Dale

Regarding the visual approach, I absolutely agree. QQ plots can also be very useful. But, if one does not have SAS-Graph or some other package to do this in, then it is difficult. Personally, I display distributions in S-Plus. Also, while the graphical approach is, in many respects, the best, it is hard to report on, especially to (say) journal editors. "JUST LOOK AT IT!!" is not acceptable (more's the pity).

This came up for me in my dissertation, I had data which was shown, by KS to be NONnormal. I used some sort of language like "none of the distributions were grossly nonnormal, all were unimodal, and none had absolute skewness above XXX or kurtosis above XXXX".

Regarding the other tests, thanks for the info! This is, to me, a fascinating area.

Peter

>>> "Dale McLerran" <dmclerra@my-deja.com> 06/29/01 02:16PM >>> Peter,

You have a point. However, rather than looking at tests of skewness and kurtosis in this situation, I would prefer the visual inspection approach of plotting a histogram of the data with a normal density curve superimposed on it. You can demonstrate right from the figure what problems the distribution has with regard to assumptions about normality. You can also use this as a basis for selecting a transformation to the data to achieve approximate normality.

Note, though, that there are a number of tests of normality which are not as sensitive to large sample effects as the tests reported by PROC UNIVARIATE. A couple of tests that I am aware of are actually constructed to test whether a distribution is Uniform(0,1). These tests then proceed by first standardizing the data to have mean 0 and variance 1 (z-statistics) and then computing the probability values under the normal distribution of the z-statistics. If normality holds, then the probability values will be distributed Uniform(0,1). Now, one test of the Uniform distribution which would not be as sensitive to sample size as a K-S test would be to count the number of observations in intervals of length 0.05 or 0.10. We know that if the Uniform(0,1) distribution holds, then the number of observations in each interval should be equal. We can perform a chi-square test of this assumption.

Another test I have recently become aware of is the Birnbaum test. As with the chi-square test above, we use the probability values which arise from the normal distribution for the z-statistics. The mean probability value is then used to compute the Birnbaum test statistic. Call the mean of the probability values Pbar. Then the test statistic is

birnbaum = sqrt(N) * (0.5 - Pbar)

The asymptotic variance estimate of the Birnbaum statistic is 1/12. For large N, we can test whether the p-values are Uniform(0,1) by comparing birnbaum/sqrt(12) to tables of the standard normal dist. Note that we are only testing whether the mean p-value differs from 0.5. If the data are symmetric but severely kurtotic, then the Birnbaum statistic would be inappropriate. A chi-square test as presented above would be preferrable.

Dale