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I have a question about interpreting changes in coefficient values between 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?

Hi Shane,
Interesting question. I guess one could look at the correlation between the various predictors prior to putting in the model as if you are trying to explain the contextual effect as being in reality compositional factors then there would be strong correlations. Even in a normal model If you have a complicated model with lots of predictors then adding further predictors is likely to move coefficients around unless the predictors are all orthogonal as the least squares algorithm essentially tries to find the best direction of the regression in the multi-dimensional space. I would be saying things like the inclusion of predictor x2 in the model reduced the value of the coefficient for predictor x1 as this would now be a predicted effect conditional on x2 being in the model however 'explained by' suggests causality which is a bit of a leap. You are right though that the logistic multilevel model is even harder to discuss due to the non-linear link function and binomial assumption of the response resulting in coefficients being affected by in a sense rescaling. Not much one can do about this.
Bill.