> On 11 Apr 2000, I had written (on converting a test from a discrete to
> a continuous space):
> > ... If the question is one that makes sense in a continuous
> > 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.
On Wed, 12 Apr 2000, Bruce Weaver wrote:
> I see your point. But one of the examples Frick gives concerns the
> existence of ESP. In the discrete space, it does or does not exist.
> For this particular example, I think one could justify using a 1-tailed
> test when moving to the continous space; and so the null hypothesis
> would encompass "less than or equal to 0", and the alternative "greater
> than 0". It seems to me that with a one-tailed alternative like this,
> the null hypothesis can certainly be true.
[I do not much care for this example of Bob's; it has for me too much
the flavor of asking how many angels can dance on the head of a pin,
which I understand was a hot topic of debate at one time.]
But surely this depends in part on which continuous space you want to
"move" to. (Which one had YOU in mind?) For some imaginable spaces, one
can imagine detecting ESP both in a positive sense (some persons can
exhibit behavior most parsimoniously explained as telepathy, telekinesis,
etc.) or in a negative sense (some persons can suppress ESP-like effects
in their own, or in others', behavior); in such a space, a two-sided
test is surely appropriate.
But in any case, a continuous space would surely encompass ranges
of values reflecting not the existence/nonexistence of ESP (that's your
discrete question) but the degree to which ESP is detectable by the means
presently available (or in use in this experiment, anyway). In that
case, one cannot logically accept the null hypothesis, because evidence
cannot distinguish between "ESP does not exist" and "ESP, if practiced,
does not occur at a level detectable in the present experiment".
> > (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.
>
> This would not be a problem if the alternative was one-tailed, would it?
Sorry, Bruce, I do not see your point. How does 1-tailed vs. 2-tailed
make a difference in whatever the underlying probability distribution is?
(2-sided tests are conventionally symmetrical on both sides, for no good
reason except that few investigators can perceive, let alone quantify,
differential risks associated with error in the two directions; but
there is nothing to prevent one from carrying out an unsymmetrical test
(at, say, the 5% level, assigning 1% of the error probability to the
left-hand tail and 4% to the right-hand tail) if one had persuasive
reasons for so doing. (Actually, you don't need persuasive reasons to DO
it, only to support your arguments with colleagues and editors that a
non-standard procedure like this is defensible.) A one-sided test is
merely an extreme form of an asymmetric two-sided test.)
-- 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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