> Jerry Dallal wrote:
>
>But, if the null hypothesis is that the means are the same, why
>isn't(aren't) the sample variance(s) calculated about a pooled
>estimate of the common mean?
I looked at this some years ago. The answer is straightforward: it
would be logically valid to do so but you would lose a *lot* of power. A
hypothesis test is essentially a proof by contradiction; in such an
argument you are permitted to run with the hare and hunt with the
hounds, changing sides as often as you like. Thus, at any stage, you
may appeal to the null hypothesis or to the data; any inconsistency
between the two, no matter how byzantine the argument, is evidence
against the null.
If you think about the two-sample-T as a two-level ANOVA (a roughly
correct idea), the pooled estimate of the mean gives you the SST; the
usual method gives you the SSE. As you expect the SSTr to be nonzero,
you have
SSE < SST
and substituting one for the other is a Bad Thing. In an extreme case:
A B
10 20
11 21
12 22
one method estimates the SD as 1, the other as 5.55.
-Robert Dawson
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