Regarding significance tests for 2x2 tables.
I do not know what to make of all the debating this
way and that way in the literature.
Q: Under what circumstance does one use the "exact test"
conditional on the "marginals being what they are" ?
I have seen the distinction made (supposedly helpful)
between -- e.g. -- case/control studies, where the
marginal totals are chosen by the experimentor, and
are therefore NOT random variables; and retrospective
analysis of risk-factor and disease incidence data,
where the marginal totals are NOT chosen by the
experimentor, and are (therefore) regarded as random
binomial variates.
Supposedly, then, one ought use the "conditional" exact
test (i.e. Fisher's) in the former case, whereas one
ought use an "unconditional" exact test in the latter case.
But it seems that the marginals in either case can be
thought of as either "random" or "determined", depending
on how one "thinks of the problem". In other words,
in the retrospective incidence/risk-factor data, the
marginals look like "random" quantities if you think
of the researcher having simply "gathered up" whatever
washed up at his doorstep; but they look like "fixed"
quantities if you think of the researcher having
been somewhat partial to a particular ratio of cases
to controls. Neither instance seems utterly clear-cut,
and moreover, why should I, when confronted with a
single 2x2 table have to worry about what the researchers
state of mind was when he either "gathered" or "selected"
the observations?
To put it another way, when I want to compute a "p-value"
which somehow expresses the "extremity" of the single
observed 2x2 table, I need to consider the universe
from which that table was drawn. There are different
ways of construing the universe ... corresponding to
"conditional" or "unconditional" tests. But the universe
is a fiction, for all I have on hand is a *single*
2x2 table. Now the researcher's methodology (if I
knew of it) would tend to cause me to favor one universe
over the other. (Is this an observation drawn from
a family having fixed marginals or not? )
But this strikes me as an unstable basis for choice of
a statistical test. As I said above, a single table
with preselected marginals can be treated as an instance
of a selection with *random* marginals, that just happens
to be the same! So the choice of the appropriate
statistical test seems to rest upon an optical illusion
of the sort where two images are combined in such a
way that the visible one depends on the angle of viewing.
How does one correctly view the choice of test?
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