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Hierarchical Causal Models
[Submitted on 10 Jan 2024 (v1), last revised 10 Jul 2026 (this v · 2024-01-11 · via stat updates on arXiv.org

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Abstract:Causal questions often arise in settings where data are hierarchical: subunits are nested within units. Consider students in schools, cells in patients, or cities in states. In these settings, unit-level variables (e.g., a school's budget) may affect subunit-level outcomes (e.g., student test scores), and subunit-level characteristics may aggregate to influence unit-level outcomes. In this paper, we show how to analyze hierarchical data for causal inference. We introduce hierarchical causal models, which extend structural causal models and graphical models by incorporating inner plates to represent nested data structures. We develop a graphical identification technique for these models that generalizes do-calculus. We show that hierarchical data can enable causal identification even when it would be impossible with non-hierarchical data--for example, when only unit-level summaries are available. We develop estimation strategies, including using hierarchical Bayesian models. We illustrate our results in simulation and through a reanalysis of the classic "eight schools" study.

Submission history

From: Eli Weinstein [view email]
[v1] Wed, 10 Jan 2024 18:56:44 UTC (133 KB)
[v2] Wed, 26 Jun 2024 16:56:12 UTC (130 KB)
[v3] Fri, 10 Jul 2026 07:46:08 UTC (164 KB)