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pooled t test vs "unpooled" t test
Posted:
Nov 4, 1996 8:39 PM
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To this observation
>> -- My guess would be that if a stat-program presented just one
>> "two-sample t-test", it *would* be the "Student's t-test" which pools
>> the variances. (The t in "t-test" is not capitalized.)
Bob Hayden commented
> Minitab only pools the variances if you explicitly ask it to. I
> think Minitab's default is the correct default. You lose very little
> if the variances ARE the same and you gain a lot of they are not.
In "What is Statistics?" (Chapter 1 in the MAA's Notes Number 21,
_Perspectives on Contemporary Statistics_), David Moore says:
"The pooled-sample t test ... is somewhat robust against unequal sigmas
if the sample sizes are equal, but not otherwise." He uses T_p for the
pooled statistic. Then, arguing for the "unpooled" statistic, labeled T,
he asserts that "A substantial literature ... demonstrates the accuracy
of this [i.e., the "unpooled" T] approximation for even quite small
samples, and demonstrates in addition that when in fact sigma_1 =
sigma_2, using T sacrifices very little power relative to T_p." (p.15)
It's this kind of "expert testimony" that is missing from so many
textbooks at the elementary level--the kind of testimony that would help
us make wiser decisions about the relative merits of various procedures,
when working with students.
(This debate about the two t tests is not all one-sided, of course. I
believe I recall Paul Velleman arguing that a good reason to teach the
pooled test is that it generalizes readily to ANOVA.)
Here's another comment by David Moore whose essence is missing from most
elementary textbooks: "... the F ratio for comparing variances is almost
worthless. ... The reason for this ... is that no data are exactly
normal. ... the F ratio is so sensitive to even small departures from
normality as to be almost useless." (same source, p. 14)
==============================================
Bruce King
Department of Mathematics and Computer Science
Western Connecticut State University
181 White Street
Danbury, CT 06810
(kingb@wcsub.ctstateu.edu)
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