Re: Need help understanding Homogeneity of Variance please

stats newbie wrote:
Hi, I was hoping someone would be able to explain the assumption of
homogeneity of variance. What is it and why should it be addressed?
What are the consequences of not having homogeneity of variance. I
hope I have posetd this in the correct group. Thanks,

Since the other responders (including sci.stat.consult) have answered
in terms of tests of hypotheses (and have been valid I think), I
thought I would take a different viewpoint ... the relative importance
of these viewpoints will vary depending on what you are actually
trying to do.

I assume you are working with a regression-like problem, although
similar things apply in most other cases. In the theory behind
regression, the assumption of homogeneity of variance is used as part
of the argument that shows that the ordinary regression formulae give
estimates that are best possible in the sense that, out of all linear
combinations of the data that are unbiassed, the ordinary regession
formulae give results that have minimum expected squared error: these
results are either for the regression coefficients or for predictions
of the outcomes of new experiments. The "unbiassedness" condition can
be replaced by one based on invariance if you think that
"unbiassedness" is irrelevant.

If the assumption of homogeneity of variance does not hold, then
better linear combinations can be found: so, since you supposedly want
you estimates to be as good as possible, you should try to get as good
as possible a description of the variance of the regression residuals.
If you find that homogeneity of variance does not hold then you can
get a best-possible linear estimate by using weighted regression (or
there are other approaches that may lead to changing the structure of
the estimators away from being linear combinations of the original
data). However, you can still use an ordinary regression estimates if
you like (they are still unbiassed) but if you want to produce
estimates for the variance of the regression residuals, for the
regression parameters or for the predictions from the regression,
these formulae would all need to be changed from those for ordinary
regression. The importance of small departures from the assumption of
homogeneity of variance will usually be small, but if there are
definite reasons to suppose it to not hold (for example, if the
departure is large enough to be seen in the data), it would be wise to
take appropriate action.

David Jones


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