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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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.
AdamO
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+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. – gung - Reinstate Monica May 12 '14 at 17:03
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@AdamO: Suppose you take 10 measurements of blood pressure of 100 people over time. In this case, there would be 100 random intercepts? – guy May 12 '14 at 17:06
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@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. – AdamO May 13 '14 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.
Sergio
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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. – Sergio May 13 '14 at 09:01
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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! – jsk May 13 '14 at 09:10
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1GEE 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). – Sergio May 13 '14 at 11:09
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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. – jsk May 13 '14 at 15:01
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Why are you claiming that a random effects model estimates $\sigma_{\alpha}^2$ but a GEE does not? Won't a mixed model also produce inconsistent estimates of the variability of the random effects if the variance model is misspecified? – jsk May 13 '14 at 16:05
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BTW: I never said that GEE does not estimate $\sigma^2_\alpha$. Are you trolling? – Sergio May 13 '14 at 16:55
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You claim gee estimates population effects. Then you imply that's in contrast to a random effects model which estimates both population effects and variability of random effects. Did I misinterpret what you were trying to say? – jsk May 13 '14 at 17:10
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Marginal models (including GEE) are "population-average models", mixed effects models (including random intercept models) are "subject-specific models". They have "different targets of inference and thereby address subtly different scientific questions". So they are not basically equivalent as you claim. According to Fitzmaurice, Laird & Ware (Applied Longitudinal Analysis, 2nd ed., pp. 341-342), not to me. – Sergio May 13 '14 at 17:46
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Indeed, my claim appears to have been worded too strongly. This quote, however, does not support the wording of your answer! The whole point seems to be that the random effects have been integrated out in the GEE, but that does not mean that a GEE is not estimating the variability of the random effects, which is precisely what you imply when you say "GEE estimates the average population effects. Random intercept models estimate the variability of these effects."! – jsk May 13 '14 at 18:36
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Integrated out??? Random effects are targets of inference in random coefficients models, are just nuisance parameters in GEE. This is what I've said. I'm not interested in continuinig this sterile discussion. – Sergio May 13 '14 at 18:37
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This is sad. I don't think we really disagree all that much in the end. I just think your answer could have been clearer :( – jsk May 13 '14 at 18:43
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Yes, integrated out. A colloquial term for marginalization. Hence the term marginal or population model, which you are clearly well aware of. – jsk May 13 '14 at 18:45