Multivariate Normality accepted at 5% significance level but rejected at 1% ??

#1
Hi,
I am new to this forum. I am trying to test whether a set of variables in the data follow multivariate normal distribution. I found Mardia's test for this and below is the summary of the results :

Mardia Skewness 58.29 <.0001
Mardia Kurtosis 2.43 0.0149
Henze-Zirkler T 7.52 <.0001

Taking 5% level of significance, I can reject the null hypothesis (data is multivariate normal). However, if i decrease the significance level to 1% and be more strict on my test, I am not able to reject the null hypothesis at 1%.

That means, out of 100 samples I draw, 95 of them will not follow multivariate normal distribution.

and on the other hand, out of 100 sample I draw, 99 of them will follow multivariate normal distribution.

How is this possible?
Could someone please explain me where I am misinterpreting the result, and whether I should reject the test?

Also, the QQ plot of each individual variable indicates normality(straight line at diagonal) but Shapiro wilk test rejects the normality everywhere.

Equation Test Statistic Value Prob
x1 Shapiro-Wilk W 0.80 <.0001
x2 Shapiro-Wilk W 0.93 0.0394
x3 Shapiro-Wilk W 0.86 0.0003
x4 Shapiro-Wilk W 0.92 0.0199


I am assuming that the null hypothesis for Shapiro Wilk test is : Data is normal.

Could someone please explain me whether I should reject or accept the normality of the data ? Also, it would be great if I can be informed of some SAS procedure to test multivariate normality.

Thanks in Advance.
 

noetsi

Fortran must die
#2
To begin with I don't think that the confidence levels mean what you reference above (I have never seen it stated that way). They deal with how confident you are of the null not how often you would come up with a given distribution or result.

To me confidence levels tell you how certain you can be that the null is not correct. At the five percent level you are sure the null is not correct (which in this case means you can assume the data is multivariate normal). But you are not confident enough to reject the null at the 99 percent confidence level (which is another way to say that your are 95 confident that the null is wrong, but not 99 percent confident).

But you probably need more theoretically grounded individuals than I to address this. I do think the way your are interpreting the confidence level is not the normal way it is done which is what is creating your confusion.