[R-sig-ME] Random Slope for Dummy Variables

[Junyan Luo]

Dear All,
I have recently run into a puzzle with a problem which needs to
include random slopes for some dummy variables. I have searched the
archive list and although I found several threads related to random
slope dummies, they did not solve my problems. To make the question
simple, consider the following scenario:
Suppose we want to study the effects on students' Math performance
(DV) from both individual and school-level covariates. One of the
individual covariates is ethnicity, and to keep it simple, let's say a
dummy variable where 1 represents Asian and 0 other wise. Now the
simple random slope model can be right as:
1) Math ~ IV1 + IV2 +...+ Asian + (Asian | School)
where IVs are other covariates. The random slope is included because
we want to model the interaction between ethnicity (or "being Asian"
in this simplified case) and schools. By now everything seems to be
fine, until I discovered that the distribution of ethnic groups among
different schools was highly even. In other words, there are a lot
schools with very few Asian students, and for individuals from those
schools, the value on the dummy is ALWAYS 0. On the other hand, there
are a few schools where almost all students are Asians, and the dummy
becomes always 1. So my questions are:
A. First, is model 1) still valid in this case? Technically there's no
problem in running it in R, but is it violating any statistical rules?
B. If model 1) is OK, what the random slope of Asian means? A random
slope suggests that there will be different slope value for different
schools, but what does this mean for schools with no Asian students,
since the dummy term will always be 0?
C. If model 1) is NOT OK, what are the alternative solutions?
Thank you very much!

REGARDS,
J. Lo