## Measures of Multivariate Kurtosis

In many applications, the manifest variables are
not even approximately multivariate normal.
If this happens to be the case with your data set, the
default generalized least-squares and maximum likelihood estimation
methods are not appropriate, and you should compute the parameter
estimates and their standard errors by an asymptotically
distribution-free method, such as the WLS estimation method.
If your manifest variables are multivariate normal,
then they have a zero relative multivariate kurtosis, and
all marginal distributions have zero kurtosis (Browne 1982).
If your DATA= data set contains raw data, PROC CALIS computes univariate
skewness and kurtosis and a set of multivariate kurtosis values.
By default, the values of univariate skewness and kurtosis are corrected
for bias (as in PROC UNIVARIATE), but using the
BIASKUR option enables you to compute the uncorrected values also.
The values are displayed when you specify the PROC CALIS statement option
KURTOSIS.
If variable *Z*_{j} is normally distributed, the uncorrected univariate kurtosis
is equal to 0. If *Z* has an *n*-variate normal distribution,
Mardia's multivariate kurtosis is equal to 0.
A variable *Z*_{j} is called *leptokurtic* if it has a
positive value of and is called *platykurtic* if
it has a negative value of . The values of
, , and should not be
smaller than a lower bound (Bentler 1985):

PROC CALIS displays a message if this happens.
If weighted
least-squares estimates
(METHOD=WLS or METHOD=ADF) are specified and the weight matrix is computed from an
input raw data set, the CALIS procedure computes two further measures of
multivariate kurtosis.

The occurrence of significant nonzero values of Mardia's multivariate kurtosis
and significant amounts of some of the univariate kurtosis
values indicate that your variables are not multivariate
normal distributed. Violating the multivariate normality assumption in
(default) generalized least-squares and maximum likelihood estimation
usually leads to the wrong approximate standard errors and incorrect fit
statistics based on the value. In general, the parameter
estimates are more stable
against violation of the normal distribution
assumption. For more details, refer to Browne (1974, 1982, 1984).

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.