Mixed model random effects term is arbitrarily correlated to dependent variable, does it bias model? 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"))

 A: The problem here is not that your random effect and your response variable are somewhat correlated, its that they are perfectly correlated. The model estimates a baseline level of the dependent variable for each random effect in the model. So, here the levels of your random effect can explain ALL of the variation in your response variable. 
Generally, when using something like zip code as a random effect in the context of disease, people are analyzing occurrence rates, given some baseline rate within that community modeled by the random effect, as a function of predictors that may influence that occurrence rate. Or, you might analyze the effect of income and water source on lead levels in blood, while using zip code of those people as a random effect, to account for the fact that there are other environmental factors that amy be geographically unique but unobserved that drive variation in baseline lead levels in blood. 
In this example, given these data, the incorporation of zip code as a random effect is inappropriate, as it cannot give you baseline information on this binary outcome. Your dependent variable and random effect are perfectly correlated, and this is why you are having trouble. 
