Re: Fisher's Exact Test and Chi-square


harriscsuiucedu@xxxxxxxxx wrote:
I'm not sure what 'conditional' means (I am obviously not a
statistician). But I do know that in the computation of both, the
marginals are computed (in FET to help determine the other contingency
tables with the same marginals, for Chi^2 to help determine the
expected cell entries). So how is one 'conditional' while the other
not?

See John Uebersax's post (it is much clearer than mine).

What's not correct about it (assuming you're talking about the '<5'
statement)?

First, that one should switch to Fisher's exact test (as that seems to
be confusing switching from unconditional to conditional with a
continuity correction). Second, the implication that observed not
expected counts are important.

Hm... I see the continuous vs discrete problem, but one could just see
contingency tables as limited to integers rather than reals (a
restriction of the data not the test), so I don't see how innaccuracy
is involved.

The probability distribution implicitly assumes continuous rather than
discrete input. Consider the simple case of a coin toss H or T. The
binomial gives an exact p value of .5 for either outcome, whereas the
chi-square or normal distribution would only give an approximation
(genearlly very accurate if p is near .5 and N is large).

How is that? Can you elaborate? This confuses me. Here it sounds like
you're saying that the probabilities depend on the marginals for
-chi^2-.but that seems to contradict your distinction at the beginning.
I'm really not following here.

Actually, I think I misinterpreted your point. I think the text book is
merely re-iterating that the chi-square statistic for a 2x2 contingency
table in the context of a test of indepedence is approximated by the
probabilities chi-square distribution with 1 d.f. (but i could be
wrong). I don't think it is saying that chi-square statistics in
general have this property. Without context its hard to interpret -
there at the least dozes and dozens of chi-square statistics. They are
probably nearly all or all only approximated by the chi-square
distribution. No statistic calculated from samples could, I think,
prefectly meet the distributional assumptions of any test (though I'm
willing hear counter-arguments).

No. Neither is an approximation of the other - they are different
distributions. However, with infinite d.f. t converges on the Normal.

Doesn't that latter sentence seem to say that for large degrees of
freedom, the t distribution is approximated well by the normal?

I think the point is a philosphical one - if I were to say this I'd say
t was an approximation of Normal, but it seems odd to think of the
distributions that way. The t distribution is the exact distribution
for sample differences in means from Normal distrubutions. I think it
may be an approxiamtion for a given purpose, but the distribution
itself isn't an approxiamtion. For example, 10 isn't an approximation
of 9 but the value 10 can be used to approximate 9 in a given context
for a given purpose.

Can you give me an idea of what such an argument would look like?
reference?

R. Macdonald (1998) in the British Journal of Mathematical and
statistical Psychology. The argument is highly technical and based on
signal detection theory. It more or less argues that if one accepts the
signal detection model and that the association is measurable on an
ordinal scale then conditional chi-squared are pure measures of
sensitivity whereas unconditional tests confound sensitivity with the
table's margianl totals.

It should be noted, though, that the history of this debate
(conditional vs. unconditional) hasn't produced consensus. I quite like
Ranald's paper because it is one the clearest, most coherent arguments
in the area - and in the absence of consensus internal consistency and
clarity is a decent fallback (at least everyone knows why you chose
Fisher's over Pearson chi-squared)

Thom

.

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