Robert is absolutely correct ... I started out just wanting to point yorgi
in the right direction, but ended up saying too much, but not enough. I
stand corrected.
On Thu, 6 Apr 2000, Robert Dawson wrote:
> David Cross wrote:
>
> > There is a standard test for comparing variances from two independent
> > samples, and it is discussed in most intro stat texts. The test statistic
> > has an F-distribution, and degrees of freedom are what you would expect
> > for the sample variances.
>
>
> No! No! Not the F test! _Anything_ but the F test! _Especially_ when -as
> Yorgi says- the distributions are far from the same shape. The F test is
> infamous for its nonrobustness against deviations from normality.
>
> Levene's test, and some other variations on the same theme, give
> somewhat more robust comparisons of spread. (Levene's test is a two sample t
> test applied to absolute deviation from the group mean. Brown & Forsythe's
> test is similar but done, perhaps more logically, on the absolute deviations
> from the group median. [The mean is the point from which total squared
> deviation is least, the median that from which total absolute deviation is
> least.]
>
> These in turn involve some odd assumptions. In particular, deviations in
> general are far from normally distributed. My experience is that, if the
> distributions _are_ normal, a square root transformation symmetrizes the
> absolute deviations rather nicely; I've done some informal simulations
> suggesting (to me, anyway) that a t test on the root-absolute-deviations
> from either the mean or the median may have good properties. Another
> interesting option is nonparametric tests; I think Lehmann mentions some
> options in _Nonparametrics_ (1975).
>
> However, when all is said and done, I would suggest that if two
> populations have very different kurtoses, there is no canonical measure of
> spread, in much the same way that if two populations have very different
> skewnesses there is no canonical measure of location. In each case, a very
> tail-sensitive measure [range, midrange] will do one thing, a robust measure
> [interquartile range, median] another, and a "sum-of-squares" measure
> (standard deviation, mean] something in between. Therefore, unless one has
> a good reason to assign meaning to one measure of spread over another, it
> may make sense to say simply "They are different shapes", provide graphics,
> and leave it at that.
>
> Also: Yorgi wrote
>
> > However, because the first
> > has much larger magnitude, it has larger variance.
>
> If it really is "because", this suggests a model in which variation
> increases with value. Such a model may sometimes be more naturally examined
> after an appropriate transformation; and after the transformation there
> might be no differemce in spread - or, possibly, even in shape. So it might
> be worth examining the situation to see whether this is true.
>
> -Robert Dawson
>
>
>
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