This is an interesting discussion. Concerning the right choice and computation
of t-tests there is still one point that is unclear to me: In the Welch t-test
we have a difference in the numerator and a standard deviation of a difference
in the denominator. Why then is the standard deviation of the difference not
computed correctly, i.e. why is the covariance between X and Y not taken into
account?
For example using the sleep data:
data(sleep)
means <- tapply(sleep$extra,sleep$group,mean) ; means
vars <- tapply(sleep$extra,sleep$group,var) ; vars
sd.welch <- sqrt(vars[1]/10 + vars[2]/10) ; sd.welch
#in sd.welch the covariance is ignored
t.welch <- (means[1]-means[2])/sd.welch ; t.welch
#verify with R-built-in t.test function:
t.test(extra ~ group, data = sleep)
However, the correlation between sleep$extra[sleep$group == 1] and
sleep$extra[sleep$group == 2] is relatively high:
cor(sleep$extra[sleep$group == 1],sleep$extra[sleep$group == 2])
Souldnt the correct standard deviation be
sd.paired <- sqrt(vars[1]/10 + vars[2]/10
-2*cov(sleep$extra[sleep$group == 1],sleep$extra[sleep$group == 2])/10)
; sd.paired
as in the paired t-test???
In other words, isnt the Welch t-test a special case of the paired t-test with
both samples assumed to be uncorrelated? And shouldnt we then teach only the
paired t-test as the most general test in class?
Thanks!
Zeno
-----Original Message-----
From: [email protected] on behalf of Albyn Jones
Sent: Tue 10/26/2010 3:51 AM
To: Ian Fellows
Cc: [email protected]
Subject: Re: [R-sig-teaching] prop.test in R
Exactly - elementary texts and methods books recommend the welch test
for the reason you mention. Curiously, those same texts recommend
using anova and regression without automatically correcting for the
possibility of non-constant variance. Why is the case of comparing
two means different from 3? Those same books will tell you that anova
is pretty robust to non-constant variance. well, the two sample
t-test is anova.
I don't use the welch test except as a conscious decision: ie I really
want to compare the means while suspecting that the variances differ.
Generally people are using the t test to certify that two populations
are different. If the variances are wildly different, that may be
much more important than a difference in means. in fact, to test for
a difference in means when the variances are wildly different is
almost always substantively silly. There was a great example a few
years ago from a psychiatric journal, comparing two medications, where
the investigators did a t-test for the means when one distribution was
unimodal and the other was bi-modal; there was no statistically
significant difference in the means, but there was a really important
difference in the distributions. The automatic use of the welch test
makes you feel that you are protected against Bad Things, when you
aren't.
albyn
Quoting Ian Fellows <[email protected]>:
> In the case of the t.test, having the default be var.equal=TRUE is
> the right way to go. There is little to no power lost by using the
> welch test, and the assumption of equal variance can be difficult to
> assess. For this reason, many introductory text books have now
> banished the equal variance t-test from their chapters (e.g. Moore's
> The Basic Practice of Statistics).
>
> Ian
>
>
> On Oct 25, 2010, at 4:05 PM, Albyn Jones wrote:
>
>> I don't know, the help file is uninformative. I'd guess the answer is
>> "the author wrote it that way". Other R functions like t.test include
>> similar unfortunate (to me) default choices, in that case
>> var.equal=FALSE (ie the Welch test) is the default.
>>
>> albyn
>>
>> On Mon, Oct 25, 2010 at 04:15:20PM -0500, Laura Chihara wrote:
>>> Yes, thank you for this reference. But according to
>>> this article, the score is better than continuity
>>> correction, so why is continuity correction the default
>>> with prop.test?
>>>
>>> -Laura
>>>
>>> On 10/25/2010 4:02 PM, Ralph O'Brien, PhD wrote:
>>>> I suggest:
>>>>
>>>> A. Agresti and B. A. Coull. Approximate is better than "exact" for
>>>> interval estimation of binomial proportions. The American Statistician,
>>>> 52(2):119-126, 1998.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> On Mon, Oct 25, 2010 at 4:38 PM, Laura Chihara <[email protected]
>>>> <mailto:[email protected]>> wrote:
>>>>
>>>> Hi,
>>>>
>>>> I have a question about prop.test in R:
>>>>
>>>> I teach students the score confidence
>>>> interval for proportions (also called
>>>> Wilson or Wilson score interval).
>>>>
>>>> prop.test(,..., correct=FALSE) gives this
>>>> interval.
>>>>
>>>> The default uses a continuity correction.
>>>> When should we use one over the other?
>>>> Is it worth going over this in class? Why
>>>> is correct=TRUE the default?
>>>>
>>>> Thanks for any pedagogical guidance here!
>>>>
>>>> -- Laura
>>>>
>>>> *******************************************
>>>> Laura Chihara
>>>> Professor of Mathematics 507-222-4065 (office)
>>>> Dept of Mathematics 507-222-4312 (fax)
>>>> Carleton College
>>>> 1 North College Street
>>>> Northfield MN 55057
>>>>
>>>> _______________________________________________
>>>> [email protected] <mailto:[email protected]>
>>>> mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-sig-teaching
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>> Ralph O'Brien, PhD
>>>> Professor, Dept of Epidemiology and Biostatistics
>>>> Case Western Reserve University
>>>> Office: 216.368.1927
>>>> Cell: 216.312.3203
>>>
>>> --
>>> *******************************************
>>> Laura Chihara
>>> Professor of Mathematics 507-222-4065 (office)
>>> Dept of Mathematics 507-222-4312 (fax)
>>> Carleton College
>>> 1 North College Street
>>> Northfield MN 55057
>>>
>>> _______________________________________________
>>> [email protected] mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-teaching
>>>
>>
>> --
>> Albyn Jones
>> Reed College
>> [email protected]
>>
>> _______________________________________________
>> [email protected] mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-sig-teaching
>
>
>
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