GEE: choosing proper working correlation structure I am an epidemiologist trying to understand GEEs in order to properly analyze a cohort study (using Poisson regression with a log link, to estimate Relative Risk). I have a few questions about the "working correlation" that I would like someone more knowledgable to clarify:
(1) If I have repeated measurements in the same individual, is it usually most reasonable to assume an exchangeable structure? (Or an autoregressive if measurements show a trend)? What about independence - are there any cases where one could assume independence for measurements in the same individual?
(2) Is there any (reasonably simple) way to assess the proper structure by examining the data?
(3) I noticed that, when choosing an independence structure, I get the same point estimates (but lower standard errors) as when running a simple Poisson regression (using R, function glm() and geeglm() from package geepack). Why is this happening? I understand that with GEEs you estimate a population-averaged model (in contrast to subject-specific) so you should get the same point estimates only in the linear regression case.
(4) If my cohort is at multiple location sites (but one measurement per individual), should I choose an independence or an exchangeable working correlation, and why? I mean, individuals in each site are still independent from each other, right?? Thus for a subject-specific model, for example, I would specify the site as a random effect. With GEE however, independence and exchangeable give different estimates and I am not sure which one is better in terms of underlying assumptions.
(5) Can GEE handle a 2-level hierarchical clustering, i.e. a multi-site cohort with repeated measures per individual? If yes, what should I specify as a clustering variable in geeglm() and what should be the working correlation if one assumes for example "independence" for the first level (site) and "exchangeable" or "autoregressive" for the second level (individual)?
I understand these are quite a few questions, and some of them may be fairly basic, but still very difficult for me (and maybe other novices?) to grasp. So, any help is greatly and sincerely appreciated, and to show this I have started a bounty.
 A: (1) You will likely need some kind of autoregressive structure, simply because we expect measurements taken further apart to be less correlated than those taken closer together. Exchangeable would assume they are all equally correlated. But as with everything else, it depends. 
(2) I think this kind of decision comes down to thinking about how the data were generated, rather than seeing how they look.
(4) it depends. For example, kids nested in schools should not, in most cases, be treated as independent. Due to social patterning, etc, if I know something about a kid in a given school, then I probably know at least a little bit about other kids in the schools. I once used GEE to look at relationships between different social and economic indicators and obesity prevalence in a birth cohort where participants were nested in neighborhoods. I used an exchangeable structure. You can find the paper here and check some of the references, including 2 from epi journals.
(5) Apparently so (e.g. see this example), but I can't help with the R specfics of doing this. 
Zeger SL, Liang KY, Albert PS. Models for longitudinal data: a generalized estimating equation approach. Biometrics. 1988;44:1049–60. 
Hubbard AE, Ahern J, Fleischer N, van der Laan M, Lippman S, Bruckner T, Satariano W. To GEE or not to GEE: comparing estimating function and likelihood-based methods for estimating the associations between neighborhoods and health. Epidemiology. 2009 
Hanley JA, Negassa A, Edwardes MDB, Forrester JE. Statistical analysis of correlated data using generalized estimating equations: an orientation. Am J Epidemiol. 2003;157:364. 
A: *

*Not necessarily. With small clusters, imbalanced design, and incomplete within-cluster confounder adjustment, exchangeable correlation may be more inefficient and biased relative than independence GEE. Those assumptions can be rather strong, too. However, when those assumptions are met, you get more efficient inference with the exchangeable. I have never found an instance when AR-1 correlation structures make sense, since it's uncommon to have measurements that are balanced in time (I work with human subjects data).

*Well, exploring correlation is good and should be done in data analysis. However, it really shouldn't guide decision making. You can use variograms and lorellograms to visualize correlation in longitudinal and panel studies. Intracluster correlation is a good measurement of the extent of correlation within clusters.

*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.

*Sites can be sources of unmeasured variation, such as teeth within a mouth, or students within a school district. There is the potential for cluster level confounders in these data, such as genetic propensity to tooth decay or community education funding, so for that reason, you will get better standard error estimates by using an exchangeable correlation structure.

*Calculation of marginal effects in a GEE is complicated when they're not nested but it can be done. Nesting is easy, and you do just as you've said.
A: (0) General comments: most of the models I see on crossvalidated are far too complicated. Simplify if at all possible. It is often worth modeling with GEE and mixed model to compare results.
(1) Yes. Choose exchangeable. My unambiguous answer is based on the most widely touted benefit of GEE: resilience of estimates to assumptions made.
If you look at studies in your field you should see that exch is the default option. It doesn't mean it is the best, but should be the first to consider. Advising exch will be the best advise without having detailed knowledge of your data.
(2) Yes, there are data driven approaches such as "QIC". This is a Stata example, but widely accepted as a reasonable option, though very rarely used in practice: http://www.stata-journal.com/sjpdf.html?articlenum=st0126)
(3) Point estimates are never the exact same (unless you are using indep correlation structure), but are usually fairly close. You can find many articles comparing simple/gee/mixed effects model estimates to get a feel for this (https://recherche.univ-lyon2.fr/greps/IMG/pdf/JEBS.pdf) Most textbooks also have a table or two for this. For an independent correlation structure you are essentially running the poisson model with robust SEs. So the estimates will be the exact same. The SE are usually larger. But sometimes robust SE are smaller (that is life: google with provide pain free explanation if interested)
(4) See (1) and (2) above.
(5) No. Or better stated, you can do anything if you put enough effort into it but it is very rarely worth the effort.                                    
A: You're using the wrong approach with a gee to do what you are doing because you don't know the structure and your results will be likely confounded.  Refer to Jamie Robinson this. You need to use long. TMLE (mark van der laan) or perhaps a gee with iptw weights.  Not accounting for correlation does underestimate variance.  Just think if all repeated measures were 100% correlated, then you would effectively have way fewer observations (essentially only n for your n subjects) and smaller n means higher variance.  
