Hi Ben,
I'll venture an answer, though I know nothing about DCM...
I think there are two issues, first, independence vs orthogonality,
and second about contrast estimability (which is a bit complicated,
but I'll try my best to explain, and maybe others can correct/simplify
what I say... it will be long -- sorry!).
A set of regressors are independent if no single one of them can be
formed from a linear combination of the others. While they are
orthogonal only if the inner products of each with all others are
zero. Orth implies indep, but not the other way around.
So, as a simple example (with rows for regressors)
[1 0 0], [1 1 0], [1 1 1]
are independent, but not orthogonal, since e.g. the inner product of
the second and third is 2. While
[1 0 0], [0 1 0], [0 0 1]
are orthogonal (and also orthonormal -- in that the inner product of
each with itself is 1).
Now, I haven't looked at the example data you refer to, but I guess if
it doesn't show the "gray, not uniquely specified" then their
regressors are all independent, even though they may not be
orthogonal. Independence can be tested by checking that the rank of
the matrix of regressors is equal to its number of columns. E.g.
rank([1 0 0; 1 1 0; 1 1 1]')
in Matlab returns 3 as hoped. While for a set with a dependent third
regressor (sum of the first two here):
rank([1 0 0; 1 1 0; 2 1 0]')
gives the number of independent regressors which is just 2.
However, it's more complicated than all this, since you can use a
generalised inverse (pinv in Matlab is one example) to invert a
non-full-rank matrix X'*X. This allows you to get a *non-unique*
solution for beta (pinv(X'*X)*X'*y), which in itself wouldn't be any
use (e.g. you can't consider the expected value E[beta] for such a
solution). However, the properties of any generalised inverse mean
that X*pinv(X'*X)*X' is invariant to the choice of inverse, and this
means that e.g. E[y] = X*beta is unique, even though beta is not.
Still with me? Okay, the important part about this is that you have
some contrast of interest which is a linear combination of the betas,
e.g. c*beta for some row vector c. Now, *if* your c can itself be made
up from a linear combination of the rows of X, i.e. c = k*X, then
because c*beta = k*X*beta and X*beta is unique, the contrast c is
uniquely "estimable".
Put simply, an estimable contrast asks a sensible question, whereas
some arbitrary combination of beta might not have a unique answer.
To test estimability, since we want c to be "made up from the rows of
X" c must be in the row-space of X, which means that
rank([X;c]) == rank(X);
(I believe the actual test used in SPM is to see if c is unchanged by
projection onto the rows of X, which I think is equivalent, but am
open to correction. See HBF2, ch8, pp9&17
http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/)
[Note also, that if all regressors are actually independent, then X is
full column rank, and *any* c will leave rank([X;c]) equal to the
number of columns -- i.e. all contrasts are estimable for independent
designs]
As an example, with
X=[1 0 0; 1 1 0; 2 1 0]';
from above, and with c = [1 0 0];
rank([1 0 0; X])
is greater than rank(X), which means this c cannot be estimated. In
other words it doesn't make sense to ask "how much beta(1) is there?"
On the other hand, with c = [-1 1 0];
rank([c;X])
equals rank(X), so this c can be estimated -- we can ask the question
"how much more beta(2) than beta(1) is present?"
So I think the answer to both your questions is, you can certainly
have non-orthogonal designs, and you can have dependent designs but
you may only ask certain questions of them, with suitable contrasts.
The particular questions that make sense for you will obviously depend
on your design, but *if* the question makes logical sense *given* the
design, then its contrast should be estimable and everything should be
okay.
Again, I know nothing about DCM, but hopefully it doesn't change any
of the stuff above... ;-)
Best,
Ged.
Xu, Ben (NIH/NINDS) [E] wrote:
> Hi,
>
> I have asked the question earlier but have not gotten a response. I'm
> trying to use DCM for an fMRI study and have a question related to the
> design matrix prior to DCM analysis.
>
> I created a design matrix (attached) with 7 regressors (i.e.,
> conditions) and an implicit baseline. The 1st regressor shares common
> input with regressor 2-5, and will be used for the "input drive" to the
> intrinsic connections. (Regressors 6-7 will not be included in the DCM
> analysis.)
>
> Questions:
> 1. As I understand, it's ok to have a non-orthogonal design matrix for
> DCM analysis. Is that correct?
>
> 2. The design matrix indicates regressors 1-5 are "not uniquely
> specified" and, therefore I can't define contrasts in "Results." I have
> tried the "attention" dataset provided at the SPM Web site
> (http://www.fil.ion.ucl.ac.uk/spm/data/). Its design matrix is
> non-orthogonal, either. However, there is no warning on regressors "not
> uniquely specified." And it allows t contrasts in Results. What's wrong
> with my design? Will it affect the DCM analysis?
>
> Any help is greatly appreciated.
>
> Ben
>
>
> ------------------------------------------------------------------------
>
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