The null hypothesis for a directional test is commonly misstated. For
example, if the alternative is "mu1 is greater than mu2," then the null
hypothesis is "mu1 is less than or equal to mu2," not "mu1 is equal to mu2."
Given proper statement of the directional null hypothesis, it should be
obvious why results in only one direction should cast doubt on the veracity
of the null. The critical region or p-value obtained in a directional test
is, as Mike notes, based on a sampling distribution derived from a
nondirectional null, but this is just a "worst-case scenario" (or "best-case
scenario," depending on your perspective) -- for example, were your null
"mu1 is less than or equal to mu2," the p value for your sample tells you
how well the null fits the data when evalued at one end of the interval
included in the null, the end where mu1 = mu2. For the rest of the range
(mu1 < m2), the p value for your sample would be smaller.
In actual practice, one commonly follows rejection of the nondirectional
null with an assertion that the direction of effect in the sampled
population is the same as it was in the sample data. This creates the
possibility of a Type III error, correctly rejecting the nondirectional null
but incorrectly asserting the direction of effect. In conventional power
analysis, the probability of this error is incorrectly included in power.
An interesting article by Leventhal and Huynh (Psychological Methods, 1996,
1, 278-292) addresses this. They conceptualize the typical test of
significance as involving three hypotheses, not two. For example, "mu1 <
mu2," "mu1 = mu2," and "mu1 > mu2."
I have posted a summary of the article at:
http://core.ecu.edu/psyc/wuenschk/StatHelp/Type_III.htm
----- Original Message -----
From: "Mike Scoles" <[EMAIL PROTECTED]>
To: "Karl L. Wuensch" <[EMAIL PROTECTED]>
Cc: "TIPS" <[EMAIL PROTECTED]>
Sent: Friday, March 02, 2001 9:40 PM
Subject: Re: Question about one-tailed tests
> Although I have taught statistics for almost 20 years, I still don't
understand
> the relevance of directional alternate hypotheses. The critical region of
the
> test statistic is determined by the statistic's distribution given that
the null
> hypothesis is true. This conditional distribution has little to do with
the
> investigator's opinion of which direction it should (or more accurately,
could)
> be false.
