Data Analysis Using Regression and Multilevel/Hierarchical Models

What is multilevel regression modeling?

Consider an educational study with data from students in many schools, predicting in each school the students' grades y on a standardized test given their scores on a pre-test x and other information. A separate regression model can be fit within each school, and the parameters from these schools can themselves be modeled as depending on school characteristics (such as the socioeconomic status of the school's neighborhood, whether the school is public or private, and so on). The student-level regression and the school-level regression here are the two levels of a multilevel model.

In this example, a multilevel model can be expressed in (at least) three equivalent ways as a student-level regression:

• A model in which the coefficients vary by school (thus, instead of a model such as y = α + βx + error, we have y = αj + βjx + error, where the subscripts j index schools),

• A model with more than one variance component (student-level and school-level variation),

• A regression with many predictors, including an indicator variable for each school in the data.

More generally, we consider a multilevel model to be a regression (a linear or generalized linear model) in which the parameters—the regression coefficients—are given a probability model. This second-level model has parameters of its own—the hyperparameters of the model—which are also estimated from data.

The two key parts of a multilevel model are varying coefficients, and a model for those varying coefficients (which can itself include group-level predictors).