GEE all clusters of size 1 have same outcome I have a large population of patients and hospitalization (>1 million) information. I'm trying to look at the relationships between various predictors and hospital re-admissions within 30 days. The data is set up as 1 record per hospitalization with a flag (binary outcome) yes/no if there was a repeat hospitalization following it. A "readmission" will also appear as its own record in the data set. 
I'm attempting to use GEE models (logistic) to account for the fact that some individuals will contribute >1 hospitalization to the data set. I planned on attempting to use a exchangeable correlation structure as a starting point.
The issue is this: Everyone who contributes only a single record (approx. 50%) to the data will necessarily not have the outcome (i.e. they only had 1 hospitalization and therefore no re-admission). When I run a GEE with exchangeable correlation structure, it estimates the correlation as 0.9999, and often fails all together depending on my model (using SAS genmod). When I run it only within those with 2+ records, the estimated correlation is 0.6. 
My question is: Can you use a GEE model when all clusters of size=1 have the same outcome value? What can be done?
Edit: The analysis I'm attempting is basically the same one as described in this paper: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4581528/
 A: This seems like a bad idea as is. Your major problem seems to be class imbalance. One way to get around this is to fit your models many times, each time including all the positive examples (readmissions) and an equal number of randomly chosen negative examples (no readmission). Here's an example of a project that used that tactic:
http://www.nature.com/ng/journal/v46/n3/full/ng.2892.html?foxtrotcallback=true
GEE with exchangeable correlation structure may not be the best choice of model for time-series data such as you have, but this tactic will at least let you get started looking at your data.
A: In comments, I suggest survival analysis and you reply that you only care about a binarized target and not about the number of days to readmission. For future reference, I'd like to explore that and push back a bit...
First, you do care about the number of days, since "readmission" means "readmission within 30 days". You're just binarizing before analysis rather than after. There are many advantages to a survival analysis:


*

*What if "readmission" is redefined to "readmission within 45 days", "90 days", etc? With survival analysis, you already have the answer without re-analysis, and you can answer other questions more easily as well.

*It's generally a net loss to throw away information in an analysis, such as binarizing up front.

*To some extent, it feels like you're going to have a censorship problem. Even if you carefully eliminate patients whose first visit was less than 30 days before your data is pulled, you'll still have issues with patients who drop out. Depending on your data, it could be patients who move away (physically or switch insurance plans), or it could be patients who are readmitted but to a different hospital that your records don't cover. Survival analysis is designed for this situation.

*If you only care about readmission from first-ever admission, you can throw out re-readmission records and use simple survival analysis. If you care about any readmission after any other (re-)admission, you could use recurrent event survival analysis. Not sure if a logistic GEE formulation is that flexible.
On the flip side, if you (or your audience or collaborators) are very familiar with GEE and you'd waste time trying to come up to speed on survival analysis, perhaps you need to go with what you know. (Acknowledging that you're ignoring censorship, etc.) Especially with more complex versions of survival analysis, you can make mistakes due to misunderstanding.
At any rate, if you're using R, I'd recommend the survival package's vignettes, which are interesting and pithy and share insights from someone who knows what they're doing.
A: I suggest modeling the number of re-admissions using a Poisson or negative binomial model and request the empirical sandwich standard error estimates.  You would have one row per subject and for each subject you would have the tally of number of re-admissions.  For many this would be zero, for others this would be 1, 2, and so on.  Your generalized linear model would be cross-sectional and inference would be on the exposure-adjusted mean number of re-administrations.  If you want to report mean number of visits instead of mean number of re-admissions you would simply add 1 to the point estimate and confidence limits.
