Things are not always what they seem.
Consider the following data:
A B A/B Log A Log B Log A - Log
3 1 3 .477 0 .477
1 3 .333 0 .477 -.477
2 2 1 .301 .301 0
The t test for the difference of logs obviously
gives a value of zero, while the t for the hypothesis
that the mean ratio is 1 has a positive value.
This seems to show that the statement that the
two tests are "precisely, 100% identical"
is incorrect.
In the above example, I created the data to "balance"
the ratios. The example demonstrates that a ratio in
(3) one direction does not balance the same ratio in the opposite
direction.
Key points to remember (and are the source
of this apparent paradox) are:
1. The expected value of X/Y is in general NOT E(X)/E(Y),
even when X and Y are independent.
2. The expected value of F(X), where F is a non-linear function,
is in general NOT F(E(X)).
Hence, when making statements about ratios or nonlinear
transforms, one must be exceedingly careful.
These facts would seem to suggest that a t-test on ratios
(suggested by Howell?) need have no relationship to the goal of Dr.
Smith's analysis, because the statistic might reject a null hypothesis
when the ratios are "perfectly balanced."
Rich's log procedure seems preferable to me at first glance,
because "balanced ratios" do result in a t of zero.
One more point. Dr. Graham reports t(12). Although the value
and associated plevel are correct, the degrees of freedom are
actually 11 (i.e., one less than the number of pairs of
observations, and so typical usage would suggest that one should
write
t(11) = -2.337
--Jim Steiger
-------------------------
James H. Steiger
Professor, Dept. of Psychology
University of British Columbia
Vancouver, B.C., Canada V6T 1Z4
email: [EMAIL PROTECTED]
On Tue, 03 Apr 2001 15:35:43 -0400, Rich Ulrich <[EMAIL PROTECTED]>
wrote:
>
>Doing that one-sample t-test on the ratio is not a bad idea.
>
>But it is not a new idea, either. It is, precisely, 100% identical to
>doing a repeated measures test on the logarithm of the raw numbers.
>Which is the same as the paired t-test.
>
>
>On 2 Apr 2001 11:53:11 -0700, [EMAIL PROTECTED] (Dr
>Graham D Smith) wrote:
>
>> I would like to start a discussion on a family of procedures
>> that tend not to be emphasised in the literature. The procedures
>> I have in mind are based upon the ratio between two sets of
>> scores from the same sample.
>[ ... snip, detail ]
>
>> My feeling is that the t test for ratios should have a similar
>> status and profile as the repeated measures t test (on
>> differences). I suspect that the t test for differences is often
>> used when the t test for ratios would be more suitable. So
>> why is the procedure not more widely used? Perhaps this
>> is only a problem within psychology where ratio level data
>> is not commonly used.
>[ snip, rest ]
>
>Logarithms (if that is what is appropriate) is a more general start
>to a model. Building directly on ratios is not as convenient.
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