HIDDEN-SUBGROUP CAUSAL EFFECTS: EXACT SIGN NONIDENTIFICATION AND RECOVERY WITH CALIBRATED PROXIES
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2026
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| _version_ | 1866902074085605376 |
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>This paper develops an exact sign-nonidentification and recovery frontier for hidden-subgroup causal effects under binary treatment and binary outcome, with explicit extensions to bounded outcomes, finite latent-class models, and multi-valued treatments. The target throughout is the hidden-subgroup average treatment effect profile under consistency and latent ignorability.</p> <p>The first result is an exact impossibility theorem: every interior observable 2x2 law of treatment and outcome is compatible with one latent decomposition in which all subgroup average treatment effects are positive and another in which all are negative. No statistic of the aggregate table can repair that failure, and every n-sample test based only on the observable data has minimax error at least one-half on the interior class. The theorem extends immediately to finite-K latent-class models by a cloning argument, and to each arm of a multi-valued treatment by an exact binary arm-versus-control reduction.</p> <p>The second result is an exact causal decomposition of the observed treatment-control contrast: the marginal contrast equals a treated-mixture weighted average causal effect plus a confounding-capacity term equal to the covariance between treatment propensity and baseline potential-outcome risk divided by the marginal treatment variance. Under oscillation budgets on propensity spread and baseline risk spread, the confounding term is bounded by a sharp radius equal to the product of those budgets divided by four times the marginal treatment variance, with the constant one-quarter proved sharp by an explicit two-point extremizer. The decomposition and radius extend verbatim to bounded outcomes and to each arm of a multi-treatment problem. Under a common subgroup effect the identified set is exactly the closed interval centered at the observed contrast with radius equal to this confounding-capacity constant, on the central interior strip, and a separate citable theorem gives the resulting three-regime sign classification: certified positive, unresolved, or certified negative.</p> <p>The extremal theory is closed at the distributional level. A finite-support extremal representation theorem shows that for any finite family of functionals of the distribution of the causal effect profile, the sharp identified set is attained by finite-support latent laws via Caratheodory's theorem applied to the continuous image of the compact parameter space. A Prokhorov compactness result then shows that every feasible law of the causal effect profile is a weak limit of finite-support extremizers, and the full distributional identified set is compact in the Prokhorov metric.</p> <p>The heterogeneous two-parameter identified set in the plane of subgroup contrasts is characterized as a union of explicit affine pieces: parallelograms when the posterior treatment-control gap is nonzero, and line segments when it is zero, indexed by the admissible posterior weight region. The auxiliary coordinates are eliminated in two stages via a two-variable semialgebraic kernel, and cylindrical algebraic decomposition via the Tarski-Seidenberg theorem yields the exact identified set as a semialgebraic subset of the plane. The full finite-K heterogeneous identified set is also shown to be semialgebraic by the same argument.</p> <p>The third result specializes the Miao-Geng-Tchetgen Tchetgen rank condition to the finite-K binary model: in a finite-K latent-class model with one observed proxy and a known channel matrix, the latent cell vectors are exactly identified if and only if the channel matrix has full column rank, with the square invertible case being only the smallest special case. A strengthened converse is proved: without channel calibration, one proxy does not identify subgroup sign in general, and this failure persists even when each candidate unknown channel is itself full-rank and informative, proved by an explicit continuous interpolation between opposite-sign witnesses and a corresponding family of calibrated channels that induce the same joint observable law.</p> <p>The fourth result compares information structures. On a paired latent-sign decision problem, the proxy-resolved experiment strictly Blackwell-dominates the aggregate experiment because the aggregate law is identical under the two paired models while the proxy-resolved laws are distinct under any invertible channel. A worked binary example with a symmetric proxy channel demonstrates that a single calibrated noisy proxy bit can substantially reduce Bayes risk where the aggregate table is completely uninformative. In the strictly positive binary-proxy pair, a polynomially corrected lower bound using the method of types matches the Chernoff upper bound at the exponential scale, establishing the exact Chernoff exponent of the optimal Bayes risk. The sample-size requirement for a target Bayes risk threshold is given explicitly in terms of the Chernoff information. The proxy condition number is shown to control both the stability of channel inversion and the collapse of the testing separation: as the channel approaches singularity the Chernoff exponent vanishes and the required sample size explodes.</p> <p>A worked calibration to published summary statistics from a pretrial release study makes the frontier concrete, reporting the exact common-effect identified intervals under varying budget assumptions and the exact finite-sample separation quantities under a simulated calibrated risk screen proxy.