We know that the residuals in ANOVAs have to be normally distributed
if we are to believe the confidence limits or p values provided by
the analysis, and we know that we are supposed to at least view the
residuals for non-normality. But how much of a departure from
non-normality does there have to be before you stop trusting the
analysis?
The Wilks-Shapiro statistic for non-normality appears to be
equivalent to "proportion of variance in the residuals explained by
normality", which would make it an effect statistic for normality.
If it really is an effect statistic not biased by sample size, let's
forget about its p value. The real question is: how small does the
W-S have to be before things go awry with the analysis? Does anyone
know of any simulations to answer this question, using various
non-normal distributions? And what about using skewness and kurtosis
in the same manner?
I did a Web search first at google.com (Wilks Shapiro test
normality). Got lots of hits, but the only thing that came close was
some really old postings at:
http://sobek.colorado.edu/LAB/STATS/normality.
A supplementary question: Some time ago I saw someone defaulting to
a non-parametric analysis when the sample size was small. My first
reaction was: "that's precisely when you don't want to use
non-parametrics, because they have less power than parametrics with
small samples. (With large sample sizes parametrics and
non-parametrics have the same power, for normally distributed
residuals.)" Ah yes, but now I see that defaulting to
non-parametrics for small sample sizes is the *safe* way to go,
because with small samples you can't be sure the residuals in the
population are normal, even when the residuals in the sample look
perfectly normal. Of course, if there is no reason to suspect
non-normality of residuals, it's reasonable to use parametrics, even
if the residuals in the small sample look non-normal (because, you
would reason, they look non-normal only because of sampling error).
Comments?
Will
--
Will G Hopkins, PhD FACSM
University of Otago, Dunedin NZ
Sportscience: http://sportsci.org
A New View of Statistics: http://newstats.org
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