Gelman & Hill's 'no', 'complete' and 'partial' pooling in the context of longitudinal data In Gelman and Hill's Data Analysis Using Regression and Multilevel/Hierarchical Models, they present a very compelling idea of 'random' effects offering a kind of compromise between no-pooling (i.e. including an indicator for every group variable and estimating a distinct intercept for each) and complete pooling (not including any group level predictors and thus forcing groups to have the same intercept). They say that including a 'random' intercept term (i.e. allowing the intercept term to have its own probability distribution) is an ideal compromise - as it serves to shrink group level effects towards the population level average when the group level data is scarce or noisy.
A common use of the 'random intercept' model, however, is in longitudinal studies in which the same individual is measured multiple times. In this case every group is a single individual. Here I don't see how the variance in the group level intercept can possibly be estimated - and the idea of 'partial pooling' no longer makes clear sense to me. Can anyone clarify the distinctions in the way random intercepts are understood between these two contexts? If a random intercept model in a longitudinal study with subjects measured at multiple time points can't be understood as 'partial pooling', how should they be better understood?
 A: It's helpful to start with the equation of the multilevel model, which applies whether the data is cross-sectional (multilevel) or person-period (longitudinal):
At level 1 (within cluster): $y_{ij} = \beta_{0j} + e_{ij},  e_{ij}\sim N(0, \sigma_e^2)$
and at level 2 (between cluster): $\beta_{0j} = \gamma_{00} + u_{0j}, u_{0j}\sim N(0, \sigma_u^2)$
In the longitudinal context, $\gamma_{00}$ is the grand mean estimated from all the observed data points - the average value of the outcome y. If not all individuals are measured the same number of times, then this is becomes a weighted mean of the outcome. The random intercept $u_{0j}$ is how much each person's mean outcome value deviates from the grand mean. The spread of person mean deviations around $\gamma_{00}$ can be summarized by a variance estimate ($\sigma_u^2$).
Partial pooling is determined by the number of repeated observations and the level 1 and level 2 variances (essentially how much of the total variation is at the person level). So if person A had outcome data on two occasions and person B had outcome data on 5 occasions, the $u_{0j}$ prediction for person A is going to be pulled back toward $\gamma_{00}$ moreso than the prediction for person B.
Translating these ideas back and forth between the multilevel (groups as clusters) and the longitudinal (persons as clusters) cases takes time and effort, but is a critical part of fully understanding mixed effects models. If something still isn't clear, please post a comment.
