Data Analysis Using Regression and Multilevel/Hierarchical Models

Multilevel modeling can be thought of in two equivalent ways:

• We can think of a generalization of linear regression, where intercepts, and possibly slopes, are allowed to vary by group. For example, starting with a regression model with one predictor, yi = α + βxi + ∈i, we can generalize to the varyingintercept model, yi = αj[i] + βxi + ∈i, and the varying-intercept, varying-slope model, yi = αj[i] + βj[i]xi + ∈i (see Figure 11.1 on page 238).

• Equivalently, we can think of multilevel modeling as a regression that includes a categorical input variable representing group membership. From this perspective, the group index is a factor with J levels, corresponding to J predictors in the regression model (or 2J if they are interacted with a predictor x in a varying-intercept, varying-slope model; or 3J if they are interacted with two predictors X(1), X(2); and so forth).

In either case, J−1 linear predictors are added to the model (or, to put it another way, the constant term in the regression is replaced by J separate intercept terms). The crucial multilevel modeling step is that these J coefficients are then themselves given a model (most simply, a common distribution for the J parameters αj or, more generally, a regression model for the αj's given group-level predictors). The group-level model is estimated simultaneously with the data-level regression of y.