GLM with a custom link function

Dear all,

I would like to use a custom link function in glmfit. The function is a special case of Cohen's (1973) model, and is written as:

v(t) = (1-exp(-(l-mu).*t))./(1-(0.2./l).*exp(-(l-mu).*t))

the above equation is technically the inverse link function. The link function is:

log(((mu.*t)-l)./(l.*(t-1)))./(l-mu);

My code specifying the input data and link functions is below. I am holding the parameter mu constant (mu = 0.2) and want to estimate the parameter l. I know I have not coded this properly, and the error message I receive "Input argument "l" is undefined"

How do I estimate my parameter "l"?

Thank you

%% Format data for glmfit

in = load('age_prev.txt');

a_prev = zeros(432,2);

a_prev(1:144,1:2) = in(:,1:2);

a_prev(145:288,1:2) = in(:,3:4);

a_prev(289:432,1:2) = in(:,5:6);

a_prev=delnanrows(a_prev);clear in

x = a_prev(:,1);y = [a_prev(:,2) ones(length(a_prev),1)];

%% Write link functions

clink = @(t,l) log(((0.2.*t)-l)./(l.*(t-1)))./(l-0.2);

cder = @(t,l) 1./((t-1).*((0.2.*t)-l));

cinv = @(t,l) (1-exp(-1.*(l-0.2).*t))./(1-(0.2./l).*exp(-1.*(l-0.2).*t));

mylinks = {clink cder cinv};

%% GLMfit

% x is an Nx1 array of the independent variable (time)

% y is an Nx2 array: column 1 is # of successes, column 2 is # of trials

[b,dev,stats] = glmfit(x,y,'binomial','link',mylinks,'constant','off');

Answer 1

[Peter Perkins]

Tal, the link function in a GLM maps the expected value to the linear predictor X*beta, and the inverse link does the opposite. Your function

v(t) = (1-exp(-(l-mu).*t))./(1-(0.2./l).*exp(-(l-mu).*t))

involves l, mu, and t, so you're gonna have to explain what's what. It has to be of the form

linearPredictor = g(expectedValue)

If, as it appears, your link function has other unknown parameters, those have to be estimated separately from the regression coefficients. McCullagh&Nelder's Generalized Linear Models goes into this kind of thing, or perhaps the paper you cite has a method tailored to your model.