Causal inference using regression has an inherent multilevel structure—the data give comparisons between units, but the desired causal inferences are within units. Experimental designs such as pairing and blocking assign different treatments to different units within a group. Observational analyses such as pairing or panel study attempt to capture groups of similar observations with variation in treatment assignment within groups.

Multilevel aspects of data collection

Hierarchical analysis of a paired design

Section 9.3 describes an experiment applied to school classrooms with a paired design: within each grade, two classes were chosen within each of several schools, and each pair was randomized, with the treatment assigned to one class and the control assigned to the other. The appropriate analysis then controls for grade and pair.

Including pair indicators in the Electric Company experiment. As in Section 9.3, we perform a separate analysis for each grade, which could be thought of as a model including interactions of treatment with grade indicators. Within any grade, let n be the number of classes (recall that the treatment and measurements are at the classroom, not the student, level) and J be the number of pairs, which is n/2 in this case. (We use the general notation n, J rather than simply “hard-coding” J = n/2 so that our analysis can also be used for more general randomized block designs with arbitrary numbers of units within each block.)