On Mon, 10 Apr 2000, Bruce Weaver wrote in part, quoting Bob Frick:
> ... Bob Frick has some very interesting things to say about all of
> this. For example, the following is taken from his 1995 Memory &
> Cognition paper (Vol 23, pp. 132-138), "Accepting the null hypothesis":
>
> <start quote>
> To put this argument another way, suppose the question is whether one
> variable influences another. This is a discrete probability space with
> only two answers: yes or no. Therefore, it is natural that both
> answers receive a nonzero probability.
It may be (or seem) "natural"; that doesn't mean that it's so,
especially in view of the subsequent refinement:
> Now suppose the question is changed into
> one concerning the size of the effect. This creates a continuous
> probability space, with the possible answer being any of an infinite
> number of real numbers and each one of these real numbers receiving an
> essentially zero probability. A natural tendency is to include 0 in this
> continuous probability space and assign it an essentially zero
> probability. However, the "no" answer, which corresponds to a size of
> zero, does not change probability just because the question is phrased
> differently. Therefore, it still has its nonzero probability; only the
> nonzero probability of the "yes" answer is spread over the real numbers.
> <end quote>
To this I have two objections: (1) It is not clear that the "no" answer
"does not change probability ...", as Bob puts it. If the question is
one that makes sense in a continuous probability space, it is entirely
possible (and indeed more usual than not, one would expect) that
constraining it to a two-value discrete situation ("yes" vs. "no") may
have entailed condensing a range of what one might call "small" values
onto the answer "no". That is, the question may already, and perhaps
unconsciously, have been "coarsened" to permit the discrete expression
of the question with which Bob started.
(2) My second objection is that if the positive-discrete
probability is retained for the value "0" (or whatever value the former
"no" is held to represent), the distribution of the observed quantity
cannot be one of the standard distributions. (In particular, it is not
normal.) One then has no basis for asserting the probability of error
in rejecting the null hypothesis (at least, not by invoking the standard
distributions, as computers do, or the standard tables, as humans do
when they aren't relying on computers). Presumably one could derive the
sampling distribution in enough detail to handle simple problems, but
that still looks like a lot more work than one can imagine most
investigators -- psychologists, say -- cheerfully undertaking.
(One might get around this contretemps by resampling techniques;
but I suspect that a careful examination of the logic of resampling
applied to one of Bob's situations would show an implicit and possibly
hidden attribution of infinitesimal probability to the value
corresponding to "0".)
-- Don.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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