I have a logistic mixed model where a random effects term I am thinking about including is totally arbitrarily related (in a non-informative way) to the response variable. I would not include it as a predictor or a fixed effects term, because it would do a great job of predicting the response variable and not tell me anything about the system. However, I want to include it as random effects term because it may help account for variability. It's like using zip code to predict epidemic location across cities when you already know the epidemic only occurs in certain cities, but zip code helps control for a number of factors influencing epidemic. When I include this term as a random intercept term (1|x in lmer in R) my prediction accuracy jumps 15% and the fixed terms all lose significance. Is the correlation between the response and random terms helping the model perform better?
Made up example code (pretend district is subset of zip code):
zip<-c(54403,54403,54403,80404,80404,80405,93513,93513,93514,30411,30412, 30413,60414,60415,601415) city_has_epi<-c(yes,yes,no,no,no,no,yes,yes,yes,no,no,yes) avg_income<-c(50000,40000,35000,80000,90000,400000,50000,40000,35000, 80000,90000,400000) hospitals_no<-c(23,46,66,29,44,54,23,46,66,29,44,54) district<-c(1,2,3,4,5,6,7,8,9,10,11,12) model<-glmer(city_has_epi~ avg_income + hospitals_no +(1|zip/district), family=binomial(link="logit"), data=mtr, control=glmerControl(optimizer="bobyqa"))