I am estimating a multilevel logistic regression with group predictors, but am unclear about some of the advice given by Gelman and Hill (2007) in their book. Therein, they recommend allowing every coefficient to possibly vary, given a large enough N. Does that include group predictors as well? They weren't clear, treating "varying slope" as just another complexity you can incorporate into a mixed effects model in lme4 along with group predictors (see: p. 549 in their book).
For example, I have roughly 50,000 observations with a binary response (plenty large N). Predictors exist at two levels, such that my model looks like:
M1 <- lmer(Y ~ X1 + X2 + X3 + X4 + G1 + G2 + G3 + G4 + (1 | group), family=binomial(link="logit"))
X1:X4 are individual-level predictors and G1:G4 are group-level predictors, thus: a multilevel model. Does their recommendation of treating all coefficients as potentially variable mean including even the group predictors within the random effect, such that:
M2 <- lmer(Y ~ X1 + X2 + X3 + X4 + G1 + G2 + G3 + G4 + (1 + X1 + X2 + X3 + X4 + G1 + G2 + G3 + G4 | group), family=binomial(link="logit"))
I ran M2 and it gave sensible estimates. AIC/BIC suggest much better fit than M1. I'm just unsure if it's appropriate since, unlike individual-level predictors, the group-level predictors are not going to vary in a given group. It will obviously vary across groups, though.
Further, if this is not an incorrect way to approach it, how suspicious should I be if one of the group predictors of interest is statistically insignificant as a stand-alone fixed effect (varying intercept model like M1), but is significant as a fixed effect in a varying slope model like M2?
Thanks for any input and feedback on this topic. I really appreciate it.