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With the log-likelihood chi-square statistics I can compare two linear mixed models (Maximum Likelihood) and see which one is the better one. But the GEE gives Quasi Likelihood under Independence Model Criterion (QIC) and I don't see the degrees of freedom so I am unsure how to statistically test two models against each other and select the one with the best fit.

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With GEE, you should do Wald tests on the coefficients. The likelihood-ratio type tests are sensitive to their assumptions such as normality of the response and the random effects. GEEs trade away the MLE efficiency for robustness (against possible misspecifications of the within-cluster correlations), and if the mixed model MLE and the GEE contradict one another, I would trust the GEE, not the MLE.

If you never came across the Wald test before, read the great exposition by Buse (1982) about the relation between different asymptotic tests.

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  • $\begingroup$ Thank you. Do you mean the Wald tests that appear in the 'Tests of Model Effects'? But actually, the factors that I included in my model are included based on theoretical considerations. I am more struggling with for example which link function I should use and compare two models with different link functions in order to select the one that gives the best fit. Does this make sense at all? $\endgroup$ – user9203 Aug 24 '12 at 14:50
  • $\begingroup$ I don't think it does. If you are choosing between logit and probit, this is a very superficial choice. If you are choosing between log and logit for proportions near zero, it does not make much difference, either. If you are choosing between the identity, log and inverse for positive data, you may want to try the Box-Cox transformation instead. At any rate, GEEs try to protect you from whether you did or did not make the "perfect" decision in model specification. $\endgroup$ – StasK Aug 24 '12 at 19:57

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