Rich:
> > > > The problem is that the data I use, which is hurricane numbers,
deals
> > > > with small numbers. ^^^^^^^
> >
> > If one interprets this (as I did) as "numbers [=counts] of
hurricanes"
> > the Poisson distribution is not unreasonable. If, of course,it refers to
> > some obscure measure of hurricane strength the Poisson distribution
would
> > not apply.
> >
> > "Small numbers" implies that a normal approximation to the Poisson model
> > might not work.
>
> Oh, yeah. You are right, and she does seem to be concerned with the
> count. I apologize for missing that, and making something of it. I
> think I identified "numbers" as the hurricane gale-force numbers,
> without reading closely enough to be bothered by the other mentions --
> partly because, "small numbers" is not a particular problem of
> Poisson.
>
> Which normal approximation? You can average several Poisson estimates
> for a good estimate of the overall. You can use an add-on (say, 1/2)
> to be even more robust for doing tests, using square-roots of the
> numbers.
Poisson is a large-sample low-frequency limit of binomial, so the normal
approximation holds under the same criteria. If you accept the min{np,nq}>=5
cutoff (which depends on what you want to do with it!) then you would accept
Poisson ~ N(lambda, (sqrt(lambda))^2) for lambda >= 5. Jennifer C. had 20
hurricanes in one group, 2 in the other; so under the null hypothesis
(lambda_1 = lambda_2) she would have lambda^ = 11 for each group, large
enough to infer lambda >= 5.
> However, 5 vs. 5 is not a powerful design.
The number of years doesn't matter - Poisson is additive over its
spatial (or temporal or whatever) extent, so we can think of it as 5 years,
one 5-year period, or sixty months.
Without doing the work, I would guess that there is enough power here to
blow H_0 right out of the water... for observations so different.
> And, as I said, she seems
> to have in her hands a t-test between means,
It won't fly as a t test; if she averaged 2 hurricanes over 5
low-pressure years the data look something like {0,0,1,0,1}. I wouldn't
t-test on that "on a sizable bet or to please an old chum", as Bertie
Wooster put it.
However, if the Poisson model is valid within a year - and that is
admittedly a big if,as there are independence questions - you have extra
information about spread that can be used to get a more powerful test,
tailored to a Poisson model.
You are, OTOH, right that the ratio is not (prima facie, anyway) the
right thing to be looking at.
-Robert
