Hi
On 10 Feb 2001, Donald Burrill wrote:
> On Thu, 8 Feb 2001, jim clark wrote in part:
> 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).
No, not if we are using non-directional to refer to the
hypothesis about Mus, Rhos, or whatever. The appropriate
critical value to test Ha: Mu1<>Mu2 or Ha: Rho<>0 is F_.05.
That is why t_.025 = sqrt(F_.05). One way to think about it is
that the t distribution is folded over to produce the F
distribution, putting both tails of the t in the upper tail of
the F distribution.
> > 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...)
Afraid I won't be acknowledging either of those things. The most
concrete way to verify my original statement is to see that t_.05
(i.e., critical value for a directional alternative
hypothesis) is equal to sqrt(F_.10). The confusion is, again,
not differentiating clearly between one- vs. two-tail and
directional vs. non-directional.
> 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.
But isn't the point for this discussion that the variance
component can be non-zero if mu1>mu2 _or_ mu<2 (or, for
regression, if rho>0 _or_ rho<0)? The MS_treatment in ANOVA
could care less about the direction of the differences among the
means, and the MS_regression could care less about whether r is
positive or negative. Directional vs. non-directional is
referring to the direction of differences or sign of the
correlation (or regression) coefficient, not whether the variance
component is large or small. The variance component is expected
to be large in either case.
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
============================================================================
=================================================================
Instructions for joining and leaving this list and remarks about
the problem of INAPPROPRIATE MESSAGES are available at
http://jse.stat.ncsu.edu/
=================================================================
