Most powerful GoF test for normality It's well established that both Anderson-Darling and Shapiro-Wilk have a much higher power to detect departures from normality than a KS-Test.
I have been told that Shapiro-Wilk is usually the best test to use if you want to test if a distribution is normal because it has one of the highest powers to detect lack of normality, but my limited experience, it seems that Shapiro-Wilk gives me the same result as Anderson-Darling every time.
I thus have two questions:  


*

*When does the Shapiro-Wilk test out-perform Anderson-Darling?

*Is there a uniformly most powerful lack of normality test, or, barring that possibility, a normality test that out-performs nearly all other normality tests, or is Shapiro-Wilk the best bet?

 A: If the only criterion is most powerful then nothing beats SnowsPenultimateNormalityTest which is in the TeachingDemos package for R.  However that test has an unfair advantage in the power competition and some may consider it less capable in other areas, for one, it is of the class of functions for which the documentation is probably (hopefully) more useful than the function itself.
What is more important is to consider what it means when these tests of normality reject the null, or fail to reject the null.
A: In generall I would advice to use more than just one test. Through statistic programms like R it isn't as much efford as just typing one line per test. 
The Shapiro-Wilks and the Anderson-Darling-Test are both tests which operate on comparing the distribution functions or analyse the variance. For example the Cramér-von-Mises or the Jaques-Bera-Test use comparisms between kurtosis and skrewness of a function. In some cases it is possible that the results between both test variants could differ.
Depending on the situation of test every test has it's might and weakness. Hence it's advisable to try different tests on the same data set. If the data set is clearly normaly distributed the results shouldn't differ verry much.
