On Thu, 8 Feb 2001, jim clark wrote in part:
> We all agree that it is confusing, but I do believe that the use
> of one-tailed and two-tailed to refer to directional vs.
> non-directional hypotheses (rather than uniquely to one or two
> tails of a distribution) is very wide-spread and quite common.
There would not be a problem if the hypotheses in question were STATED.
It's this sloppy habit of saying "F test" or "chi square test", with no
hint of WHICH "F test" or "chi square test" one is talking about, that
impedes communication.
> That is probably what led to the posting that initiated this
> thread.
Yes. "I thought the chi-square test was always two-sided", or words to
that effect, the querent wrote. He she or they have not, in all the
correspondence since, said what the hypothesis being tested was.
I had written:
> > It is still possible to use the F _statistic_ to test the null
> > hypothesis that Var1 = Var2, in circumstances where it is entirely
> > possible that Var1 < Var2, Var1 = Var2, or Var1 > Var2. In such
> > cases _both_ tails of the F distribution are of interest, not just
> > the upper tail.
--- and Jim replied:
> Right, but if one calculates F_larger/F_smaller, then one is only
> looking at the upper tail of the F distribution even though one
> is doing a non-directional test (i.e., two-tailed in the
> vernacular). The appropriate critical value for a
> non-directional test would be F_.05.
Whoops! Not if you want to test at the usual 5% level! For a
non-directional test of the null hypothesis that two variances are
equal, the critical value would be F_(alpha/2).
> If you made a directional hypothesis and predicted which variance was
> going to be larger (as implied in F's use for anova and regression),
> then you would compare the obtained value of F to F_.10, not F_.05.
I'll agree with you if you halve those subscripts! (Or acknowledge that
you wanted to test at the 10% level...)
You state that using F in ANOVA and regression imply that one had a
_prediction_ of which variance would be larger. This is not how I
understand the idea of "predicting", which I take to imply that one could
have predicted something in the opposite direction. In ANOVA the null
hypothesis _of interest_ is commonly expressed as "all the means are
equal" (in some language or other), vs. "some of the means differ", and
the alternative hypothesis is indeed non-directional -- in the metric of
the subgroup means. But the hypothesis actually _tested_ (using F) is
the null hypothesis that a particular variance component is zero, vs. the
alternative that it isn't, and since a variance component cannot be
negative, the alternative really is that the variance component in
question is positive: thus in the metric of variances the alternative
hypothesis is one-sided. This is a matter of algebra, not of
"predicting" the direction of an effect.
However, perhaps others are more willing to use "predict" in
this rather sloppy (from my point of view ;-) fashion.
> You are using the upper tail (i.e., one-tail) of the distribution to
> test a directional (i.e., "one-tailed") hypothesis.
Yes. Because a result in the _lower_ tail would tend to confirm the
null hypothesis, not reject it.
> Like Don, I hope that language can become clearer on these
> issues, but my suspicion is that it will be a long, long time
> before one- vs. two-tailed stops meaning directional
> vs. non-directional alternative hypotheses for most people.
I have no problem with that. I just wish that people would say what
they're talking about: if it's a hypothesis test that is of concern,
what is the hypothesis and what is the test statistic, for example.
To say only "chi-square test" or "F test" or "z test" is simply
insufficient.
-- Don.
----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 (603) 535-2597
Department of Mathematics, Boston University [EMAIL PROTECTED]
111 Cummington Street, room 261, Boston, MA 02215 (617) 353-5288
184 Nashua Road, Bedford, NH 03110 (603) 471-7128
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