How does one compare the Goodness of Fit for different models of some given binomial data using Generalized Linear Mixed Models.
Specifically we want to know whether a model with a logit link gives a better fit than a model with a probit link.
Is it legitimate to compare -2loglikelihood values for the 2 models using the different link functions?
If not, what would be a good Goodness of Fit comparison measure?
Models with different link functions are not nested, and therefore you cannot use standard statistical tests (e.g., a likelihood ratio test) to compare them. You could consider using information criteria, such as the AIC and the BIC, though given that there is no other difference between the two models, it boils down to comparing the likelihoods.
Two other alternatives could be (1) to see which model predicts better, e.g., using ROC analysis or by calculating the squared prediction error, and (2) go for a Bayesian approach and use the idea of posterior predictive checks.
