On Mon, 8 May 2000 [EMAIL PROTECTED] wrote:
> Do the null and alternative hypotheses have to be complements?
In the usual hypothesis-testing paradigm, in which onke argues by
elimination, the set of hypotheses considered must be both mutually
exclusive and exhaustive. If that's what you mean by "complements",
then yes.
> In other words, can one set up a problem like:
>
> H0: d = 0
> H1: d <= 0
These are neither mutually exclusive ("=" occurs in both!) nor exhaustive
(">" is not accounted for). One would expect, if the alternative
hypothesis H1 includes d < 0, that the SET of hypotheses would be
H0: d >= 0
H1: d < 0
The "=" sign must appear in the null hypothesis ("=" is the only
condition that permits you to construct a sampling distribution for the
test statistic; whence some writers, notably Lumsden, refer to the null
hypothesis as "the model-distributional hypothesis"). If the test is to
be one-sided, _that_ side is expressed in the alternative hypothesis, and
the _other_ side must be contained in the null hypothesis. For the
one-sided set of hypotheses displayed above, one rejects only if the
observed value of d (or of the statistic estimating d ) is
sufficiently negative to be unlikely if the null hypothesis (d = 0, which
yields the sampling distribution of d ) were true.
> when d can take positive values?
Not sure I understand the question. If you mean that d can only take
positive values, i.e., it cannot take negative values, then you cannot
possibly observe evidence that would lead you to accept H1 under any
circumstances.
> Or must one couch the problem as:
>
> H0: d=0
> H1: d<0 OR d>0
This formulation at least has the advantage of permitting conclusions to
be drawn on the basis of empirical evidence.
> If you can go with the one-tailed test, does choosing it imply a
> different Type II loss function than in the two-tailed test?
Of course it does. In the first place, under H0: d >= 0, positive
values of the statistic estimating d dop not provide evidence against
H0, however far they may be from 0. In the second place, for a given
level of significance, you don't have to be so far from 0 (or in general
from the value specified in the null hypothesis) to reject H0 as you do
for a two-sided test; but you do have to be in the negative direction
(in the case we're discussing), you cannot reject H0 in the positive
direction (as would be possible in a two-sided test).
> Are there any other hidden assumptions underlying the choice of a
> one-tailed test in this context?
Dunno how to deal with this one. I wouldn't have described those
consequences as "hidden assumptions"; they seem quite open and obvious
to me -- nothing hidden about them, and "assumptions" doesn't apply.
Nor do I perceive what other "contexts" you envision in using the phrase
"in this context".
I don't know if this helps to clarify anything for you, but a
one-sided test is merely the most extreme form of an asymmetric two-sided
test. (Conventionally, because we usually can't think of a good reason
to behave otherwise, in a two-sided test we divide the significance level
(nominal probability of a Type I error) into two equal parts, and assign
one part to each tail of the sampling distribution. There's nothing in
statistical theory that prevents us from dividing it into two UNequal
parts, and assigning one part to one end of the distribution and the
other part to the other end; this would even be rational behavior when
the risk associated with a type I error in one direction be more severe
than the risk associated with a type I error in the other direction, and
we thought we could quantify the relative risks.)
-- DFB.
------------------------------------------------------------------------
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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