A recent query asked about tests for multivariate normality. It has long
seemed to me that the term "tests for normality" (even for univariate ones)
is a misleading name since they are really tests for NON-normality, and as I
understand matters, most have only low power to find that (unless sample
size
is very large).
As with any significance test, failure to reject the null hypothesis does
not provide evidence FOR that hypothesis. In this application, failing to
demonstrate non-normality does not (or at least should not) leave you at all
confident that you *have* normality. If normality is important for some
reason, then in my view, one should use normal probability plots, empirical
density plots, and possibly estimates of skewness and kurtosis, along with
judgment, to decide whether the data at hand are "normal enough" or not.
I suggest that anyone using this kind of test should generate 1000 or so
datasets (of sample size comparable to their own) with normality that is
marginally unacceptable in the way suspected for one's data. Then apply the
proposed test to those datasets, and check whether it performs adequately.
Note that "tests for homogeneity of variance" are (il)logically similar
---they are really tests for non-homogeneity, and a non-significant result
provides little evidence that "all is well." Isn't the ratio of the
largest
to the smallest variance more relevant than a p value if skedasticity is at
issue?
Dave Parkhurst
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