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

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.