Multivariate Kurtosis vs Multivariate normality in AMOS

[S.Faridah Syed Alwi]

<FONT face="Default Sans Serif,Verdana,Arial,Helvetica,sans-serif" size=2><DIV>Hi,</DIV><DIV>&nbsp;</DIV><DIV>I would like to ask how to assess&nbsp;multivariate normality in AMOS.&nbsp;I need to get clarification on the following:</DIV><DIV>&nbsp;</DIV><DIV>(1)&nbsp;I'm confused when Hair et al., (1998) mentioned the cut-off point for multivariate normality is 3. However, this is impossible as multivariate kurtosis in the multivariate normality assessment&nbsp;frequently shows more 10 when involve more than 40&nbsp;items?</DIV><DIV>(2) Does Multivariate Kurtosis and Multivariate Normality is the same?</DIV><DIV>(3) Can I check for&nbsp;multivariate normality in&nbsp;AMOS, not other software?.</DIV><DIV>&nbsp;</DIV><DIV>I would appreciate any&nbsp;thought on these issues.&nbsp;<BR></DIV><DIV>Thanks</DIV><DIV>&nbsp;</DIV><DIV>Sh Faridah</DIV><DIV>Manchester Business School</DIV><DIV>Booth Street West, Manchester<BR>M15 9BU, United Kingdom<BR>Tel: [phone] (Office)<BR>Tel: [phone] (Research Degree Office)<BR> <BR></DIV></FONT>

[Jason Cole]

Hello Sh,



I'll see if I can send you on the right track. Here's some relevant
information for your questions.



(1) I'm confused when Hair et al., (1998) mentioned the cut-off point
for multivariate normality is 3. However, this is impossible as
multivariate kurtosis in the multivariate normality assessment
frequently shows more 10 when involve more than 40 items?



Recall that there are many different assessment of normality and
multivariate normality. I am not familiar with Hair et al.'s paper, but
I can say that the cut off is standard WHEN the test uses a normalized
coefficient (i.e., the normality estimate divided by the normality
error). This is a common technique, and fits with what you will likely
see referenced much on SEMNET, Mardia's coefficient of multivariate
normality (or sometimes called multivariate kurtosis). When I see a
score of 10 or more I think you may be examining one of two things:
either (A) VERY abnormal data (not too surprising in studying disease
and psychological measures in the general populations, for example) or
(B) examining the Mahalanobis coefficient. This is used to examine the
influence of any one case of the multivariate normality rather than
getting an overall assessment of multivariate normality.





(2) Does Multivariate Kurtosis and Multivariate Normality is the same?



Well, that depends of the source. In tradition normality discussion,
one will label normality to describe the overall fit of a distribution.
Kurtosis is a specific measure of normality that examines the peakedness
of a distribution (rather than, say, skewness, which examines the skew
of the tails on a distribution). Nevertheless, some authors use these
terms more interchangeably.



(3) Can I check for multivariate normality in AMOS, not other software?



Certainly, AMOS provides multivariate normality checks. In the Analysis
Options, under output, check the option for tests for normality and
outliers. This will provide two outputs. The first box is an
examination of normality. It provides univariate normality for each
dependent manifest variable (recall that normality of independent
variables is not a problem for assumption issues) and then the last line
provides Mardia's estimate for multivariate normality (and it is a
standardized value). The second table provides an examination of
Mahalanobis tests for each case (outlier examination). The help file in
AMOS provides more information on interpreting these values, but it is
not really any different that interpreting a typical Mahalanobis
formula.



HTH,



Jason







____________________________________



Jason C. Cole, PhD

Senior Research Scientist & President

Consulting Measurement Group, Inc.

Tel: 866 STATS 99 (ex. 5)

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________________________________

From: Structural Equation Modeling Discussion Group
[mailto:***@BAMA.UA.EDU] On Behalf Of S.Faridah Syed Alwi
Sent: Friday, July 08, 2005 9:27 AM
To: ***@BAMA.UA.EDU
Subject: Re: Multivariate Kurtosis vs Multivariate normality in AMOS



Hi,



I would like to ask how to assess multivariate normality in AMOS. I need
to get clarification on the following:



(1) I'm confused when Hair et al., (1998) mentioned the cut-off point
for multivariate normality is 3. However, this is impossible as
multivariate kurtosis in the multivariate normality assessment
frequently shows more 10 when involve more than 40 items?

