Nonparametrics.txt
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Received: by ECUVM1 (Mailer R2.08) id 1501; Fri, 31 Jan 92 19:40:49 EST
Date: Fri, 31 Jan 1992 18:39:45 CST
Sender: "STATISTICAL CONSULTING"
From: thisted@GALTON.UCHICAGO.EDU
Subject: Re: DATA TRANSFORMATION THEORY ???
To: "Karl L. Wuensch"
>On Fri, 31 Jan 1992 Evan Cooch writes:
>>
>> QUESTION: IF YOU CAN'T USE THE SAME TRANSFORM FOR ALL YOUR DATA,
>> SHOULD YOU TRANSFORM AT ALL?
To which Bill Knight replies:
>
>Question: If you can't use the same transform for all your data,
>SHOULD YOU USE ANOVA AT ALL?
>
>(With data that nasty, nonparametrics look good don't they?)
>
And the envelope, please.....
NO!
***** nonparametrics don't solve the problem at all!!! *****
There seems to be a common misconception that nonparametric methods are
assumption-free methods. Take, for example, the simple two-sample Wilcoxon
test ( = Mann-Whitney). There are some pretty restrictive assumptions,
which include not only independence of observations (fairly standard), but
also...
EQUAL VARIANCES. And equal skewness. And equal kurtosis. And, well, you
get the picture.
The null hypothesis is that the X's and Y's come from the same
distribution. The alternative [here comes the assumption] is that the X's
and Y's come from the same distribution after the Y's are all shifted by a
constant amount, say M.
Note: unequal variances, or shapes, invalidates the assumption.
Remarks.
(1) Actually the Wilcoxon procedure is applicable under a less restrictive
set of conditions, but under those assumptions it is no longer a test of
equal medians.
(2) Yes, I know that the underlying distribution might not have a variance
(or higher moments); I'm trying to keep things simple.
(3) Exactly the same remarks apply to the Kruskal-Wallis test (one-way
anova based on ranks) --- except for the difference in medians, the groups
all have the *same* distribution, including identical variances (if any).
(4) Most nonparametric methods have similar assumptions. As a result,
nonparametric methods are not
(a) A free lunch
(b) Necessarily more applicable in practice
(c) Safer than doing normal statistics
(d) Less prone to error than other methods, and,
most importantly, they are
(e) NOT ASSUMPTION-FREE.
(5) An excellent book on nonparametrics (in part because it spells out the
assumptions in considerable detail), is Nonparametric Statistical Methods,
by Myles Hollander and Douglas Wolfe (Wiley).
Bill did ask the right question:
>Question: If you can't use the same transform for all your data,
>SHOULD YOU USE ANOVA AT ALL?
Despite it's name, anova is essentially examines whether the means of the
groups are plausibly the same. As such, it is most appropriate
CONCEPTUALLY when the groups under study are such that the main important
way in which they might differ is through a shift in the means. If that
isn't the case, it is unlikely that anova will come close to answering your
question. It is often (but not always) the case that transforming the data
so that the spreads are roughly comparable makes comparison of means more
readily interpretible. Occasionally, this transformation makes the
distributions more symmetric as well, which also makes it easier to think
in terms of means. Transformations are not a panacea but, often as not,
they do help out.
If the groups seem to have different distributional shapes, rethink the
idea of shoving the data through an anova program. By thinking carefully,
you may realize that there is a different question (i.e., not equality of
means) that you really should be asking.
Here is a (true) example. One indicator of whether a prescription drug has
abuse potential is whether people like the drug. Liking means choosing the
drug more often than a placebo when repeatedly given a choice. This and
other measures can be combined to produce what we'll call a Liking score.
Let's suppose that scores around zero are neutral, negative scores mean
dislike, and positive scores liking. An investigator studied three drugs
and a placebo, studying many subjects in a controlled setting. [The actual
study was blocked, so that each subject was exposed to each drug; for
exposition suppose that subjects were randomized to four treatment groups.]
The mean scores for all four compounds were near zero. Anova is nowhere
close to significant. Let's all breathe a sigh of relief. But as it turns
out, the scores for all subjects were all very close to zero for placebo
and two of the drugs, but there was a very high variance for drug number 3.
