aq.plot(mtcars[c("mpg","disp","hp","drat","wt","qsec")])

outliers # show list of outliers

## Univariate Normality

You can evaluate the normality of a variable using a **Q-Q plot**.

`# Q-Q Plot for variable MPG `

attach(mtcars)

qqnorm(mpg)

qqline(mpg)

Significant departures from the line suggest violations of normality.

You can also perform a Shapiro-Wilk test of normality with the **shapiro.test(***x***)** function, where *x* is a numeric vector. Additional functions for testing** **normality are available in **nortest** package.

## Multivariate Normality

MANOVA assumes **multivariate normality**. The function** mshapiro.test( )** in the **mvnormtest** package produces the Shapiro-Wilk test for multivariate normality. Input must be a numeric matrix.

```
# Test Multivariate Normality
```

mshapiro.test(M)

If we have *p* x 1 multivariate normal random vector

then the squared Mahalanobis distance between ** x** and μ is going to be chi-square distributed with

*p*degrees of freedom. We can use this fact to construct a

**Q-Q plot**to assess multivariate normality.

`# Graphical Assessment of Multivariate Normality`

x <- as.matrix(mydata) # n x p numeric matrix

center <- colMeans(x) # centroid

n <- nrow(x); p <- ncol(x); cov <- cov(x);

d <-
mahalanobis(x,center,cov) # distances

qqplot(qchisq(ppoints(n),df=p),d,

main="QQ Plot Assessing Multivariate Normality",

ylab="Mahalanobis D2")

abline(a=0,b=1)

## Homogeneity of Variances

The **bartlett.test( ) **function provides a parametric K-sample test of the equality of variances. The **fligner.test( ) **function provides a non-parametric test of the same. In the following examples **y** is a numeric variable and **G** is the grouping variable.

`# Bartlett Test of Homogeneity of Variances`

bartlett.test(y~G, data=mydata)

# Figner-Killeen Test of Homogeneity of Variances

fligner.test(y~G, data=mydata)

The **hovPlot( )** function in the **HH** package provides a graphic test of homogeneity of variances based on Brown-Forsyth. In the following example, **y** is numeric and **G** is a grouping factor. Note that G **must** be of type factor.

`# Homogeneity of Variance Plot`

library(HH)

hov(y~G, data=mydata)

hovPlot(y~G,data=mydata)

## Homogeneity of Covariance Matrices

MANOVA and LDF assume homogeneity of variance-covariance matrices. The assumption is usually tested with** Box's M**. Unfortunately the test is very sensitive to violations of normality, leading to rejection in most typical cases. **Box's M** is not included in **R**, but code is available.