Re: F-test for comparing variances?


This usually comes up in an intro course in the context of a between
groups t-test. The two group t-test is a special case of oneway
analysis of variance, which is a special application of ordinary
regression. It is an approach to testing whether the residuals (errors)
are homogeneous. Later you should be introduced to better methods for
testing this.

variance (a) is by convention the larger of the two variances under
consideration, so it does not "happen" to be larger.

-----

By and large, these tests are built on what we take to be the degree to
which different samples (groups of cases) have means that vary simply by
the vagaries of sampling.

To get an image of what "two samples from the same pop" means, try this.
Keeping in mind that a percentages is the mean of a dummy(zero, one)
variable. Take 2 sets of 10 pennies. Shake and throw throw them on
different surfaces. Keep two lists of the number of heads in each pile.
Note that many of the list entries are not 5, but counts farther away
from 5 in either direction occur a smaller and smaller percentage of all
the throws.

The null hypothesis is that the two sets of counts are from the same
population. Any apparent differences are attributed to the vagaries of
sampling. Given what we know about how sample results in general vary,
are the results sufficiently different that we should go with the
alternative hypothesis?

If the results are not sufficiently different we keep or retain the the
null hypothesis. We do not "accept" H0.
Keeping H0 means that the [status quo, default, current, presumed]
practice, understanding, or policy (p/u/p) prevails. We stick with the
p/u/p represented by H0, until we have sufficient reason to believe
otherwise.

This is strongly analogous to the way that under US and British Common
Law, a person is presumed innocent until prove guilty. The burden of
proof is on the party asserting the alternative.

In both scenarios, the alternative is "proven to a reasonable degree" or
"not proven to a reasonable degree".

Aside <I have heard that in Scotland, a jury can return "not proven". I
don't know if they have this instead of "not guilty", or in addition to
it.>

When the values of F or chi-squared are small, that means that the
evidence against the default null hypothesis is very weak.

IMHO, it is unfortunate that terms like status quo, default, current, or
presumed are not often enough associated with the "null" hypothesis.

Hope this helps.

Art Kendall
Social Research Consultants


cody wrote:

Hi, I'm a grad student who tutored someone in an intro course last week. According to an example shown in their textbook, if you have two normally distributed populations, you can test for equality of variance by

F = variance(a)/variance(b), where df = [n(a)-1, n(b)-1].

It makes sense to reject H0 if variance(a) happens to be bigger than variance(b), therefore we get a large F statistic. But what if variance(a) is smaller than variance(b)? Can the F-test be used to reject if the value is too small?

I'm confused on situations where F- and chisquare-statistics have small values. It seems the protocol is only to reject if they are large?

Thanks



.

Relevant Pages

  • Re: F-test for comparing variances?
    ... This usually comes up in an intro course in the context of a between groups t-test. ... The two group t-test is a special case of oneway analysis of variance, which is a special application of ordinary regression. ... We stick with the p/u/p represented by H0, until we have sufficient reason to believe otherwise. ... IMHO, it is unfortunate that terms like status quo, default, current, or presumed are not often enough associated with the "null" hypothesis. ...
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  • Re: F-test for comparing variances?
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