# Specifying the correlation structure of an unevely spaced time series in GEE with geepack

I have counts of plants from different sites over a number of years. In each census year, all sites were surveyed, but the gaps between census years vary (between 1 and 4 years between consecutive surveys). I would like to use a GEE to account for the correlation structure of the data (but really, my purpose is just in estimating the long-term population trend, and some estimate of its uncertainty).

I think I can specify a model like this:

library(geepack)
model1<-geeglm(Count~Year,id=Site,family=poisson,data=mydata,corstr="ar1",waves=Year)


But I am not sure if this is quite right--I think the correlation parameter estimate it gives is the correlation between any two consecutive counts, i.e. without accounting for the time between counts. My question relates to a previous one below, but I could not comment on it. The waves argument makes no differences to my results (the data is already ordered by site, and time within site). Can anyone point me in the right direction? Or comment on how much I should worry about this?

How to analyze GEE with unevenly spaced observations?

3 approaches come to mind, only one allows you to keep your plan on using the AR1 cor structure for these data. I would heavily advise you to generate some variograms for these data to really look at what the time dependent variation in these data is. I'd also like to know how many sites are in this analysis.

1. Use multiple imputation. Technically speaking, measurements that aren't collected at every year are missing data. You can impute these values using some MCMC models for the plant growths, although it introduces a great deal of dependence upon identifying the correct imputation model.

2. Use an exchangeable correlation structure instead. If the variogram does not have a big disparity between the kernel and the nugget, then you see that the time dependent variation is of large influence upon the values. An important thing to remember about specifying correlation structures in GEE versus mixed models: the GEE is always consistent for the marginal mean model. That means that specifying the wrong correlation structure will not give you biased estimates, just inefficient ones (even though they may be different).

3. Model time as a fixed effect. Using some semi-parametric modeling strategies like smoothing splines or fractional polynomials to model time, you can control for time in the same way that random intercepts / indicator variables control for site dependent variation.

You raise a good point. Don't be too surprised that you are getting the same results with the two different correlation structures. Using a camera lens analogy, Walter Stroup (2013, p.438) describes GEE-type (i.e. R-side) repeated measures models as "lesser quality wide-angle lens[es]" whose "greater depth of field makes them capable of a useable image even when somewhat out of focus." On the other hand, G-side GLMMs are more like a "high quality telephoto lens" that is "very sharp when in focus but with a shallow depth of field ..."

GEE-type models do not correspond to any real-world probability process that gives rise to the data. Instead, they provide a technique for estimating the mean of the marginal distribution. As @AdamO points out, they are consistent for estimating the marginal mean and therefore are (to some degree) robust to correlation structure misspecification. If you experiment with different correlation structures, you will find that you get roughly similar results. If I were you, I'd be more concerned about overdispersion than the correlation structure.

If you fit a GLMM and use integral estimation (i.e. Laplace or quadrature) instead of pseudo-likelihood, then you can fit different correlation structures and compare them on the basis of fit statistics (AIC, BIC, etc.). See here for available packages.

The GLMM will be more sensitive to the correlation structure and it will also target a different parameter than your GEE. Instead of the mean of the marginal distribution, it will target a parameter closer to the median of the marginal distribution, but it is based on a real-world probability process.

• Thank you for the lens analogy. This is the perfect way of explaining exactly why GEE is good at handling unmeasured sources of variation, despite that you get improved efficiency if you correctly identify such variation. Jan 10 '14 at 17:43