I only said "this behaviour MAY introduce some sort of undesirable noise", and wasn't 100% certain.

As an example of the type of effect I was thinking of, try putting the attached tstat.m function somewhere where MATLAB can find it, and pasting the following into MATLAB.

N     = 100;

% Design matrix consisting of constant term, group difference, some
% covariate and an interaction between covariate and group
X     = [ones(N*2,1) [ones(N,1); -ones(N,1)] [1:N, 1:N]'/(N/2)-(1+1/N)];
X     = [X X(:,2).*X(:,3)];


Nrand = 10000;
p = zeros(Nrand,1);
for i=1:Nrand,
    beta0 = [randn(3,1).*[1 0.5 0.5]'; 0];   % Interraction term set to zero
    y     = exp(X*beta0 + randn(N*2,1)*0.5); % Assume 1st order kinetics
    p(i)  = tstat(X,y,[0 0 0 1]);            % Test for interaction
    % plot((1:N*2)',y,'.',(1:N*2)',X*inv(X'*X)*X'*y,'-')
end

hist(p,100); % Should be flat

Even though the mechanism by which the data were generated does not include any interaction, there appears to be one because of the nonlinearity.  For this reason, I'm slightly sceptical of the importance of claims such as those about male brains atrophying faster than female brains with age.  My take on it is that the atrophy rate could be the same in both groups, but brains that are slightly larger to begin with will lose more tissue.

Best regards,

-John

On 11 March 2014 23:25, Chou Paul <[log in to unmask]> wrote:

Dear John

Sorry to bother you, I don't understand why the Jacobian determinants introduce some sort of undesirable noise into the statistical analysis. Could you explain this concept more detail for me ? 

Thank you !

Best

Paul

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Date: Tue, 11 Mar 2014 11:59:56 +0000
From: [log in to unmask]
Subject: Re: [SPM] Divergence rate vs Jacobian rate in Longitudinal Registration
To: [log in to unmask]

The Jacobian changes are easier to explain in a paper, so I would generally suggest using those for now.  In practice, because longitudinal deformations are pretty small, it probably doesn't make so much difference which you use.

Both measures are included because I actually don't know which is likely to be most useful over the long term.  Having both options makes it possible to compare the effectiveness of the two different types of volume change representation.

When some form of groupwise registration is run, one of the things we can expect is that these divergences should be zero on average.  This is not the case for the Jacobian determinants.  If an average of Jacobian determinants is computed for a population of subjects that have been aligned together, this average does not turn out to be uniformly 1.  I figured that this behaviour may introduce some sort of undesirable noise into the statistical analysis, which is one reason why I kept the option of using the divergence.

Divergence represents the rate of volumetric change according to the registration model ( http://en.wikipedia.org/wiki/Divergence ).  Consider the following simple exponential growth model, where the volume of a structure atrophies at a constant rate.

dt    = 1; % Time interval

r      = -0.3; % Growth rate

vol1 = 1; % Volume at first time point

vol2 = vol1*exp(r*dt); % Volume at second time point

jac_rate = (vol2 - vol1)/dt

The divergence rate is analogous to "r", whereas the volume change rate computed from the Jacobian determinants is analogous to (vol2 - vol1)/dt.  If r or dt are sufficiently small, then r and jac_rate are similar.  In practice, the registration is slightly more complicated than this simple first order kinetic model, but similar principles apply.

Best regards,

-John

Best regards,

-John

On 10 March 2014 20:36, Shashwath Meda <[log in to unmask]> wrote:

Dear John & SPM'ers - I am in the midst of analyzing some longitudinal data via SPM12b and am trying to understand the differences between the divergence rate and the jacobian rate images that are produced via the pairwise longitudinal registration module. I am interested in looking at how volumetric changes correlate with drinking behavior in a cohort of normal adults. Which of the above metrics would be the best to look at from an morphometrics perspective?

TIA

--

Best,

Shashwath