What question did this study set out to answer?

To investigate hidden-subgroup causal inference when both treatment and outcome are binary, focusing on the role of latent variables.

April 4, 2026Open Access

Hidden-Subgroup Causal Effects, Simpson Reversal, and Proxy Recovery Frontiers

Key Points

To investigate hidden-subgroup causal inference when both treatment and outcome are binary, focusing on the role of latent variables.
Established an impossibility theorem for interior observable 2x2 laws of (X,Y).
Derived an exact causal decomposition for aggregate contrasts considering latent group structures.
Handled multi-valued treatment cases through binary reduction methodologies.
Performed a proximal extension to include proxies and assess selection distortion effects.
Utilized Blackwell comparison to formalize the information hierarchy in causal inference.
Identified that every observable 2x2 distribution has contrasting latent causal decompositions.
Quantified minimax error in aggregate tests related to nonidentified classes at a rate of at least 1/2.
Derived closed-form intervals for causal effects under various latent group conditions.
Showed that controlled proxy specification can precisely identify latent variables with bounded error rates.
Illustrated empirical applications with the Dobbie-Goldin-Yang pretrial release study.

Abstract

This paper studies hidden-subgroup causal inference under binary treatment and binary outcome when the stratifying variable is latent. Under consistency and latent ignorability, the paper establishes a sharp impossibility theorem: every interior observable 2x2 law of (X,Y) admits one latent causal decomposition with all subgroup average treatment effects positive and another with all subgroup effects negative. Hence the noiseless aggregate table does not identify even the uniform sign of hidden-subgroup causal effects, and every n-sample test based only on the aggregate has minimax error at least 1/2 on the nonidentified interior class. For arbitrary latent group structure, the paper derives an exact causal decomposition of the aggregate contrast into a treatment-weighted causal term and a confounding covariance term. Under oscillation budgets on treatment propensity and baseline risk, the confounding term is bounded by a sharp radius attained by a binary two-point extremizer on the central strip. For every finite number of latent groups sharing a common subgroup effect, the exact identified interval is derived in closed form, with explicit boundary-truncation formulas on the budget-saturating branch where positivity and outcome constraints bind. The multi-valued treatment case is handled by exact arm-versus-control binary reduction; the bounded-outcome case extends the population-level theory verbatim. A proximal extension with one observed proxy, bounded selection distortion, and controlled proxy misspecification establishes a finite-support extremal representation theorem: the sharp identified set for any finite family of effect functionals is attained by finite-support latent laws. In the known-channel setting, an invertible proxy channel identifies the latent cell vectors exactly. Under epsilon-level misspecification, recovery error scales at rate kappa(M) times epsilon. The paper closes by formalizing the information hierarchy via Blackwell comparison: the experiment based on (X,Y,Z) strictly Blackwell-dominates the aggregate (X,Y) experiment for the latent-sign decision problem, with an explicit exponential finite-sample separation rate governed by the Chernoff information between the proxy-resolved joint laws. A worked empirical illustration calibrates the identification frontier to the Dobbie-Goldin-Yang pretrial release study.

Hidden-Subgroup Causal Effects, Simpson Reversal, and Proxy Recovery Frontiers

Key Points

Abstract

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