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Date:         Fri, 7 Jul 2006 12:09:43 +0530
Reply-To:     Madan Gopal Kundu <Madan.Kundu@RANBAXY.COM>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Madan Gopal Kundu <Madan.Kundu@RANBAXY.COM>
Subject:      Re: Test of Normality
Comments: To: David L Cassell <davidlcassell@MSN.COM>
Content-Type: text/plain; charset="us-ascii"

Dear David,

Thanks for your so informative reply. I have one more question to you. Suppose I have a data which can be divided based on different treatment group. If the data within a or few treatment groups comes to be non normal whereas the total data follows normal distribution then can we apply ANOVA on that data?

Thanks and Regards, Madan

-----Original Message----- From: SAS(r) Discussion [mailto:SAS-L@LISTSERV.UGA.EDU] On Behalf Of David L Cassell Sent: Friday, July 07, 2006 11:58 AM To: SAS-L@LISTSERV.UGA.EDU Subject: Re: Test of Normality

Madan.Kundu@RANBAXY.COM wrote: >Hi, > >We know there is a number of statistical test to test the normality of a >distribution. some of these are: >1. Kolmogorov Smirnov test >2. Shapiro -Wilk's test >3. D'Agostino-Pearson Omnibus test >4. Cramer-vol Mises >5. Anderson-Darlling > >Now my questions are: >which test I should use and in what situation? >Which one is most powerful in what situation? >Which one is most conservative in what situation? > >Any comments on this topic will be appreciated. > >Thanks and Reards, >Madan Kundu >

Okay, first of all, you left out a bunch. And SAS does not do all known normality tests. There's also Filliben, Durbin, studentized range, Lilliefors, binned chi-square, skewness test, kurtosis test, and half a dozen that I'm not thinking of at the moment.

Then there's the question of what you want to *test*. Just saying a 'test of normality' is not all that helpful. Some tests have directional alternative hypotheses (like 'if non-normal the distribution is skewed to the right') and others do not. It turns out that matters.

The details of comparions of these critters are really extensive. Here are some papers you can look up for details:

Shapiro, Wilk, and Chen. 1968. A comparative study of various tests for normality. Journal of the American Statistical Association. 63 (December): pp. 1343-1372.

Pearson, D'Agostino, and Bowman. 1977. Tests for departure from normality: Comparison of powers. Biometrika 64 (August): pp. 231-246.

Here are more you can look up:

Gastwirth and Owens. 1977. Biometrika.

D'Agostino and Rosman. 1974. Biometrika.

Fama and Roll. 1971. JASA.

D'Agostino. 1973. Communications in Statistics.

Chen. 1971. JASA.

Dumonceaux, Antle, and Haas. 1973. Technometrics.

Uthoff. 1973. Annals of Statistics.

There are a million different combinations people have looked at. One-sided alternative or not. Symmetric alternative or not. Long- tailed or not. Contaminated distribution or not (and what *kind* of contamination: scale? location?). You could write a master's thesis just on a review of all these different options, and all the papers written since these seminal ones.

My bottom line: use Shapiro-Wilk.

My 2nd line: the goodness-of-fit tests always lose to something else.

Now.. why are you obsessing about tests of normality?

HTH, David -- David L. Cassell mathematical statistician Design Pathways 3115 NW Norwood Pl. Corvallis OR 97330

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