It depends on whether your are testing based on means or testing based
on sums of squares. The former is of course, the z or t test, with the
t test being preferred from theoretical aspects. The latter is of
course the F test or variations on the chi-square test.
Way back in ancient history, R. C. Geary (Biometrika 34, pages
209-242) explored the effects of skewness and kurtosis on the
distribution of means (t distributions) for small samples. The work is
somewhat hard to draw generalizations from.
For sums of squares involving two independent samples drawn at random
from the same universe, the variance of z is proportional to (B2-1)/4,
where B2 is the kurtosis of the universe. This is just an expansion of
Fisher's approximate formula for normal samples on page 228 of Fisher
(Statistical Methods...). z here is Fisher's one-half the log of the
ratio of the two variances. Kurtosis is then the primary concern on
tests using Chi-square or F tests.
Geary shows that the distortions in the normal t distribution due to
kurtosis is slight, even for small samples.
In reality, large skewness values and large kurtosis values occur at
the same time, under normal sampling. Therefore it is not unreasonable
to look at kurtosis values rather than skewness values to estimate
departures from the standard mean or sum of squares distributions
derived from normal distributions.
DAHeiser
----- Original Message -----
From: Will Hopkins <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Sent: Friday, January 19, 2001 1:31 PM
Subject: Re: Normality assumption for ANOVA (was: Effect statistics
for non-normality)
> Yes, I was wrong about the need for normality of the residuals. I
somehow
> had the idea that estimates of the precision of estimates come
directly
> from normality of the individual errors, but it just ain't so.
Estimating
> the confidence limits for the mean of a sample is the way to see how
the
> central limit theorem smooths out a nasty distribution of
> residuals. According to Bill Ware and Paul Swank, the distribution
of the
> variance of the mean takes a bigger sample to settle down than the
> distribution of the mean, but I can't really see how that matters,
unless
> you can make it the basis of the kind of test for non-normality I am
> looking for.
>
> So when I plot residuals (Y) against predicteds (X), the scatter in
the Y
> direction can look quite discrete (as in residuals from a Likert
scale with
> only a few levels) and skewed (as in responses piled up at either
end of
> the Likert scale, or as in Robert Dawson's example of Poisson
distributions
> with small means). All that matters is that I have enough
observations
> for the central limit theorem to smooth out the "graininess". It's
really
> cool to learn that I may not have to use logistic regression for
Likert
> scales, but how do I know whether I have enough observations?
Someone
> suggested some function of the sample size and the third and/or
fourth
> moments. Anyone know of any simulations done on anything like that?
>
> While we're on the subject of residuals vs predicteds... We are
supposed
> to check for substantial curvature in the plot (which would indicate
the
> model needs refining) and substantial non-uniformity in scatter for
> different predicted values (heteroscedasticity, which biases the
estimates
> towards the observations with more scatter and also stuffs up the
> confidence limits). The rule for these two problems seems to be: if
you
> can see it on the plot, you should do something about it. Anyone
got
> anything more quantitative than that? I guess you have to make your
own
> decision about curvature, based on what you know from clinical
experience
> about what effects are substantial. (You could use Cohen's scale of
> magnitudes as a default.) But what about the non-uniformity of
> scatter? How big does a difference in variance between groups or
between
> either ends of the residuals vs predicteds have to be before the
associated
> bias is a concern?
>
> Will
>
>
>
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