# Do GEE and GLM estimate the same coefficients?

In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are

$$\sum_i^n (\frac{\partial \mu_i}{\partial \eta_i}) \frac{y_i - \mu_i}{Var(Y_i)}x_{ij} = 0, \quad (j = 1, ..., p)$$

and in the quasi-likelihood case, we just let $$Var(Y_i) = v(\mu_i)$$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.

Does this imply that GEE and GLM will have the same parameter (say $$\beta$$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)

If the estimated coefficients are not the same, then what is the difference?

Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):

library(geepack)
library(lme4)

#get book data from
mydf<-data.frame(subj=mydf$$subj, week=as.factor(mydf$$week), fev=mydf$$fev) #Make 5th level the reference level to match book results mydf$$week<-relevel(mydf$week, ref="5") #Fit GLM Mixed Model mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf)) mixed.model$coefficients

Estimate Std. Error     t value
(Intercept)  6.99850  0.2590243 27.01870247
week1        2.81525  0.2439374 11.54087244
week2       -0.15025  0.2439374 -0.61593680
week3        0.00325  0.2439374  0.01332309
week4       -0.04700  0.2439374 -0.19267241

#Fit a gee model with any correlation structure.  In this case AR1
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
gee.model$coefficients [Estimate Std.err Wald Pr(>|W|) (Intercept) 6.99850 0.2418413 8.374312e+02 0.0000000 week1 2.81525 0.2514376 1.253642e+02 0.0000000 week2 -0.15025 0.2051973 5.361492e-01 0.4640330 week3 0.00325 0.2075914 2.451027e-04 0.9875090 week4 -0.04700 0.2388983 3.870522e-02 0.8440338][1]  UPDATE As Mark White pointed out in his comment, I did indeed previously fit a "single-level" Mixed Effects GLM. Since you didn't specify whether you wanted a "fixed effects" or "random" effects GLM model, I just picked "random" since that's the model fit in the book I selected from. But indeed, Mark is right that the coefficients do not necessarily agree in multilevel models, and someone provided a nice answer about that question previously. For your reference, I've added a "fixed" effects GLM model below using lm. #Fit Traditional GLM Fixed Effect Model (i.e. not Random effects) glm.fixed<-summary(lm(fev~week, data=mydf)) glm.fixed$coefficients
Estimate Std. Error     t value     Pr(>|t|)
(Intercept)  6.99850  0.2590243 27.01870247 7.696137e-68
week1        2.81525  0.3663157  7.68531179 7.287752e-13
week2       -0.15025  0.3663157 -0.41016538 6.821349e-01
week3        0.00325  0.3663157  0.00887213 9.929302e-01
week4       -0.04700  0.3663157 -0.12830465 8.980401e-01


Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.

You also added a comment which asked, "And does this remain the case when we choose a non-linear link function?" Note first that this is a different question since non-linear link functions generally aren't General Linear Models but Generalized Linear models. In this case, the coefficients do not necessarily match. Here's an example again in R:

#Fit Generalized Linear Mixed Effects Model with, say, Binomail Link
nlmixed.model<-summary(lme4::glmer(I(mydf$$fev>mean(mydf$$fev))~week+(1|subj), family="binomial", data=mydf))
nlmixed.model$coefficients #Fit GEE model with, say, Binomial Link nlgee.model<-summary(geeglm(I(mydf$$fev>mean(mydf$$fev))~week, id=subj, waves=week, family="binomial", data=mydf)) nlgee.model$coefficients

• And does this remain the case when we choose a non-linear link function? – Marcel Jan 10 at 0:34
• The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/… – Mark White Jan 10 at 2:30
• @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said. – Marcel Jan 10 at 2:34
• @Marcel, I've updated the question to address your comments as well as Mark's. I've also included Mark White's excellent post on multilevel models in the body of the answer for future users who might stumble upon the post. – StatsStudent Jan 10 at 14:27
• @StatsStudent in your updated answer, you are using lm, which is not a generalized linear model, it is ordinary least squares. OLS regression's coefficients have a population averaged interpretation, which implies that they will have the same coefficients as a model estimated with generalized estimating equations - in fact it reduces to the same case under constant variance. My question was regarding the coefficients for a generalized linear model fit with ML and that of a generalized linear model fit with GEE - and if THOSE two models have the same estimated coefficients. E.g., a logit link – Marcel Jan 10 at 19:17