Random intercept models

Random intercept models are the simplest form of random-effects models (aka multilevel models or partial-pooling models). They allow you to say that each group has a different average outcome, but that the relationship between the outcome and the covariates is invariant across groups.

Multilevel (hierarchical) models also provide a framework for interpreting the change in an outcome variable as it relates to covariates at the level of both the individual observation and of the group. We will look at both cases here, in relation to reading test scores among grade-one students.

We will use all of the grade-one data in the Tennessee STAR data set, which has over 6000 students in more than 300 classes.

Random intercept-only model

To begin, we will use the same random intercept model (with no covariates) as we used in the previous lab:

$$ \begin{aligned} S_{ik} &\sim \mathrm{Norm}(\mu_{ik},\sigma)\\ \mu_{ik} &= \alpha_k\\ \\ \alpha_k &\sim \mathrm{Norm}(\gamma,\eta)\\ \\ \sigma &\sim \mathrm{Cauchy}(0,50)\\ \gamma &\sim \mathrm{Norm}(500,100)\\ \eta &\sim \mathrm{Cauchy}(0,50) \end{aligned} $$

Even without predictor variables, this model shows us that there is a significant amount of between-class variation (as estimated in phi).

Including student-level covariates

We will start by seeing how students' age affects their test scores, taking into account inter-class variation.

Age has a negative relationship with test scores: or every increase of one standard deviation increase in age, the model predicts that student will score about five points lower.

Including class-level covariates

Very often, it is important to be able to look at the relationship between an outcome variable and a covariate that only varies at the group level. This is exactly the case in the STAR data if we are interested in the effect of the experimental treatment. Do smaller classes lead to better test scores? A model that estimates this effect, allowing between-class variation inn average scores, be written mathematically (omitting priors for now) as

$$ \begin{aligned} S_{ik} &\sim \mathrm{Norm}(\mu_{ik},\sigma)\\ \mu_{ik} &= \beta_{0k} + \beta_{1}Age_i\\ \\ \beta_{0k} &= \gamma_{0} + \gamma_1Small_k+ \eta_{0k}\\ \eta_{0k} &\sim \mathrm{Norm}(0, \phi_0) \end{aligned}$$

To determine how to specify this model in brms, note that we can substitute $\beta_{0k}$ into the individual-level linear model and rearrange, giving us

$$ \begin{aligned} S_{ik} &\sim \mathrm{Norm}(\mu_{ik},\sigma)\\ \mu_{ik} &= \beta_{0k} +\gamma_1Small_k+ \beta_{1}Age_i\\ \\ \beta_{0k} &\sim \mathrm{Norm}(\gamma_0, \phi_0) \end{aligned}$$

The classroom-level covariate can be specified as a student-level covariate in this case, so the changes to the formula are quite minimal: