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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?

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  • $\begingroup$ i don't understand what is not clear to you, mind you elaborate? The average of each individual is modeled as a random intercept. These intercepts are partially pooled together so that each intercept should be estimated more precisely since we take into an account that all subjects are a random sample from the same distribution $\endgroup$
    – rep_ho
    Sep 25, 2020 at 11:06

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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.

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    $\begingroup$ Thanks for laying this out so clearly - its really helpful. The equation as expressed is much more intuitive to me than most I've seen. My question is just: how would things change if we add a 'time' variable in (i.e. estimate β1j)? γ00 would then come to indicate the mean value of Y at time = 0, correct? And u0j the variation about that mean. If this is correct and the intercept term is now limited to time = 0, what becomes of the partial pooling? All participants only have one observation at time 0. I really want to get my head around this, i've struggled a good while! Cheers $\endgroup$
    – Lachlan
    Sep 25, 2020 at 6:11
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    $\begingroup$ @Lachlan, good questions! $\gamma_{00}$ will be conditional time==0 when you add time to the model. However, what we are predicting in these models, and what gets pooled is $u_{0j}$, which represents, for each person, how much their scores at any time point deviate off the predicted mean value for the sample. This deviation is constant across measurement occasions, and thus uses all observations about an individual in pooling (or not). $\endgroup$
    – Erik Ruzek
    Sep 25, 2020 at 19:20
  • $\begingroup$ @Lachlan , hey dude, could you please have a look at this question stats.stackexchange.com/questions/601314/… . It is similar to this one, which you have already answered before stats.stackexchange.com/questions/583916/… . I need once again your enthusiasm about the stuff you are an expert on) $\endgroup$
    – Rustam
    Jan 10, 2023 at 5:30

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