Random intercepts model - one measurement per subject I have repeated measurements on several hundred subjects.  I plan to fit a random intercepts model.  Most subjects have 3 to 4 measurements but about 20% only have one measurement.
For the subjects with one measurement, should they share the same intercept or should they each have their own intercept as well?  I'm not sure if there is enough information for the latter.
I plan on doing the modelling in BUGS where I can control the intercepts for each subject.
 A: I have not used BUGS but I believe that conceptually you should allow an random intercept for all of your subjects. Having said that, I believe you will find that there is very little difference in the model fits whether you include or exclude the single-observation groups (assuming you are not using a glmm where in that case issues of over-dispersion come into play). Heteroscedasticity between subjects might be an issue but you will probably be unable to present subject-specific variances but other than that you should be OK.
I believing looking at some basic literature regarding unbalanced designed mixed models will give you better understanding of the issues that are entailed: The following two papers are considered classics will probably serve as a good first reads: 


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*Harville, D. A. (1977). Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems. Journal of the American
Statistical Association 72 (358), 320-338   

*Jennrich, R. I., & Schluchter, M. D. (1986). Unbalanced Repeated-Measures Models with Structured Covariance Matrices Biometrics, 42 (4), 805-820
A: I think your data can be viewed as multilevel data, where measurements are nested in subjects. For 20% of your cases measurements at 2-3 time points are missing. Maximum likelihood (ML) fitting of a mixed-effects model will assume that these observations are missing at random (MAR). 
Having said this, if you fit a multilevel model, each subject will have its own intercept to allow hetorogeneity in initial scores at t=1. I would additionally include a random slope of the time indicator(s) to allow for between subject differences in the change across time.
About the quantity of missing information: 20% missing observations seems still fair enough to allow ML fitting. If you want to be really sure that the missing data do not cause any problem, you could use a multiple imputation (MI) model before estimation of the multilevel model. Also multiple imputation will assume data are MAR, but the Bayesian nature of the method may be easier to accomodate in BUGS. An alternative in R is the package mice.
However, if you have only missing data in the dV the big advantage of multilevel modeling is that its ML algorithm allows missing data in model fitting. If the ML algorithm does not have any convergence issues, the estimates from MI and ML should be very similar.
