Thanks for Tony's comments. (And thanks to Marcello for forwarding the
original, which the server refused twice.)
Tony's measure W is itself asymmetric as it is bounded below by 0, 1 if
the data are symmetric in its sense, and unbounded above, so logging
should usually help.
The quotation from George Box he alludes to is in
G. E. P. Box. 1953.
Non-Normality and Tests on Variances.
Biometrika 40: 318-335
This was the paper that introduced robustness as a technical term.
Nick
[email protected]
Lachenbruch, Peter
This is a welcome update (I hadn't known about it before). I've found
that bad asymmetry (skewness) is the biggest hazard with many tests
relying on normality. Another measure of asymmetry is based on lmoments
which is available in a program from Nick Cox (we have to control that
guy :-) ).
I've found a simple descriptive measure is W=(P90-P50)/(P50-P10) which
should be 1 if the data are symmetric. In some simulations
(unpublished) I found that ln(W) is more closely normal than W. I
forget the variance, but bootstrapping would work nicely.
I use asymmetry as an indicator - I'd be reluctant to use it to
determine what test to use. At once time I compared a couple of
strategies for determining whether to use a normal test, a ranksum test,
or a test for normality followed by a ranksum if p<0.05 or a z test
otherwise. Basically the ranksum won when data weren't normal.
Lachenbruch, P.A. (1991) "The Performance of tests when observations
have different variances," Journal of Statistical Computation and
Simulation, 40:83-92.
More generally, preliminary tests of normality are usually unnecessary
as (I think) Box said regarding testing for unequal variance in ANOVA
doing such a test "is like going to sea in a rowboat to see if it's OK
for the Queen Mary to sail"
Marcello Pagano
The package -omninorm- on SSC by Kit Baum and myself has been updated.
-omninorm- requires Stata 9 (except for the older version, -omninorm7-,
frozen as was, which requires only Stata 7). To install, use -ssc-.
The update is in one sense minor, but still notable for people
interested in what -omninorm- does, provide an omnibus test for
univariate or multivariate normality.
-omninorm- implements a test proposed by Doornik and Hansen in a 1994
working paper (highly accessible and much cited). That working paper has
now been written up as a standard journal paper, which should satisfy
anybody queasy about using ideas that have not received the
sanctification of peer review for a journal. The 2008 reference has now
been included in the help file.
Doornik, Jurgen A. and Hansen, Henrik. 1994. An omnibus test
for univariate and multivariate normality. Working Paper,
Nuffield College, University of Oxford. See
http://ideas.repec.org/p/nuf/econwp/9604.html or
http://www.doornik.com/research/normal2.pdf
Doornik, Jurgen A. and Hansen, Henrik. 2008. An omnibus test
for univariate and multivariate normality. Oxford Bulletin
of Economics and Statistics 70: 927-939.
This test is widely used in other statistical software.
At least one of us most of the time, and both of us some of the time,
regard normality tests as over-rated. If you care about normality, draw
a plot. If you care about normality, wonder why, as normality is less
often an assumption, and less often an important assumption, than is
frequently asserted or believed.
That said, if you are going to test for normality, the arguments in the
Doornik and Hansen paper should lead you to treat their test as a good
competitor, and at least in some senses as superior to those previously
implemented in Stata. What's more, most of the alternatives only provide
univariate tests.
Nick
[email protected]
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