The correct interpretation was *not* that none of these drugs had abuse
potential---the anova answer. Rather, drug #3 elicits strong reactions in
many subjects, so that a third of subjects hate it, a third don't care, but
a third REALLY like it. A drug that a third of the population really likes
may well be said to have abuse potential.
Note, the "statistical problem" here shows up as unequal variances. Either
ignoring the problem or hunting up an unequal-variances version of anova
would miss the answer entirely. Thinking about the data and what they mean
can lead to insight, and to better statistics.
May your days be merry and bright,
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Date: Sun, 02 Feb 92 12:29:12 EST
From: "Karl L. Wuensch"
Subject: Re: DATA TRANSFORMATION THEORY ???
To: thisted@GALTON.UCHICAGO.EDU
In-Reply-To: Your message of Fri, 31 Jan 1992 18:39:45 CST
What an excellant response you made to the short note on using
nonparametrics (which are to a some extent just another data transformation).
I'm glad I resisted making a similar response, thinking that someone might
have beat me to it. In fairness, the chap who said such data make
nonparametrics look pretty good might have been thinking about a more general
set of hypotheses than ones involving only medians, but it certainly is true
that most persons think of nonparametrics as being assumption free and
involving hypotheses about medians only.
If you have no objection, I'd like to post your message on my graduate
students "help disk." I've been harping at them about this. Your note also
makes another point I try to emphasize -- group differences in variances
should be considered more than just a nuisance. The example I use is
post-instructional achievement test scores for groups taught with different
methods. With means about the same but variances different, we should
consider that in the high variance group the method there used may have been
very effective for some subjects and very ineffective for others -- which
suggests additional research to find out if we can discriminate between those
two types of subjects, ultimately to recommend that method for one type of
student but not for others.
Karl L. Wuensch, Dept. of Psychology, East Carolina Univ.
Greenville, NC 27858-4353, PSWUENSC AT ECUVM1 (BITNET)
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Date: Tue, 4 Feb 92 18:57:57 CST
To: "Karl L. Wuensch"
From: thisted@galton.uchicago.edu
Subject: Re: DATA TRANSFORMATION THEORY ???
> What an excellant response you made to the short note on using
>nonparametrics (which are to a some extent just another data transformation).
Thanks for the vote of confidence!
>
> If you have no objection, I'd like to post your message on my graduate
>students "help disk."
Be my guest. (Thanks for asking, though)
> Karl L. Wuensch, Dept. of Psychology, East Carolina Univ.
> Greenville, NC 27858-4353, PSWUENSC AT ECUVM1 (BITNET)
Ronald Thisted Departments of Statistics and
(312) 702-8332/8333 Anesthesia & Critical Care
r-thisted@uchicago.edu The University of Chicago
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Date: Thu, 27 Jan 1994 20:55:18 -0500
Sender: Stat-l Discussion List
From: Dick Adams
Organization: Merrick School of Business
Subject: Re: Kruskal-Wallis test assumptions
To: Multiple recipients of list STAT-L
(PSWUENSC@ecuvm1.bitnet) Karl L. Wuensch writes:
> Right or wrong (if wrong I trust someone on this list will set me
> straight). I think of the "nonparametrics" as testing the null
> hypothesis that 2 or more populations are identical in all aspects
> and, if one is willing to assume that they are identical in dispersion
> and shape, then if they are not identical then a rejection of the null
> indicates that they differ in location. Apparently many of these
> tests are especially sensitive to differences in location, but
> differences in dispersion or shape might lead to rejection of the null
> even in the absence of differences in location.
IMHO, this is an accurate representation.
> I've met many a researcher who has been taught that a Kruskal-Wallis
> is equivalent to a parametric ANOVA except that one does not need
> worry about non-normality and heterogeneity of variance. Their
> samples can indicate that the populations differ radically in shape
> and dispersion, but if they can reject the K-W null they will surely
> restrict their conclusions to differences in location. Ask them if
> they made any assumptions about shape or dispersion and they will
> opine that K-W has no assumptions of any kind.
Many of us have been confronted by this dysfunctionality.
OFF THE RECORD:
The good news is that I had the priviledge of being educated in the
Promised Land of North Carolina (at the intersection of the yellow
brick roads of 15-501 and 54) and you have the priviledge of teaching
there (East of BB&T).