From: WLTPIRIE@VTVM1.BITNET (Walt Pirie)
Newsgroups: bit.listserv.edstat-l
Subject: Fisher's r to z
Date: 12 Nov 92 14:47:58 GMT
Reply-To: Walt Pirie <WLTPIRIE@VTVM1.BITNET>
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU

I'm not sure I understand some of Larry's points, below. e.g. The
point that the t-test is "fairly robust" refers to testing means.
But in lots of distributions, the mean is not a "parameter of the
distribution" in the sense of being an explicit parameter. Similarly
the correlation coefficient is well defined for many bivariate
distributions in which it is not an explicit parameter. So the question
of robustness seems equally valid. That the distribution of rho is
non-robust in the sense of not yielding approximately valid tests
under non-normality I think is a given.

What makes the use of rho questionable is its special property for
bivariate normal distributions where linear dependency and dependency
are synonymous, unlike any other distribution family.

"Rank Correlation" (e.g. Kendall or Spearman) on the other hand, is
a misnomer in the strict sense, because they are not measures of
purely linear relationships as is rho, but of general monotone
relationships. That's why some are more careful to use the word
association rather than correlation in those contexts.
I am in agreement that correlation is used or misused far too
often. Because any strong monotone association will have a "linear
component" the usual tests of correlation will generally display
some power, aside from the robustness issue. The really questionable
practice is in ascribing meaning to the estimate of rho if normality
is in question. And therefor the use of confidence intervals is
equally questionable. They are implying a linear structure that
is probably inappropriate.
Walt.


     I'm a little confused about the discussion of robustness concerning
correlation coefficient inferences.  As I understand robustness, it refers
to, essentially, how well a test does when the assumptions about the
population are violated.

     For example, to do a one sample t-test we must assume the data comes from
a normal population.  As I understand it, the t-test works pretty well
even if the data isn't from a normal population, so we consider it a
fairly robust test.

     On the other hand, if we are even talking about a correlation, we
basically MUST ASSUME we are talking about a bivariate normal distribution.
(How many distributions have rho as a parameter?  The bivariate normal does,
and I can't think of any others off hand.)

     So my training suggests that you shouldn't look at a corrlation unless
you have a bi-variate normal.  Thus robustness can't be an issue.

     On the other hand, we do nonparametric correlations all the time.
ranks aren't bivariate normal.  Hmmmm.....

     My point is that basically a correlation has no meaning without the
assumption of bivariate normality, so maybe people look at correlation
a whole heck of a lot more than they should.

     Any discussion?


Larry Ries
Dept. of Statistics
University of Missouri
Columbia

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Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU