JiscMail Logo
Email discussion lists for the UK Education and Research communities

Help for SPM Archives


SPM Archives

SPM Archives


SPM@JISCMAIL.AC.UK


View:

Message:

[

First

|

Previous

|

Next

|

Last

]

By Topic:

[

First

|

Previous

|

Next

|

Last

]

By Author:

[

First

|

Previous

|

Next

|

Last

]

Font:

Monospaced Font

LISTSERV Archives

LISTSERV Archives

SPM Home

SPM Home

SPM  2006

SPM 2006

Options

Subscribe or Unsubscribe

Subscribe or Unsubscribe

Log In

Log In

Get Password

Get Password

Subject:

Re: DCM design matrix

From:

Ged Ridgway <[log in to unmask]>

Reply-To:

Ged Ridgway <[log in to unmask]>

Date:

Thu, 17 Aug 2006 16:23:05 +0100

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (124 lines)

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
>
>
> ------------------------------------------------------------------------
>

Top of Message | Previous Page | Permalink

JiscMail Tools


RSS Feeds and Sharing


Advanced Options


Archives

May 2024
April 2024
March 2024
February 2024
January 2024
December 2023
November 2023
October 2023
September 2023
August 2023
July 2023
June 2023
May 2023
April 2023
March 2023
February 2023
January 2023
December 2022
November 2022
October 2022
September 2022
August 2022
July 2022
June 2022
May 2022
April 2022
March 2022
February 2022
January 2022
December 2021
November 2021
October 2021
September 2021
August 2021
July 2021
June 2021
May 2021
April 2021
March 2021
February 2021
January 2021
December 2020
November 2020
October 2020
September 2020
August 2020
July 2020
June 2020
May 2020
April 2020
March 2020
February 2020
January 2020
December 2019
November 2019
October 2019
September 2019
August 2019
July 2019
June 2019
May 2019
April 2019
March 2019
February 2019
January 2019
December 2018
November 2018
October 2018
September 2018
August 2018
July 2018
June 2018
May 2018
April 2018
March 2018
February 2018
January 2018
December 2017
November 2017
October 2017
September 2017
August 2017
July 2017
June 2017
May 2017
April 2017
March 2017
February 2017
January 2017
December 2016
November 2016
October 2016
September 2016
August 2016
July 2016
June 2016
May 2016
April 2016
March 2016
February 2016
January 2016
December 2015
November 2015
October 2015
September 2015
August 2015
July 2015
June 2015
May 2015
April 2015
March 2015
February 2015
January 2015
December 2014
November 2014
October 2014
September 2014
August 2014
July 2014
June 2014
May 2014
April 2014
March 2014
February 2014
January 2014
December 2013
November 2013
October 2013
September 2013
August 2013
July 2013
June 2013
May 2013
April 2013
March 2013
February 2013
January 2013
December 2012
November 2012
October 2012
September 2012
August 2012
July 2012
June 2012
May 2012
April 2012
March 2012
February 2012
January 2012
December 2011
November 2011
October 2011
September 2011
August 2011
July 2011
June 2011
May 2011
April 2011
March 2011
February 2011
January 2011
December 2010
November 2010
October 2010
September 2010
August 2010
July 2010
June 2010
May 2010
April 2010
March 2010
February 2010
January 2010
December 2009
November 2009
October 2009
September 2009
August 2009
July 2009
June 2009
May 2009
April 2009
March 2009
February 2009
January 2009
December 2008
November 2008
October 2008
September 2008
August 2008
July 2008
June 2008
May 2008
April 2008
March 2008
February 2008
January 2008
December 2007
November 2007
October 2007
September 2007
August 2007
July 2007
June 2007
May 2007
April 2007
March 2007
February 2007
January 2007
2006
2005
2004
2003
2002
2001
2000
1999
1998


JiscMail is a Jisc service.

View our service policies at https://www.jiscmail.ac.uk/policyandsecurity/ and Jisc's privacy policy at https://www.jisc.ac.uk/website/privacy-notice

For help and support help@jisc.ac.uk

Secured by F-Secure Anti-Virus CataList Email List Search Powered by the LISTSERV Email List Manager