# What does the change in intercept mean if you allow for clustering in your data?

When I estimate a logistic binomial regression with an allowed to vary effect on a second and a third level, the overall intercept represents the average log-odds for any level one unit in any higher level unit:

$\text{logit}(\pi_{ijk}) = \beta_0 + \mu_{jk} + \mu_{k}$

Where $\beta_0$ is the fixed intercept and $\mu_{jk}$ and $\mu_{k}$ are the random cluster effects.

$\beta_0$ is estimated as -3.525 in this case (or 2.9% probability of success: $\pi_{ijk} = \frac{e^{\beta_{0}}}{1+e^{\beta_{0}}}$).

However, the model where no random effects are allowed (or the general mean):

$\text{logit}(\pi_{ijk}) = \beta_0$

gives -2.932 log-odds or 5% probability of success.

Does the decrease of overall intercept mean there were more clusters with low probability than with high probability?

PS: I also switch from MQL to PQL when going to the random intercept model (since PQL is not possible for the fixed effect only model). It could have to do with that too.