MODEL Procedure

The NORMAL option in the FIT statement performs multivariate and univariate tests of normality.

The three multivariate tests provided are Mardia’s skewness test and kurtosis test (Mardia 1970) and the Henze-Zirkler test (Henze and Zirkler 1990). The two univariate tests provided are the Shapiro-Wilk W test and the Kolmogorov-Smirnov test. (For more information about the univariate tests, see the "Goodness-of-Fit Tests" section of the "The UNIVARIATE Procedure" chapter in the Base SAS Procedures Guide.) The null hypothesis for all these tests is that the residuals are normally distributed.

For a random sample , , where d is the dimension of and n is the number of observations, a measure of multivariate skewness is

where S is the sample covariance matrix of X. For weighted regression, both S and are computed by using the weights supplied by the WEIGHT statement or the _WEIGHT_ variable.

Mardia showed that under the null hypothesis is asymptotically distributed as . For small samples, Mardia’s skewness test statistic is calculated with a small sample correction formula, given by where the correction factor k is given by . Mardia’s skewness test statistic in PROC MODEL uses this small sample corrected formula.

A measure of multivariate kurtosis is given by

Mardia showed that under the null hypothesis, is asymptotically normally distributed with mean and variance .

The Henze-Zirkler test is based on a nonnegative functional that measures the distance between two distribution functions and has the property that

if and only if

where is a d-dimensional normal distribution.

The distance measure can be written as

where and are the Fourier transforms of P and Q, and is a weight or a kernel function. The density of the normal distribution is used as

where .

The parameter depends on as

The test statistic computed is called and is approximately distributed as a lognormal. The lognormal distribution is used to compute the null hypothesis probability,

where

Monte Carlo simulations suggest that has good power against distributions with heavy tails.

The Shapiro-Wilk W test is computed only when the number of observations (n) is less than 2,000, while computation of the Kolmogorov-Smirnov test statistic requires at least 2,000 observations.

The following is an example of the output produced by the NORMAL option:

proc model data=test2;
   y1 = a1 * x2 * x2 - exp( d1*x1);
   y2 = a2 * x1 * x1 + b2 * exp( d2*x2);
   fit y1 y2 / normal ;
run;