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FSL  December 2008

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Subject:

Re: general covariate in group analysis

From:

Mark Jenkinson <[log in to unmask]>

Reply-To:

FSL - FMRIB's Software Library <[log in to unmask]>

Date:

Wed, 10 Dec 2008 18:40:51 +0000

Content-Type:

text/plain

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text/plain (175 lines)

Hi Hans,

You still want to demean, as otherwise you cannot
interpret gamma1 as "the mean response".

Consider these three cases (for 5 subjects for simplicity):
(a) beta = [50 45 55 40 60] ; X2 = [ 100 90 110 80 120 ]
(b) beta as above ; X2 = [ 200 190 210 180 220 ]
(c) beta as above ; X2 = [ 0 -10 10 -20 20 ]
In the last case I have demeaned X2.

The results of the GLM for gamma1 and gamma2 would be:
(a) gamma1=0, gamma2=0.5
(b) gamma1=-50, gamma2=0.5
(c) gamma1=50, gamma2=0.5

Now only in the last case is gamma1 a good estimate of the
group mean BOLD.  In the other cases it is *highly* dependent
on RT mean.  In general people want to remove any potential
variation *away from the mean* that the RT might induce.
Normally you are still interested in whether there is an
average BOLD effect over the group, and this is what using
the demeaned X2 gives you.

Hope this helps.
All the best,
    Mark


Hans Tissot wrote:
> Hi Mark,
>
> Thanks for your help! I understand your argument. However, I am still 
> confused about your interpretation. 
>
> Going back to the maths in your e-mail:
>
> beta = gamma1 * ones(m,1) + gamma2 * X2 + error
>
> beta = observed data
>
> gamma1*ones(m,1) = mean response (without any contribution from RT effect)
>
> gamma2 * X2 = contribution of RT covariate (contribution of RT effect 
> alone)
>
> (1) You correctly point out that gamma1 will correspond to the 
> "intercept" i.e., the response predicted by the linear model for a 
> subject with zero RT. I interpret it as asking the question "Is the 
> mean response WITHOUT any contribution from RT effect = 0"? i.e., we 
> are measuring gamma1 alone without any contribution from RT.
>
> (2) If we demean X2, then we are testing ( gamma1 + gamma2 * X2_bar ) 
> = 0. [X2_bar = mean(X2)]
> This is a MIXED test where we ask the question "Is the mean response + 
> average contribution of RT covariate = 0"?
> Now imagine a situation where gamma1 = 0 but gamma2 * X2_bar > 0 
> (observed beta purely driven by RT contribution). 
> Then (gamma1 + gamma2 * X2_bar = 0 + gamma2 * X2_bar > 0) even though 
> gamma1 = 0. This could be a misleading test, since the point of 
> including a covariate is to tease out effects while accounting for the 
> effect of covariate.
>
> I think it would be much better to ask a question that tests for (A) 
> gamma1 = 0? and (B) gamma2 = 0? separately to tease out contributions 
> from each effect separately without MIXING gamma1 and gamma2 in the 
> same question.
>
> What do you think?
>
> Cheers, Hans.
>
> ------------------
> Hans Tissot
> McLean Hospital,
> Belmont, MA, USA
>
>
> On Wed, Dec 10, 2008 at 11:19 AM, Mark Woolrich 
> <[log in to unmask] <mailto:[log in to unmask]>> wrote:
>
>     Hi Hans,
>
>     For the sake of argument let's say the covariate is reaction time
>     (RT), and we fit a GLM: beta=gamma1*X1 + gamma2*X2, where X1 is
>     all ones and X2 is the RT covariate.
>
>     If you do not demean the RT covariate X then the  gamma1 estimate
>     will correspond to the "intersect", i.e. the response predicted by
>     the linear model for a subject with a zero RT.  If you demean the
>     covariate then the gamma1 estimate will correspond to the group
>     mean, i.e. the response predicted by the linear model for a
>     subject with a RT equal to the group average RT. The latter is
>     what most people want to interpret their gamma1 as typically.
>
>     As you can see this is consistent with the maths you had in your
>     email where mu corresponds to the intersect, and mu + gamma *
>     X_bar is equal to the group mean.
>
>
>     Cheers, Mark.
>
>     ----
>     Dr Mark Woolrich
>     EPSRC Advanced Research Fellow University Research Lecturer
>
>     Oxford University Centre for Functional MRI of the Brain (FMRIB),
>     John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK.
>
>     Tel: (+44)1865-222782 Homepage:
>     http://www.fmrib.ox.ac.uk/~woolrich
>     <http://www.fmrib.ox.ac.uk/%7Ewoolrich>
>
>
>
>
>
>     On 10 Dec 2008, at 15:27, Hans Tissot wrote:
>
>         Dear FSL experts,
>
>         I am a new user of FSL, so my question might be very basic. I
>         am trying to do a higher
>         level analysis (group average) with an additional covariate X.
>         The model that I am
>         thinking of in my mind is:
>
>         beta = mu + gamma * X + error
>
>         where:
>         beta = contrast of interest from first level analysis
>         mu = unknown group average
>         gamma = unknown multiplier of X
>         X = additional covariate
>
>         Approach A:
>
>         I want to test the null hypothesis mu = 0 while accounting for
>         the covariate X. My first
>         thought was to create a design matrix [ones(m,1), X] (assuming
>         X is a column vector
>         with m rows) and use the contrast [1,0].
>
>         Approach B:
>
>         However, I have come across multiple postings on this list
>         about demeaning the covariate
>         before putting it in the design matrix.
>
>         The model above can also be written as:
>         beta = mu + gamma * X_bar + gamma * (X - X_bar) + error
>         where X_bar = mean of m values in X
>
>         If we demean X before including it in the design matrix and
>         then use contrast [1,0] we
>         will be testing the hypothesis mu + gamma *X_bar = 0 (NOT mu = 0).
>
>         Question:
>
>         I have a feel Approach A above makes more sense for testing mu
>         = 0, but I am not sure
>         given the many postings on FSL list about demeaning the
>         covariate. I would like to get an
>         expert opinion from FSL developers about which is the way to go.
>
>         Thanks so much for your help.
>
>         Hans.
>
>         ------------------
>         Hans Tissot
>         McLean Hospital,
>         Belmont, MA, USA
>
>

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