From: wvenable@attunga.stats.adelaide.edu.au (Bill Venables)
Newsgroups: bit.listserv.edstat-l,bit.listserv.stat-l
Subject: Re: Diff in R squares
Date: 24 Nov 92 10:38:33
Organization: Department of Statistics, University of Adelaide
NNTP-Posting-Host: attunga.stats.adelaide.edu.au
In-reply-to: nichols@spss.com's message of 20 Nov 92 19:49:28 GMT


In article <EDSTAT-L%92112013540483@NCSUVM.CC.NCSU.EDU> Dennis
> Roberts <DMR@PSUVM.BITNET> writes: Can someone help me out quickly: two
> random groups, multiple regressions on sam e variables in both, and R
> squared for first, and R squared for second. How to do a hypothesis test
> of 0 difference between the two population R squared value s???? Thanks
> in advance.

>>>>> "David" == David Nichols <nichols@spss.com> writes:

David> Just a generalization to multiple predictors of the case of testing
David> two univariate correlations for independent groups. Can be tested as
David> a homogeneity of regressions in an analysis of covariance setup.

>>>>> "Frank" == Frank Dane <FDANE@UGA.BITNET> writes:

Frank> The easiest way to test R^2 differences between groups is to include
Frank> a grouping variable in the equation.  If the grouping variable
Frank> (dummy or contrast coding) is "significant", beta will provide some
Frank> measure of the influence of the grouping variable.  This apparoach
Frank> also includes the advantage of testing for interactions between the
Frank> grouping variable and the other variable(s) in the equation.

Both of these responses are wrong, in my opinion, although both suggest
something sensible, whereas I cannot think of a context where the original
question is really going to be all that appropriate.

David's response is a test for *equality of [conditional] regressions*,
which, if retained will certainly guarantee equality of multiple correlation
coefficients, but this is too strong.  Equality of MCRs can easily happen
without the regressions being even vaguely equal.  The reason the two are
equivalent in the simple linear regression case is because you are only
dealing with one regression coefficient and one correlation coefficient.

Frank's response, if I read it correctly, addresses the question of
equality of regressions, assuming parallel regressions.  This is also too
strong for similar reasons.

Reading the original question, however, if the two groups are two "random"
groups why would you want to test equality of anything at all?  Leaving that
aside, the question amounts to a test that the regressions each account for
an equal proportaion of the total variation.  I think Dennis's best bet is
to look at some of the classical work on the approximate distribution of the
MCR, such as Hotelling's paper in JRSS (B), 1951, and J. Gurland JRSS (B)
(1968) and J. Gurland and R. Milton, JRSS (B) (1970) and piece together an
approximate procedure.  Constructing a likelihood ratio test for this
particular hypothesis, for example, would be computationally very difficult.

Bill.
--
___________________________________________________________________________
Bill Venables, Dept. of Statistics, | Email: venables@stats.adelaide.edu.au
Univ. of Adelaide, South Australia. | Tel: +61 8 228 5412  Fax: ...232 5670
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