I think that would be appropriate, as your data resemble the data we had in a forensic, multiple measure study over time, where each case varied in the number of observations and there were sometimes missing data.

We consulted a statistical service available to us through the university and were guided through using GEE for this purpose. If you have such a service available, I think it would be useful to get an expert statistical consultation on this question.

I've linked the study abstract in case you are interested. 

http://www.tandfonline.com/doi/abs/10.1080/14999013.2013.857740

Ok. Thank you very much Michael!!!

Yes, you can.  There are a couple of things to consider though.  The GEE model will give you a "population averaged" or marginal results.  That is, the average increase in Y for a one-unit increase in X.  This would be like estimating the average differences between persons with X=0 vs. persons with X=1. Frequently, this approach is sufficient to address the question, accounting for the correlated nature of the data.  However, if you're question is more "subject specific", i.e., subject-specific change, then you want to use a mixed model approach.  (The mixed approach would address the question, what would the estimated Y be if an individual changed from X=0 to X=1).

The other issue you may have to consider is your measure of time.  Is your time measure "fixed" at specific periods? e.g., baseline plus 4 measures every 6-months for 2 years.  (e.g., some subjects were followed for 2 years and have 5 measures, some were followed for 2 years and have 3 measures (e.g., baseline, 18 months, and 24 months;) some 3 measures because they were only followed for 1 year, etc.)   OR, is time more subject-specific, and measured at irregular time intervals, e.g., some have 4 measures in 12 months, some have 4 measures in 24 months, etc,  all at irregular time intervals?)

This will determine how you model time.   I guess the question is how complex a model do you want to have?

Paul