JISCMail - SPM Archives

As others have pointed out, orthogonalising the confounding covariates w.r.t. those of interest would be a very bad idea. If the variance in the data is explained by the confounding variables, then in order to obtain a good fit via the orthogonalised confounds, the multiple regression needs to use some of the variance of the effects of interest to explain it. This simulation should show that this leads to an incorrect distribution under the null hypothesis.....

N = 10000; % Number of simulations
P = zeros(N,1); % Simulated p values

c = [1 0 0 0]; % Contrast vector

for n=1:N

X1 = randn(10,1); % Effect of interest

X2 = [randn(10,2) ones(10,1)]; % Confounds

% Data partially explained by confounds

y = X2*randn(3,1)+randn(10,1);

% Orthogonalising confounds w.r.t. effects of interest

X2 = X2 - X1*((X1'*X1)\X1'*X2);

% Design matrix

X = [X1 X2];

if true

% T statistic

      b = X\y;
      nu = size(X,1)-rank(X);
      r = sum((y-X*b).^2)/nu;
      t = (c*b)/sqrt(r*c/(X'*X)*c');
      p = 1-spm_Tcdf(t,nu);

else

% F statistic

      X0 = X*null(c);
      nu = [rank(X)-rank(X0), size(X,1)-rank(X)];
      t1 = y'*(y - X0*(X0\y));
      t2 = y'*(y - X*(X\y));
      F = (t1-t2)/t2 * nu(2)/nu(1);
      p = 1-spm_Fcdf(F,nu);

end

P(n) = p;

end

hist(P,100)

title('This histogram should be uniform');

xlabel('P-value');

ylabel('Frequency')

Best regards,
-John

On 5 July 2016 at 14:24, Marko Wilke <[log in to unmask]> wrote:

Dear All,

I have a question that has recently come up in a discussion. Background is, we are doing a VBM study of three groups, say A/B/C. We want to perform an analysis on "modulated" GM maps, as derived by a DARTEL procedure. As such, it is usually recommended to include a global covariate, either total GM volume or total intracranial volume, as a covariate. Our groups, however, due to the nature of the underlying condition, have different globals, i.e., group A has higher globals than group B and/or C.

I understand that including the globals will change the interpretation of the resulting group differences. It may also, due to the group difference in the globals, "take away" some effects that may exist between the groups because the variance may be shared.

One idea that then came up was whether it is possible to use an orthogonalization on total GM / TIV, w.r.t. group status. This seemed like a worthwhile idea at the time as the aim was to not explain group differences by this orthogonalized variable that are already explained by group. We tried it and the effect was substantial, to say the least.

I am not sure, though, that this is a good (or even valid) idea from a statistical point of view. For one, I have seen several mails in the archives that mention that othogonalization of a nuisance variable is not a good idea. One could of course argue that it is not really a nuisance variable, it "only" changes the interpretation of the results by (interpreted in a very naive way) rescaling. But then again, I could be totally wrong and not see the forest for the trees. So as always, your insights into this matter are much appreciated.

Cheers
Marko

--
____________________________________________________
Prof. Dr. med. Marko Wilke
Facharzt für Kinder- und Jugendmedizin
Leiter, Experimentelle Pädiatrische Neurobildgebung
Universitäts-Kinderklinik
Abt. III (Neuropädiatrie)

Marko Wilke, MD, PhD
Pediatrician
Head, Experimental Pediatric Neuroimaging
University Children's Hospital
Dept. III (Pediatric Neurology)

Hoppe-Seyler-Str. 1
D - 72076 Tübingen, Germany
Tel. +49 7071 29-83416
Fax +49 7071 29-5473
[log in to unmask]

http://www.medizin.uni-tuebingen.de/kinder/epn/
____________________________________________________