Hi Mirko,
Actually I think this is a very good question, and it's one I haven't
been able to fully understand... I hope you don't mind me sending my
reply via the list -- I thought it would be good to see if anyone else
has any comments.
Several books define contrasts as linear combinations whose weights
sum to zero. Christensen's "Plane Answers..." (cited a few times in
the SPM literature/code) distinguishes between "estimable functions"
and contrasts, also suggesting that the latter should sum to zero.
The SPM literature (e.g. HBF2 chapter 8, see below) seems to define
contrasts as estimable functions, and doesn't mention any sum-to-zero
constraints.
My understanding of things is that an estimable function is invariant
to the choice of pseudo-inverse (given a rank-deficient design matrix)
and hence gives uniquely-defined "con" results even when the
individual betas are not uniquely estimable. In practice, I understand
estimable functions/hypotheses as simply ways of specifying the null
hypothesis, so e.g. if you have two columns/betas A,B then a contrast
of [1 0] is the null hypothesis A<=0 (with alternative hypothesis A>0
since SPM t-contrasts are right-tailed), and [1 -1] would test whether
A>B. Another example is that if a design matrix contains a constant
(ones) column and a covariate, then [0 1] gives a t-statistic that can
be equivalently transformed to the correlation coefficient between the
response and covariate (see a recent mailing list thread).
What I don't really understand is what the benefit of the sum-to-zero
constraint on contrasts (e.g. in Christensen's terms) as opposed to
more general estimable functions is. In particular, I can't really see
the key motivating point of Christensen's section 4.2 (though he is
far from the only author who defines contrasts as summing to zero).
I had an off-list email from Will Penny, in which he suggested that
the sum-to-zero constraint was necessary in an ANOVA-with-constant
model, since the individual effects are not estimable in such a case.
While this makes sense, I don't think it fully explains the
distinction between contrasts and estimable hypotheses (in e.g.
Christensen) since estimable functions, by definition, must be chosen
to be appropriate for ANOVA-with-constant models (i.e. a test of a
single unidentifiable beta of a with-constant model would not be an
estimable function; but it is possible to have estimable functions
whose elements do not sum to zero).
To give a concrete example of the difference between being estimable
and summing to zero:
X = [ones(6,1) kron(eye(3),[1;1])]; % ANOVA-with-constant
spm_SpUtil('isCon',X,[1 0 0 0]') % gives 0, can't test mean
spm_SpUtil('isCon',X,[0 1 0 0]') % 0, can't test individual beta
spm_SpUtil('isCon',X,[0 -1 1 0]') % 1, can test level2-level1
% now the following sums to zero but isn't estimable:
spm_SpUtil('isCon',X,[-1 1 0 0]') % 0, can't test level1-mean
% while the following doesn't sum to zero, but is estimable
spm_SpUtil('isCon',X,[3 1 1 1]') % 1, though hard to interpret
I hope someone can shed some more light on this (or at least read
Christensen's section 4.2 more intelligently than I've been able to!)
Ged.
P.S. Some further reading for those who don't think this email is long
enough already (!)
http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch8.pdf
http://www.jiscmail.ac.uk/cgi-bin/webadmin?A2=ind04&L=SPM&P=R237670
http://www.jiscmail.ac.uk/cgi-bin/webadmin?A2=ind0602&L=SPM&P=35712
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