linear mixed effect model, can I have fixed effect and random effect from a same source?

Question

Say I sampled 50 schools and take 20 students from each school. I randomly give 10 in each school concentration enhancement treatment.

For each school, I also have the school's teacher-student ratio.

Now I want to model the students' performance,

For school $i$ and student $ij$, $S_{ij}$ is the student's score, $E_{ij}$ is indicator of the treatment, and $T_i$ is the teacher-student ratio. I want to set up the model like this $$S_{ij} = \mu+\alpha T_i + \beta E_{ij} + U_i + \epsilon_{ij}$$ to mean that schools have a random effect and that effect is partially explained by teacher-student ratio. $\alpha$ and $\beta$ are parameters of interest.

Is this a valid model? I haven't seen a mixed effect model with explanatory variable and random effect explaining variance from a same source.

The problem I am worried is that, if $U_i$ were to be included as a fixed categorical variable, then the model is not determined, $\alpha$ can be anything and $U_i$ alone will fit for all between school variance.

Answer 1

Yeah that would be valid. All that would happen is that the random effect for school would represent all the variance between schools that can't be accounted for by knowing the teacher-student ratio of the schools.

If you compare the model with and without the teacher-student ratio variable then I'd expect you'd find that the model with that fixed effect is a better fit to the data.

Models like this are not uncommon in psychology, where, say, you want know whether people find one task more difficult than another, and you expect that the difficulty a person experiences in both tasks is partially influenced by some individual difference (e.g., IQ).

Answer 2

Yes, that is true. you mean students' performance, after treatment not only depends on treatment but also it may have better effects with some teachers.