I think that your colleague made at least one error here. The unpooled t is
not necessarily more conservative. If it had been the group with the
smaller variance (.02) that had the larger sample sizes, the resulting t
would have been considerably larger. The important thing is whether the
large variance is paired with the large sample size, or whether it is
paired with the small sample size.
On a different note, if you lay out the arithmetic with a pencil and paper,
you see that for the pooled t the larger variance gets five times as much
weight in computing the pooled estimate as does the smaller variance. That
will make the test conservative. In the unpooled situation the larger
variance receives somewhat less weight in the result.
To take a different approach, we could use a randomization test. If we
apply (Monte Carlo) randomization to the scores (rather than to the ranks
as in Wilcoxon's test), the probability is approximately .10, which falls
between the pooled and unpooled t. I have included a screen shot of the
result, where the statistic that is calculated is the unpooled t. The
statistic actually doesn't matter--I could have plotted the mean of the
larger (or smaller) sample or any of several other statistics, and the
probability would be the same. I just chose t for its greater familiarity.
You can see that the Empirical Sampling Distribution of t certainly does
not look like the distribution Student derived.
Dave Howell
p.s. This is an interesting problem. I don't have experience with studies
that have two observations in one group, although that does happen in lots
of legitimate studies. What do people normally do in those cases?
At 02:16 PM 11/3/00 -0500, Richard Lehman wrote:
>A colleague sent me this note.
>
>
>>A statistics question.
>>
>>Temperatures taken from different portions of a stream:
>>
>>Portion 1
>>16.9
>>17
>>15.8
>>17.1
>>18.7
>>18
>>
>>mean = 17.25
>>variance = 0.995
>>
>>Portion 2
>>18.3
>>18.5
>>
>>mean = 18.4
>>variance = 0.02
>>
>>Do these portions have different temperatures?
>>
>>Obviously the variances are unequal and a 2-sample [unequal variance]
>
> t = 2.74 w/ 5 df p = 0.037.
>
>>No problem.
>
>
>>But (and this is where I am perplexed), a pooled [equal variance]
>
> t = 1.54 w/ 6 df p = 0.17.
>
>>Why is the less conservative pooled t giving a lower t-value? Are the
>>variances so uequal (and the one so close to zero) that the formula is
>>messed up?
>
>
>
>
>I have a hunch that his hunch is probably correct, if inexactly formulated.
>
>Any insights you can offer? Thanks.
>
>
>--------------------------------------------------------------------------
>
>Richard S. Lehman [EMAIL PROTECTED]
>Professor of Psychology (That's R-underscore-Lehman)
>Department of Psychology
>Franklin & Marshall College Voice (717)291-4202
>PO Box 3003 FAX (717)291-4387
>Lancaster, PA 17604-3003
> "I'd rather be blowing glass."
>
>
>
>
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