Hi
I'll take Don's points out of order.
On 8 Feb 2001, Donald Burrill wrote:
> On Tue, 6 Feb 2001, jim clark wrote in part:
> > One can get a large F by either Mu1>Mu2 or Mu1<Mu2 (or by positive or
> > negative R, ...). Therefore the one-tail of the distribution
> > corresponds (normally) to a two-tailed or non-directional test.
> > However, there is absolutely nothing wrong with making the necessary
> > adjustment to make the test directional (i.e., equivalent to the
> > one-tailed t-test), and therefore referring to it (confusingly,
> > of course) as a one-tailed test.
>
> On this point, one must agree with Thom: such a use of language can only
> be confusing, as you acknowledge. "Newspeak", it was called in "1984".
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.
That is probably what led to the posting that initiated this
thread. Hence, I am not sure that it is very helpful to wish
that things were otherwise, although such wishing might
eventually get us to a better terminology.
Don began by commenting on:
> On Tue, 6 Feb 2001, jim clark wrote in part:
> > The problem is that one-tailed test is taken as synonymous with
> > directional hypothesis (e.g., Ha: Mu1>Mu2). This causes no
> > confusion with distributions such as the t-test, because
> > directional implies one-tailed. This correspondence does not
> > hold for other statistics, such as the F and Chi2.
>
> The statement is not correct. The correspondence certainly holds for
> F and chi-square _statistics_.
I am not denying that F and other statistics have two tails and
that they might be used for specified purposes. I thought that
my example of Ha:Mu1>Mu2 here (or rho>0 vs. rho<0, or beta>0 vs.
beta < 0, ...) made clear the sense in which I was using
directional hypothesis and the correspondence that I was
referring to. In general when one calculates a t-test for
differences between means, correlation or regression
coefficients, or whatever, the positive and negative tails will
correspond to the rejection region for distinct directional
hypotheses (commonly called one-tailed tests) and both tails will
correspond to the rejection region for a non-directional
(commonly called two-tailed) test. The perfectly equivalent F
test, however, as normally calculated, will only consider the
upper end of the F distribution, irrespective of whether the
alternative hypothesis is directional or not. That is why t_.025
= sqrt(F_.05).
> 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.
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. 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. You are
using the upper tail (i.e., one-tail) of the distribution to test
a directional (i.e., "one-tailed") hypothesis.
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.
Best wishes
Jim
============================================================================
James M. Clark (204) 786-9757
Department of Psychology (204) 774-4134 Fax
University of Winnipeg 4L05D
Winnipeg, Manitoba R3B 2E9 [EMAIL PROTECTED]
CANADA http://www.uwinnipeg.ca/~clark
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