I have a question about interpreting changes in coefficient values across logistic multilevel models. I would like to see if an apparent contextual effect (level 2 coefficient) is explained by compositional factors (level 1 coefficients). My planned modelling strategy is to set up a model with level 2 fixed effects and then introduce level 1 fixed effects to see if the value of the level 2 coefficient diminishes. However, my rather limited understanding is that with a logistic multilevel model the introduction of new independent variables leads to changes in the coefficient values through not only changes in the model, but also rescaling.

My question is whether it is indeed possible to make any interpretation of changes in coefficient values in Logistic Multilevel Models when new coefficients are introduced to a model?