How many observations per subject are necessary to fit a random slope in a mixed model? I am working on a project that collected data retrospectively on subjects.  There are subjects with multiple points of follow up per person, anywhere from 1 to 3 measurements.  The timing of such measurements is extremely variable.  Most subjects have 1 measurement, but many have 2 or 3.  We are interested in the population trend over time, and would like to use all possible data.
Is it appropriate to use a mixed model with a random slope for time to analyze data with this structure?  I see no intuitive problem with fitting a random intercept model.  However, for the people with a single measurement, how is the random slope estimated?  If most subjects have only 2 or 3 measurements, is the random slope even possible (or recommended) or is this overfitting the data?
 A: In a basic mixed effects model, 
$$E[Y_{it}|X_{it}] = \alpha + \sigma_i^2 + \beta X_{it}$$
clusters having just one observation contribute influence to both the estimated variance of the random effect and the slope of the fixed effect. This is because the random intercept is never actually estimated. While some numerical solvers produce estimates for random intercepts, they are actually post-hoc statistics calculated after joint estimation of the random effect variance and the fixed effect slope. 
If you fit mixed effects models with unbalanced designs, it's important to verify the normality of these estimates (this can be a strong and influential assumption when there are a small number of clusters). As an example, suppose I run a health care clinic and we're verifying the management of AIDS in subjects on antiretroviral therapies, such as effivirenz. If I combine prevalent cases at baseline and incident cases during follow-up, my analysis is now sensitive to the distribution of incidence. For instance, suppose 70% of my cases were diagnosed two years ago, and have had successful management of disease while 30% of my cases are incident and have high viral loads before starting therapy. I now have an uneven bimodal distribution of random intercepts (viral load at "visit 1") and my fixed effect is biased toward the null (when it's actually suggestive that it's effective in managing disease).
A GEE on the other hand makes no assumption about the distribution of random effects and is consistent for the population averaged effect estimate: $\beta_M$ (M for marginal) rather than $\beta_C$ (C for conditional). These models are related to one another, but on average $\beta_M \leq \beta_C$ yet tests of inference about $\beta_M$ can often be of higher power. 
