Consider a random intercept linear model. This is equivalent to GEE linear regression with an exchangeable working correlation matrix. Suppose the predictors are $x_1, x_2,$ and $x_3$ and the coefficients for these predictors are $\beta_1$, $\beta_2$, and $\beta_3$. What is the interpretation for the coefficients in the random intercept model? Is it the same as the GEE linear regression except that it is at the individual level?
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2 Answers
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GEE and Mixed Model Coefficients are not usually thought of as the same. An effective notation for this is to denote GEE coefficient vectors as $\beta^{(m)}$ (the marginal effects) and mixed model coefficient vectors as $\beta^{(c)}$ (the conditional effects). These effects are obviously going to be different for non-collapsible link functions since the GEE averages several instances of the conditional link across several iterations. The standard errors for the marginal and conditional effects are also obviously going to be different.
A third and oft overlooked problem is that of model misspecification. GEE gives you tremendous insurance against departures from model assumptions. Because of robust error estimation, GEE linear coefficients using the identity link can always be interpreted as an averaged first order trend. Mixed models give you something similar, but they will be different when the model is misspecified.
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$\begingroup$ +1, your point about differences, even for linear models, w/ model misspecification is a nice one. A small worked example illustrating this would be a really great addition, should you be interested in providing one. $\endgroup$ May 12, 2014 at 17:03
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$\begingroup$ @AdamO: Suppose you take 10 measurements of blood pressure of 100 people over time. In this case, there would be 100 random intercepts? $\endgroup$– guyMay 12, 2014 at 17:06
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$\begingroup$ @guy there are any number of ways of analyzing such data. Certainly, if you are interested in average levels of BP and conditioning out intracluster variability, then a random intercept model is a fine choice. Sometimes, you need to handle effects of time with random slopes, AR-1, or fixed effects which adds another wrinkle. So in general, the answer depends upon the question. $\endgroup$– AdamOMay 13, 2014 at 17:03
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GEE estimates the average population effects. Random intercept models estimate the variability of these effects. If $\alpha_j=\gamma_0+\eta_j$, $\eta_j\sim\mathcal{N}(0,\sigma^2_\alpha)$, random intercept models estimate both $\gamma_0$ (which is the average population intercept and, in normal linear models, is equal to the one estimated by GEE) and $\sigma^2_\alpha$.
If the intercept is modeled by second-level predictors, e.g. $\alpha_j=\gamma_0+\gamma_1 w_j+\eta_j$, a random intercept model can estimate how the intercepts vary at the individual level, i.d. according to economic, demographic, familiar etc. factors, to the 'group' to which a specific individual belongs.
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$\begingroup$ In GEE $\sigma^2_\alpha$ is just a nuisance parameter, in random intercept models $\hat{\sigma}^2_\alpha$ makes feasible subject-specific inference. See this paper. $\endgroup$– SergioMay 13, 2014 at 9:01
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$\begingroup$ What do you think the off-diagonal parameter of the exchangeable correlation matrix corresponds to? It's $\sigma_{\alpha}^2 / ( \sigma_{\alpha}^2 + \sigma_{\epsilon}^2)$ where $\sigma_{\epsilon}^2$ is the variability of the error term. It may be a nuisance, but it is still estimated! $\endgroup$– jskMay 13, 2014 at 9:10
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$\begingroup$ Could you say that GEE consistently estimates $\sigma^2_\alpha$? $\endgroup$– SergioMay 13, 2014 at 9:25
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1$\begingroup$ GEE is appealing because provides consistent estimates of the fixed effects even if the variance models is misspecified, but without the 'true' variance model you can't get consistent estimats of random effects. Furthermore, while fixed effects require second order moments, consistent estimates of random effects would require fourth order moments (here, page 139). Last but not least, the choice of a working matrix is tipically aimed to reduce the number of... nuisance parameters (Lang Wu, Mixed Effects Models for Complex Data, p. 340). $\endgroup$– SergioMay 13, 2014 at 11:09
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$\begingroup$ This seems to be missing the current point of comparing a linear mixed model with a random intercept to a GEE with exchangeable correlation. Both models will have inconsistent estimates of the variance without the true variance model. All I'm really interesting in arguing about is your claim that gee with exchangeable correlation doesn't measure the variability of the random effects. $\endgroup$– jskMay 13, 2014 at 15:01