Is it possible that GEE and mixed effect GLM give contradicting answers? If so, which one should be trusted? Is it possible that GEE and mixed effect GLM give contradictive answers in significance of covariates? I assume both GEE and GLM selects same covariates. If so, which one should be trusted?
From asymptotic viewpoint, GEE and GLM should give same answer. However, I am always in finite sample case. The same answer is not guaranteed.
 A: Do you mean inference rather than "answer", in terms of the supposed contradictions? Then yes for a number of reasons. A non-comprehensive list:

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*There is not a perfect 1-1 correspondence between the types of models GEE and LME fit. Consider variance structures versus random effects. For instance, a single random intercept cannot induce a negative correlation within subjects in the same cluster, but the WLS estimator with exchangeable correlation can. Even when positive intraclass correlation is met, the estimates induced by random effect variance may not be the same as those from WLS.

*The GEE is not a maximum likelihood procedure, and so handles missing and unbalanced data differently. LME places relatively more weight on subjects with sparser data.

*GEE is, in general, more robust to model specification whereas LME is, in general, more efficient with smaller sample sizes.

*When considering marginal models for response, such as the famous example of educational attainment versus age in a sample with a strong cohort effect, the LME is powered to detect individual level effects, or conditional effects, whereas the GEE is powered to detect population-average effects.

