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_. What it seems not to hold for is
certain particular hypothesis tests for which those statistics are the
commonly used test statistics. The "large F" Jim speaks of below celarly
refers to an analysis of variance (and one with only two groups, at
that!). In that context, while the hypotheses _of interest_ are the
null hypothesis that the several means Mu_j are identical, vs. the
two-sided alternative hypothesis that some of them are different, the
formal hypothesis tested by the F statistic is the null hypothesis that a
certain variance component equals zero, vs. the alternative hypothesis
that it does not equal zero; and since a variance component cannot be
negative, the _test_ is one-sided, in the metric of variances: one
rejects only for F sufficiently greater than 1 for the result to be
improbable under the null hypothesis.
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.
Similarly, one may use Chi-square to test the null hypothesis that a
variance has a specified value, and wich to reject if the evidence leads
one to believe that the true value is LESS, OR if the true value is
GREATER, than the value specified.
> 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".
-- Don.
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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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