Dennis Roberts wrote:
> the discussion of comparing variances brings to mind the following ... and
> is related to the post i just sent re: hyp testing
>
> let's assume that we are interested whether there is some difference in
> treatment effects ... as measured by means ... our null is the mu1 = mu2
>
> now, we use the 'standard' t test ... and forgive me, pooled variances ...
> where the assumption is that this is a test of MEANS ... not differences
in
> variances ... so we assume equal variances.
>
> but, given the data ... we suspect that there might be a difference in
> variances so ... we do the (not preferred method as has been mentioned a
> few times) classic F test ... first s square on top divided by second s
> square on the bottom ....
>
> HOWEVER ... as has been pointed out ... this test assumes for proper
> interpretation that the populations are normally distributed ...
> SOOOOOOOOOOOOOO ... given simple dotplots of the samples of data ... we
> think that there could be some non normality going on here ...
> SOOOOOOOOOOOO ... we look for a test of normality ... and of course, when
> we find one, there will be assumptions for IT too
>
> thus, what we are really interested is the difference in population means
> ... BUT, before we can look at this ... we have to check the equal
variance
> assumption ... BUT ... before we can look at this we ... need to check on
> the normality assumption of IT ....
"She swallowed the cat to catch the bird, she swallowed the bird to
catch the spider, she swallowed the spider to catch the fly... I don't know
_why_ she swallowed the fly."
First off, the chain is unnecessary. Even with pooled variance the t
test is generally better than (eeeew!) the F test; and there is no reason to
use the pooled t test.
Secondly, the chain just does not work. In almost every case, if test T
(testing H)makes assumption A, A is _more_ restrictive than not-A. In other
words, if I tell you that a naturally arising test (or other inference
method) depends on one of the following assumptions:
(*) The data are iid from a Beta-distributed population
(**) The data are iid from a population whose distribution is anything
but Beta
(***) The data are not iid, but can have any nontrivial dependence.
you will surely be able to guess which one. The trouble is that if we want
to do a second test S to test assumption A, we will have to use A as the
null hypothesis - and then we cannot prove it, only disprove it or fail to
disprove it. The process should thus be:
do test S
reject: A is false, do not do T
fail to reject: A is unproved, so do not do T.
Curiously, researchers don't like this! However,
do test S
reject: A is false, do not do T
fail to reject: A is undecided, do T
reject: report rejection of H
Fail to reject: report non-rejection of H
is not logically valid. If there is not enugh data to determine whether T is
valid, such a process leads to T being done and the results reported!
Conversely, if there are enormous amounts of data, T may never get done
[despite the fact that in many cases that the large sample size will
increase robustness]
Now, I suppose that in principle, the hypothesis test S could - under
some circumstances - be replaced by two one-sided tests. EG, for the t test
test S+ : is there evidence that V1/V2 is less than some appropriate
upper bound?
test S- : is there evidence that V1/V2 is greater than some appropriate
lower bound?
If the answer is "yes" in both cases, do T.
or by an interval estimate:
Construct a 95% CI for V1/V2.
If the interval lies entirely within a specified range, do T.
I have never seen this done or suggested - in the F-before-T case, for
obvious reasons of relative robustness!
-Robert Dawson
===========================================================================
This list is open to everyone. Occasionally, less thoughtful
people send inappropriate messages. Please DO NOT COMPLAIN TO
THE POSTMASTER about these messages because the postmaster has no
way of controlling them, and excessive complaints will result in
termination of the list.
For information about this list, including information about the
problem of inappropriate messages and information about how to
unsubscribe, please see the web page at
http://jse.stat.ncsu.edu/
===========================================================================
