Herman Rubin wrote:
> Least square and similar procedures are good despite lack of
> normality, but they are not maximum likelihood, provided that
> one does not disturb the underlying linear structure. Any
> attempt to make the observations more normal is likely to
> destroy any useful structure. Normality is NOT usually
> important, and probably never occurs in nature.
The assumption that there *is* an underlying linear structure is
unwarranted; there may not be, and this is why transformation often
works. To take one example: if one is measuring an AC signal, should one
look at the amplitude of the signal, the power, or the level in
decibels? The power is the square of the amplitude, and the level in
decibels its logarithm.
How about acidity/alkalinity? The pH is a logarithmic transformation;
actual ion concentration is "more real" but average ion concentrations
will always be dominated by the most alkaline measurements if there is
any range to speak of.
I will agree with Herman on the irrelevance and nonexistence of _exact_
normality. I hope he will agree with me that _approximate_ normality
exists and is useful. If not, perhaps we can at least agree that
_symmetry_, which makes
location well-defined, is useful even if it does not exist in its exact
form.
By "making location well-defined" I mean that most obvious measures of
location coincide, so that we do not have to decide "location is the
mean" or "location is the median" or what-have-you.
There are only a few good reasons to choose the mean (not including
"all the other kids are doing it" and "I only know two measures of
location and my calculator doesn't do medians".)
(1) You have an actual interest in the sum of the observations. A
waiter has more interest in the mean tip than in the median tip because
it predicts the total better.
(2) You have reason to believe in a parametric model for which the
sample mean is a sufficient statistic (NOT the same as a parametric
model for which the mean is a parameter.)
(3) You have an actual interest in minimizing the squared differences
of the data from the estimate. Of course, if this is because of maximum
likelihood, you then have a normal distribution...
Otherwise - and if no considerations like that suggest something else
instead - you don't really want to be making such a choice. For a
symmetric distribution you don't have to.
Moreover, a nonlinear transformation does not destroy any information -
provided it's 1-1 and you don't forget what you did. It's a different
way of looking at the same data. If you can find a "viewpoint" from
which you can define location without ad-hockery, that seems to me to be
a good choice.
-Robert Dawson
.
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