# GEE standard errors lower than standard errors assuming independence

I have a categorical dataset where the outcome is nominal (with three categories). There are 300 observations, and each individual contributes two observations to the dataset.

When I analyzed the data assuming independence (using multinomial regression through the nnet package in R), my standard errors for $$\beta$$ coefficients were slightly larger than when modeling with a GEE with a time-exchangeable odds ratio dependence structure between the correlated observations (through the nomLORgee) function.

This was a very surprising result to me. Shouldn't standard errors assuming independence be smaller than when a dependence structure is assumed? Are there ever cases where it can be backwards, like this?

All the coefficients from the GEE-based estimation were almost the exact same as the multinomial coefficients, but in the opposite direction (negative where the multinomial coefficients were positive). I am also quite confused by this result. Is there a fundamental difference in the modeling here?

UPDATE: I have found this reference, where pages 47-50 (pages 5-7 on the pdf) show that the standard errors of the GEE estimates are smaller. However, I do not understand their reasoning on page 50: "Standard errors are smaller because regressor (time) is changing within an individual." Any clarification would be extremely helpful.

Consider linear regression with the simplest possible model: outcome $$Y$$ and a single binary predictor $$X$$

Suppose we have two observations on each of $$n$$ people, and that there is correlation within people, so we consider GEE.

Case 1: $$X$$ is constant within each person (eg, native-born vs immigrant)

In this case, having two observations on the same person gives us less information about differences in $$X$$ than if observations were independent. The second observation on each person is correlated with the first and we already have some of the information. The true (GEE) standard error will be larger than the incorrect (linear regression) one.

Case 2: $$X=0$$ for the first observation on each person and $$X=1$$ for the second observation on each person.

In this case, having two observations on the same person gives us more information about differences in $$X$$ than if the observations were independent. The two observations on the same person are matched; they differ less for reasons other than $$X$$, so it's easier to see the relationship with $$X$$. The true (GEE) standard error will be smaller than the incorrect (linear regression) one.

The same principle applies for your multinomial models: the incorrect standard errors assuming independence can be too large or too small.