## RE: AI-GEOSTATS: Summary: Large sample size and normal distribution

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• ... is a common argument in simulation experiments, that because you can do an infinite number of replicate simulations, somehow the differences detected are
Message 1 of 5 , Aug 11, 2003
> Hi,
>
> I'm not sure i agree with the idea that a test can be too powerful. This
is a common argument in simulation experiments, that because you can do an
infinite number of replicate simulations, somehow the differences
detected are not real. In fact, the differences are real. They may not
be biologically (or geologically or whatever field you are in)
significant, but they are still real. That is why it is better to decide
first on the magnitude of difference that you consider significant.

The null hypothesis is always false although it might be false by a very
small quantity, that is the trivial fact that the very large sample size
illustrates in the common test of significance. The conclusion to be drawn
from this is not that we must set in advance the amount of difference that
we would find significant (a rather restrictive strategy which will be
violated very often because it is nonsensical), but rather that the only
sensible strategy is to compare hypotheses one against another. This can
be done on an evidential basis by evaluating the likelihood ratio, the
likelihood of the data under one hypothesis divided by the likelihood of
the data under another hypothesis. By constructing the whole likelihood
function (in the case of a single parameter) any pair of hypotheses can be
tested for the value of the likelihood ratio.

> Now, in the case of deviation from normality, I suppose you wouldn't
have much intuition about what is significant, but the relevant question
is what is the effect of small deviations from normality on your test or

Perhaps a better question is what the data say about a given hypothesis
for the mean versus another value for the mean assuming the normal
distribution is true? If the variance is unknown there is a simple
solution only for the normal and a few other cases, by orthogonalization,
and then the two parameters can be assessed separately. For comparing two
different models, say normal versus lognormal, a likelihood based
approach, the Akaike Information Criterion, is available although i am not
sure that Akaike's approach is fully in agreement with the likelihood
principle.

Ruben

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