Students Nested Within Classes and Schools

Why distinguish classes from schools?

Students in the same class share a teacher and classroom environment; students in different classes at the same school may still share school policies and resources. A model that represents only one of these groupings can miss an important part of the dependence in the data. A three-level nested model assigns a separate random intercept to each school and to each class within its school, while the students are the level-1 observational units.

The nesting is substantive: each class belongs to one school, and each student belongs to one class. The model therefore distinguishes variation between schools from additional variation between classes within schools. It does not automatically assign a random intercept to every student. For a single response per student, the student is the observational unit; in a Gaussian model the level-1 residual accounts for remaining variation, while in a binary model the conditional response distribution accounts for it.

Model specification and assumptions

For an outcome from an exponential-family distribution, write \(\mu_{ijk}=E(Y_{ijk}\mid u_k,v_{jk},x_{ijk})\). A basic three-level generalized linear mixed model is

Here \(g\) is the link function, \(\beta_0\) and \(\beta_1\) are fixed effects, and \(x_{ijk}\) is a predictor whose value may differ among students. The school effect shifts the linear predictor for the whole school; the class effect adds a further shift for one class within that school. The model commonly assumes that school and class effects are mutually independent, that the class effects are independent across classes, and that the random effects are independent of the predictors. Conditional on the random effects and predictors, responses are assumed independent and to follow the specified response distribution.

For a binary response, a common choice is the Bernoulli distribution with a logit link. For a count response, a Poisson distribution with a log link is one possible choice. For a continuous response, the Gaussian identity-link model is a special case; it usually includes a student-level residual, such as \(\epsilon_{ijk}\sim N(0,\sigma^2)\), independent of the random effects. The response distribution and link should be chosen for the outcome and checked against the data, not inferred from the hierarchy alone.

The random-effects formulation treats the observed schools and classes as realizations from populations of schools and classes represented by the model. If the scientific target is only a fixed, specifically enumerated set of schools, treating school effects as fixed may instead be appropriate. The design must also genuinely support nesting. If a class crosses school boundaries, or a student belongs to multiple classes in the relevant data structure, this simple nested model does not represent the memberships correctly.

Interpreting the effects on the link scale

A random intercept is an additive shift on the linear-predictor scale, not necessarily on the response scale. With an identity link, a one-unit shift in a random intercept changes the conditional mean by one response unit. With a logit link, it changes the log odds by one unit; the corresponding odds are multiplied by the exponential of that shift. With a log link, it changes the log mean and therefore multiplies the conditional mean by the exponential of the shift.

The fixed intercept \(\beta_0\) is the linear predictor for \(x=0\) when both random effects equal zero. It is not generally the overall population-average response. In nonlinear-link models, applying the inverse link to a predictor with random effects set to zero does not usually give the average response after integrating over the random-effects distributions.

Why observations share covariance

The hierarchy induces dependence because two students in the same class share both random intercepts, while students in different classes of the same school share only the school intercept. The following calculation makes the unit represented by each effect explicit. In a Gaussian identity-link model with no predictors for simplicity, write \(Y_{ijk}=\beta_0+u_k+v_{jk}+\epsilon_{ijk}\), with independent mean-zero effects and residuals, and variances \(\tau^2_{\text{school}},\tau^2_{\text{class}},\sigma^2\).

For distinct students \(a\) and \(b\), conditional residuals are independent. In the covariance, only random effects shared by both responses contribute: the covariance of a random effect with itself is its variance, while independent effects have covariance zero. Therefore,

For one response, the variance is \(\tau^2_{\text{school}}+\tau^2_{\text{class}}+\sigma^2\), because all three independent components contribute. This pattern is a consequence of the specified random-intercept structure; it is not a general guarantee that all students within a school are equally alike in every respect.

Model checks, alternatives, and limitations

Before fitting the model, verify the data structure: each student should map to one class and each class to one school for this formulation. Check that class identifiers are unique within schools, or create a combined school-class identifier. Otherwise, a class label reused in two schools could mistakenly be treated as one shared class effect. Also check whether the number of schools and classes provides meaningful information for estimating their respective variance components. Many students in only a few schools do not replace having information across more schools.

After fitting, inspect convergence warnings, estimated variance components, and whether the model's conditional response distribution and link are plausible for the outcome. A variance estimate near zero can indicate little detectable variation at that level, but it can also arise when there is limited information to estimate that component. A very small number of groups, sparse binary outcomes, or separation can make estimates unstable. Residual and predictive checks should be conducted at relevant levels, since a model that fits student-level patterns poorly may also misrepresent group-level patterns.

The main alternatives answer different questions. A model with school effects but no class effects assumes there is no additional class-level variation after accounting for school and predictors. A model with only class effects does not represent the fact that distinct classes from one school may share a school-level shift. Treating every school and class as fixed effects can be useful when interest is restricted to the observed groups, but it does not provide the same population distribution of group effects. An ordinary GLM without group effects ignores the dependence represented here and may give misleading uncertainty estimates. These approaches are not interchangeable merely because they use the same response and predictors.

The random-intercept model also has limits. It assumes a common school shift across classes and a common class shift across students, on the link scale. It does not allow a predictor's association to vary across schools or classes; that requires a different random-effects structure. The independence assumptions may also be inappropriate if group effects relate systematically to predictors. A good fit to the nesting labels alone does not establish that these assumptions hold.