How do I determine if random slopes are justified in a model? In particular, how would I determine whether to only use random intercepts, or to use both random intercepts and slopes?
 A: (I'm no expert here; I'm submitting this to try out an idea.)
Imagine a large school district conducting an experiment on the effect of sugar on children's behavior.  A randomly chosen 10 schools make Halloween Candy Corn freely available during the school day, while another set of schools doesn't.  The principal's office in each school records the number of incidents during the week that prompt a teacher to send a student to the office for discipline.
It's reasonable to expect that different schools will have different usual levels of disciplinary incidents, making it sensible to incorporate for each school a random intercept.  But is it reasonable to expect that each school will show a different relationship between sugar intake and incidents?  Probably not.  For example, one might need to imagine that students would be differently susceptible to any behavioral effect from sugar, and that such differences would occur systematically, school by school. But if it were reasonable, one would want to build in provision for random slopes as well as random intercepts.
