Hierarchical Causal Models

Abstract

Scientists often want to learn about cause and effect from hierarchical data, collected from subunits nested inside units. Consider students in schools, cells in patients, or cities in states. In such settings, unit-level variables (e.g. each school's budget) may affect subunit-level variables (e.g. the test scores of each student in each school) and vice versa. To address causal questions with hierarchical data, we propose hierarchical causal models, which extend structural causal models and causal graphical models by adding inner plates. We develop a general graphical identification technique for hierarchical causal models that extends do-calculus. We find many situations in which hierarchical data can enable causal identification even when it would be impossible with non-hierarchical data, that is, if we had only unit-level summaries of subunit-level variables (e.g. the school's average test score, rather than each student's score). We develop estimation techniques for hierarchical causal models, using methods including hierarchical Bayesian models. We illustrate our results in simulation and via a reanalysis of the classic "eight schools" study.