(2) Does Multivariate Kurtosis and Multivariate Normality is the same?

(3) Can I check for multivariate normality in AMOS, not other software?.



I would appreciate any thought on these issues.

Thanks



Sh Faridah

Manchester Business School

Booth Street West, Manchester
M15 9BU, United Kingdom
Tel: [phone] (Office)
Tel: [phone] (Research Degree Office)

[David A. Heiser]

-----Original Message-----
From: Structural Equation Modeling Discussion Group
[mailto:***@BAMA.UA.EDU]On Behalf Of S.Faridah Syed Alwi
Sent: Friday, July 08, 2005 9:27 AM
To: ***@BAMA.UA.EDU
Subject: Re: Multivariate Kurtosis vs Multivariate normality in AMOS


Hi,

I would like to ask how to assess multivariate normality in AMOS. I need to
get clarification on the following:

(1) I'm confused when Hair et al., (1998) mentioned the cut-off point for
multivariate normality is 3. However, this is impossible as multivariate
kurtosis in the multivariate normality assessment frequently shows more 10
when involve more than 40 items?
(2) Does Multivariate Kurtosis and Multivariate Normality is the same?
(3) Can I check for multivariate normality in AMOS, not other software?.

I would appreciate any thought on these issues.

Thanks

Sh Faridah
Manchester Business School
Booth Street West, Manchester
M15 9BU, United Kingdom
Tel: [phone] (Office)
Tel: [phone] (Research Degree Office)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
First of all, I don't have AMOS or any of the other SEM programs, so I don't
know how Mardia's theories have been implemented. Second, Mardia had a
brilliant insight on how to apply Fisher's skewness and kurtosis concepts
from a single variate to the multivariate situation.

Multivariate normality is a concept, not a measurable entity. Multivariate
skewness and kurtosis are measurable quantities. Mardia's multivariate
skewness differs from the univariate skewness measures of Fisher and
Pearson. However the concept that normality occurs when the calculated
skewness measures are close to zero, is still valid.

Mardia's multivariate kurtosis has similarities to the Fisher/Pearson single
variate kurtosis measures. For a normal distribution, kurtosis as a ratio of
moments comes out have a value of 3. Most single variate statistical
software returns 3 minus this value, and therefore returns a zero value of
kurtosis for the normal case. This is an area of great confusion to most
users and the terminology used by the "sources" is totally inconsistent. You
can only determine what your program does by finding the values calculated
from a reference data base.

Skewness and kurtosis have to be computed directly from the raw data base.
They cannot be obtained from the covariance/correlation matrix.

High kurtosis values have the effect of invalidating chi-square measures
(i.e. valid p values from a chi-square measure). The theories come from the
1930's. Since there is no universal non-normality distribution, you can't
determine quantitatively what a high kurtosis condition has on your exact
fit chi-square measures. One can do a lot of Monte Carlo studies on this,
but they are all based on taking some known distribution as the "non-normal"
distribution together with some reference model.

One way out of this is to do extensive bootstrapping. However this fails
when no permutation of you data fits your model.

DAH

[John Williams]

Post by Jason Cole
(1) I'm confused when Hair et al., (1998) mentioned the cut-off point
for multivariate normality is 3.

Read again. I would be very surprised if they say that. They probably
say "if the data come from a Normal distribution the kurtosis value will
be 3" or something similar.

As noted in another reply, some software subtracts 3 from the value, so
that it is reported as zero when the data are multivariate Normal.

Post by Jason Cole
(2) Does Multivariate Kurtosis and Multivariate Normality is the same?

No. Multivariate Normality is a necessary but not sufficient condition
to judge a distribution to be well-approximated by the Multivariate
Normal.