I found this discussion: GEE: choosing proper working correlation structure
Cite:
Correlation structure in GEE, unlike mixed models, does not affect the marginal parameter estimates (which you are estimating with GEE). It does affect the standard error estimates though. This is independent of any link function. The link function in the GEE is for the marginal model.
and also: Do GEE and GLM estimate the same coefficients?
Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. T
How can this be true, if: GEE with exchangeable working covariance vs. assuming independence and using Huber-White standard errors?
In general, the GEE model solves the equation
∑i=1n∂μi∂βV(α)−1(yi−μi)=0
as a function of the regression coefficients, β , where μi=xiβ is the expected values of the cluster i response, yi, given the predictors xi under the specified model. V(α)−1 is the "working" covariance matrix of the of the elements of cluster i. (note that μi=xiβ because we're dealing with a linear model but GEEs can more generally use a link function so that μi=g(xiβ))
A key point here is that when you change the working covariance, you change the estimating equation, therefore the β that solves it will be different. For example, if V was σ2 down the diagonal and 0 off the diagonal and μi=xiβ as it does here, then the GEE estimator is the least squares estimator, which will not solve that equation in the exchangeable case. So it is no surprise that you're getting different parameter estimates. It may be a coincidence that you're getting the same standard errors.
So 1) the choice of the correlation structure changes the estimating equations and the resulting beta coefficients, 2) since B depends on the correlation structure, how can it do what the GLM does, if in GLM the choice is not present?
I just run an example in R and it clearly looks like the covariance structure DOES affect the betas:
> fm4 <- gee(Reaction ~ Days, id = Subject,data = sleepstudy,corstr = "AR-M", Mv=1)
Beginning Cgee S-function, @(#) geeformula.q 4.13 98/01/27
running glm to get initial regression estimate
(Intercept) Days
251.4 10.5
> coef(summary(fm4))
Estimate Naive S.E. Naive z Robust S.E. Robust z
(Intercept) 253.5 10.71 23.67 6.36 39.88
Days 10.5 1.67 6.25 1.44 7.27
> fm4 <- gee(Reaction ~ Days, id = Subject,data = sleepstudy,corstr = "independence")
Beginning Cgee S-function, @(#) geeformula.q 4.13 98/01/27
running glm to get initial regression estimate
(Intercept) Days
251.4 10.5
> coef(summary(fm4))
Estimate Naive S.E. Naive z Robust S.E. Robust z
(Intercept) 251.4 6.61 38.03 6.63 37.91
Days 10.5 1.24 8.45 1.50 6.97
The warning below only confirms that.
> fm4 <- gee(Reaction ~ Days, id = Subject,data = sleepstudy,corstr = "unstructured")
Beginning Cgee S-function, @(#) geeformula.q 4.13 98/01/27
running glm to get initial regression estimate
(Intercept) Days
251.4 10.5
Komunikat ostrzegawczy:
W poleceniu 'gee(Reaction ~ Days, id = Subject, data = sleepstudy, corstr = "unstructured")':
Working correlation estimate not positive definite
> coef(summary(fm4))
Estimate Naive S.E. Naive z Robust S.E. Robust z
(Intercept) 252.04 7.33 34.40 10.09 24.97
Days 9.78 1.02 9.62 2.64 3.71
