Introduction to Categorical Data Analysis

314 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
to have more-alike observations than students from different schools. This could be
because students within a school tend to be similar on various socioeconomic indices.
Hierarchical models contain terms for the different levels of units. For the example
just mentioned, the model would contain terms for the student and for the school.
Level 1 refers to measurements at the student level, and level 2 refers to measurements
at the school level. GLMMs having a hierarchical structure of this sort are called
multilevel models.
Multilevel models usually have a large number of terms. To limit the number of
parameters, the model treats terms for the units as random effects rather than fixed
effects. The random effects can enter the model at each level of the hierarchy. For
example, random effects for students and random effects for schools refer to different
levels of the model. Level 1 random effects can account for variability among students
in student-specific characteristics not measured by the explanatory variables. These
might include student ability and parents’ socioeconomic status. The level 2 random
effects account for variability among schools due to school-specific characteristics
not measured by the explanatory variables. These might include the quality of the
teaching staff, the teachers’ average salary, the degree of drug-related problems in the
school, and characteristics of the district for which the school enrolls students.
10.4.1 Example: Two-Level Model for Student Advancement
An educational study analyzes factors that affect student advancement in school from
one grade to the next. For student t in school i, the response variable yit measures
whether the student passes to the next grade (yit = 1) or fails. We will consider a
model having two levels, one for students and one for schools. When there are many
schools and we can regard them as approximately a random sample of schools that
such a study could consider, we use random effects for the schools.
Let {xit1,...,xitk} denote the values of k explanatory variables that have values
that vary at the student level. For example, for student t in school i, perhaps xit1
measures the student’s performance on an achievement test, xit2 is gender, xit3 is race,
and xit4 is whether he or she previously failed any grades. The level-one model is
logit[P(yit = 1)] =αi + β1xit1 + β2xit2 +···+βkxitk
The level-two model provides a linear predictor for the level-two (i.e., school-level)
term in the level-one (i.e., student-level) model. That level-two term is the intercept,
αi. The level-two model has the form
αi = ui + α + γ1wi1 + γ2wi2 +···+γtwiℓ
Here, {wi1,...,wiℓ} are ℓ explanatory variables that have values that vary only at the
school level, so they do not have a t subscript. For example, perhaps wi1 is per-student
expenditure of school i. The term ui is the random effect for school i.