On Mon, 16 Oct 2000, Pat Cabe wrote:

> The F distribution (correct me if I'm wrong) is actually a two-tailed 
> distribution, but a skewed one. We use, under normal circumstances, the "long" 
> tail (i.e., where "between" variance is greater than "within" variance). So it 
> is possible to get a very small, less than unity F ratio that would fall out in 
> that "other" tail. Nobody does it, but one could in principle take a small 
> enough F ratio as an indication of statistical significance in the "wrong" 
> direction.

"No one does it" may be a correct assertion, but Shine (1982) did
publish a paper on how to do it, should anyone want to.  He notes
that the significance should be interpreted as either due to
chance, or a sign of "some inadequacy" in the ANOVA model used
for the data.

I also have a note of Myers (1979) on the topic. He gives an
example of a situation in which an F-ratio less than 1 may occur.
This is an experiment in which two different experimenters are
used, sharing the testing within each group. This method could
increase within-group variability above that of across-group.
I've noted in the margin of my photocopy (which, in this case, is
large enough to contain it) that another example might be if data
on males and females are collected with a large sex difference in
the measure, but the data are not separately analyzed by sex.
Since I seem to have made up that example, I may not be right
about it.


Myers, J. (1979). Fundamentals of Experimental Design, 2nd ed. 

Shine, L. (1982). Interpreting significant left-tailed analysis
  of variance F-ratios. Educational and Psychological 
  Measurement, 42, 1045--

-Stephen
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