Confused about meaning of subject-specific coefficients in a binomial generalised mixed-effects model In *A Comparison of Cluster-Specific and Population-Averaged Approaches for Analyzing Correlated Binary Data*, Neuhas, Kalbfleisch, and Hauck state:
"With the cluster-specific approach, the probability distribution of $Y_{ij}$ is modelled as a function of the covariates $X_{ij}$ and parameters $\alpha_{i}$ specific to the $i$th cluster."
I am having trouble getting an intuitive sense of what this means when coefficients are expressed as a single coefficient in a regression output.
For example in the analysis referred to in this post, where I tested the effect of week in treatment (measured at 4 time points per individual, 4, 8, 12, and 24 weeks) and experimental group (two levels: placebo vs active) on the odds of people guessing that they had been allocated to the active group, specified in a binomial generalised linear mixed effects model in the lme4 package in R like so:
glmer(guess ~ group * week + (1 | id), 
      data = w24, family = binomial())

The clusters in this model are participant id. The coefficients for the fixed effects were
Fixed Effects:
    (Intercept)           group2         weekFac2         weekFac3  
        10.2474           5.0411           2.8542          -1.8699  
       weekFac4  group2:weekFac2  group2:weekFac3  group2:weekFac4  
         0.7396           7.8657           0.8067           9.5187

I just fundamentally don't understand how you can get a single estimate that is "specific to the $i$th cluster", when there are multiple clusters/participants.
 A: I agree that this can be a little confusing. Some authors avoid setting it up in this way. The important point is that the $\alpha_{i}$ are not estimated individually, instead they are subsumed into a general model and the usual assumption is that they are normally distributed, with an unknown variance, which is to be estimated.
Focusing on the main point:

parameters $\alpha_{i}$ specific to the $i$th cluster

and translating this to something a bit more usual:
$$ y_i = X_i \beta + Z_i b_i + \epsilon_i, \text{ }\text{ }\text{ }\text{ }    i=1,...,N $$
where $b_i$ is a vector of random effects and $Z_i$ is the design matrix for the $i$th cluster, we then combine vectors $y_i$ and matrices $X_i$ into the $\Sigma n_i \times 1$ vector $y$ and $\Sigma n_i \times m$ matrix $X$, and letting $Z = \text{diag}(Z_1,...,Z_N)$ the model can be written as
$$ y = X \beta + Z b + \epsilon$$
which is the usual mixed model equation.
A: The point that is made in this paper is with regard to the conditional versus marginal interpretation of the regression coefficients. Namely, because of the nonlinear link function used in the mixed effects logistic regression, the fixed effects coefficients have an interpretation conditional on the random effects. Most often this is not the desirable interpretation that relates to groups of individuals. You may find more information regarding this issue here and here.
On the contrary, in linear mixed models and because the link function is the identity, you do not have this problem.
