# how to compute a linear mixed effect using nlmefit?

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Cyril Pernet on 13 Mar 2013
Hi,
I'd like to fit a simple linear mixed model Y=XB+Zb+e with X the design matrix of the fixed effect (always the same size but not always the same values) and Z described subjects (which I specify as random)
here is the code I used
% generate data
for subject=1:10
x(:,subject) = [1:10]+randi(30,1);
coef(subject) = (3+rand(1));
y(:,subject) = coef(subject)*x(:,subject)+3*randn(10,1)- 5*mean(x(:,subject));
end
% create X, Y, subject for nlmefit
Y = y(:);
X = [x(:) ones(100,1)];
subject = sum(kron(diag(1:10),ones(10,1)),2);
% fit the data using 'model'
model = @(Betas,X) (X*Betas)
[Betas,PSI,stats] = nlmefit(X,Y,subject,[],model,[1 0])
The error is: Error using * Inner matrix dimensions must agree. Error in @(Betas,X)(X*Betas)
in a fixed effect Betas=pinv(X)Y and the fitted data = X*Betas, and that why i defined model this way, assuming that for each subject, parameters are fitted using 'model' ?? any idea what I am doing wrong ?
Thanks Cyril
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Cyril Pernet on 14 Mar 2013
I'm looking for matrix multiplication (for fixed effect Betas = pinv(X)*Y and thus model = X*Betas) - I thought maybe using twice the same variable name confused matlab so I changed the code to
model = @(betas,X) (X*betas);
beta0 = [1 0]';
[Betas,PSI,stats] = nlmefit(X,Y,subject,[],model,beta0)
since the error was dimension issue - now for sure X*beta0 (the 1st guess for Betas) works - still I have the same error message :-(

Tom Lane on 14 Mar 2013
It looks like nlmefit invokes your model function with betas as a row vector. Try this:
model = @(Betas,X) (X*Betas(:))
Cyril Pernet on 15 Mar 2013
oops, this was indeed the issue - thx
now I got this, I'd like to generate some F test, I used the following code
Res = reshape(stats.ires,10,10);
for s=1:10
Yhat(:,s) = y(:,s) - Res(:,s);
end
SSeffect = norm(Yhat(:)-mean(Yhat(:))).^2;
SStotal = norm(Y-mean(Y)).^2;
R2 = SSeffect / SStotal;
SSerror = norm(Res(:)-mean(Res(:))).^2;
df = (rank(X)-1)+9; % add number of subjects -1?
% alternatively it can be nb of subjects - rank(X)
dfe = stats.dfe;
F = (SSeffect/df) / (SSerror/dfe);
p_val = 1-fcdf(F,df,dfe);
obviously that works, doesn't mean that's right! in particular I'm not sure
(1) if the modeled data correspond to the data minus the individual residuals (stats.ires) or the weighted residuals (stats.cwres)
(2) of the df of the model ? for each subject we fit X so df is rank(X)-1, we also fit each subject, so maybe add the number of subjects -1 ? - alternative;y, I read the is can be the number of subjects - rank(X)
No worries, if you don't know this, that's already a relieve to get the code running ..