Re: Questions concerning T-tests
From: Noli Brazil (nbbrazil_at_hotmail.com)
Date: 12/24/04
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Date: Fri, 24 Dec 2004 03:12:30 +0000 (UTC)
Thank you all for your replies and guidance. I now know how to
approach the problem. Thank you.
Noli Brazil
On 18 Dec 2004 12:33:22 -0800, George Kahrimanis wrote:
>Hello. I have read with interest the comments posted by
>Rich Ulrich (Thu, 16 Dec 2004 14:37:40 -0500) in reply to a double
>query posted by Noli Brazil (Wed, 15 Dec 2004 13:02:15 +0000).
>
>We are in agreement that no version of the t-test is robust
>wrt deviations (however tiny) from normality, when the samples are
>very large. The same goes for the K-S test of normality for each
>sample.
>
>I also agree that the Mann-Whitney test (aka "U test" or
>"Wilcoxon two-sample test") does not even apply to the first
>problem, because the two distributions are not assumed to be
>identical.
>
>Perhaps the reader has forgotten the simple answer with the
difference
>of the sample-means, which seems to me *the* correct solution. Or do
>you disagree? I repeat it here, a bit more verbosely. Each sample is
>so large that the StDev of each distribution may be assumed as
>practically identical to its estimate: s1 and s2. The corresponding
>StDevs for the two means are s1/sqrt(n1) and s2/sqrt(n2), and they
>are distributed normally, with a good approximation, regardless of
>the two underlying distributions. The StDev for the difference of
>the two sample-means is sqrt( (s1^2)/n1 + (s2^2)/n2 ) and, according
>to the null hypothesis the mean of this difference is zero. Ergo we
>have to perform a test using the normal distribution.
>
>The above solution is robust wrt using very large samples, because
>of the asymptotic normality of each sample-mean. (I admit that I
>have not considered other issues, like computational rounding off or
>the increase in probability that some of the data may be corrupt.)
>
>~~~~~~~~~
>
>It seems to me that the following reply was in jest; please correct
>me if I am wrong.
>
>>> should i use the parametric t-test with the Welch correction in
>>> addition to a non parametric test?
>
>> (yep, for a simplistic answer.)
>
>~~~~~~~~~
>
>When I have time I plan to debate the following maxim:
>> The defining characteristic of a rank-test is that it
>> tests *ranks* and not *means*.
>
>~~~~~~~~~~~~~~~~~~~~~~~
>
>Wrt the second question...
>
>>> I am comparing one member's score to the mean of 40 to 150
>>>member scores. [...] to test whether or not there is statistically
>>>significant difference between the one score and the mean of the
>group
>>>scores.
>
>... for one, this is not a comparison between two numbers but a
>comparison between a number and a sample.
>
>If normality can be assumed, then the solution given by Ray Koopman
>applies. That is, the two-sample t-test, where the first sample
>consists of a single datum.
>
>If normality cannot be assumed, then the Mann-Whitney test (mentioned
>above) applies. This is a two-sample test; the first sample now
>consists of a single datum.
>
>Without having gone through the details, I presume that the plain
>nonparametric test in my first reply (Wed, 15 Dec 2004 22:42:19
+0200,
><<a
href="news://41C0A0E8.FB22ABE9@hol.gr">news://41C0A0E8.FB22ABE9@hol.gr>>
is equivalent to the Mann-Whitney test
>as reduced in this situation.
>
>~ George Kahrimanis
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