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.