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19420323 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | HIDDEN-SUBGROUP CAUSAL EFFECTS: EXACT SIGN NONIDENTIFICATION AND RECOVERY WITH CALIBRATED PROXIES Fathi, Kevin Simpson's paradox causal inference causal identification statistics Simpson reversal latent stratum model binary mixture model nonidentifiability partial identification identification frontier confounding capacity marginal risk difference within-group contrast subgroup effect treatment effect heterogeneity unmeasured confounding proxy variable channel matrix mixture model identifiability sign certification aggregation paradox contingency table binary treatment binary outcome epidemiology methods <p>This paper develops an exact sign-nonidentification and recovery frontier for hidden-subgroup causal effects under binary treatment and binary outcome, with explicit extensions to bounded outcomes, finite latent-class models, and multi-valued treatments. The target throughout is the hidden-subgroup average treatment effect profile under consistency and latent ignorability.</p> <p>The first result is an exact impossibility theorem: every interior observable 2x2 law of treatment and outcome is compatible with one latent decomposition in which all subgroup average treatment effects are positive and another in which all are negative. No statistic of the aggregate table can repair that failure, and every n-sample test based only on the observable data has minimax error at least one-half on the interior class. The theorem extends immediately to finite-K latent-class models by a cloning argument, and to each arm of a multi-valued treatment by an exact binary arm-versus-control reduction.</p> <p>The second result is an exact causal decomposition of the observed treatment-control contrast: the marginal contrast equals a treated-mixture weighted average causal effect plus a confounding-capacity term equal to the covariance between treatment propensity and baseline potential-outcome risk divided by the marginal treatment variance. Under oscillation budgets on propensity spread and baseline risk spread, the confounding term is bounded by a sharp radius equal to the product of those budgets divided by four times the marginal treatment variance, with the constant one-quarter proved sharp by an explicit two-point extremizer. The decomposition and radius extend verbatim to bounded outcomes and to each arm of a multi-treatment problem. Under a common subgroup effect the identified set is exactly the closed interval centered at the observed contrast with radius equal to this confounding-capacity constant, on the central interior strip, and a separate citable theorem gives the resulting three-regime sign classification: certified positive, unresolved, or certified negative.</p> <p>The extremal theory is closed at the distributional level. A finite-support extremal representation theorem shows that for any finite family of functionals of the distribution of the causal effect profile, the sharp identified set is attained by finite-support latent laws via Caratheodory's theorem applied to the continuous image of the compact parameter space. A Prokhorov compactness result then shows that every feasible law of the causal effect profile is a weak limit of finite-support extremizers, and the full distributional identified set is compact in the Prokhorov metric.</p> <p>The heterogeneous two-parameter identified set in the plane of subgroup contrasts is characterized as a union of explicit affine pieces: parallelograms when the posterior treatment-control gap is nonzero, and line segments when it is zero, indexed by the admissible posterior weight region. The auxiliary coordinates are eliminated in two stages via a two-variable semialgebraic kernel, and cylindrical algebraic decomposition via the Tarski-Seidenberg theorem yields the exact identified set as a semialgebraic subset of the plane. The full finite-K heterogeneous identified set is also shown to be semialgebraic by the same argument.</p> <p>The third result specializes the Miao-Geng-Tchetgen Tchetgen rank condition to the finite-K binary model: in a finite-K latent-class model with one observed proxy and a known channel matrix, the latent cell vectors are exactly identified if and only if the channel matrix has full column rank, with the square invertible case being only the smallest special case. A strengthened converse is proved: without channel calibration, one proxy does not identify subgroup sign in general, and this failure persists even when each candidate unknown channel is itself full-rank and informative, proved by an explicit continuous interpolation between opposite-sign witnesses and a corresponding family of calibrated channels that induce the same joint observable law.</p> <p>The fourth result compares information structures. On a paired latent-sign decision problem, the proxy-resolved experiment strictly Blackwell-dominates the aggregate experiment because the aggregate law is identical under the two paired models while the proxy-resolved laws are distinct under any invertible channel. A worked binary example with a symmetric proxy channel demonstrates that a single calibrated noisy proxy bit can substantially reduce Bayes risk where the aggregate table is completely uninformative. In the strictly positive binary-proxy pair, a polynomially corrected lower bound using the method of types matches the Chernoff upper bound at the exponential scale, establishing the exact Chernoff exponent of the optimal Bayes risk. The sample-size requirement for a target Bayes risk threshold is given explicitly in terms of the Chernoff information. The proxy condition number is shown to control both the stability of channel inversion and the collapse of the testing separation: as the channel approaches singularity the Chernoff exponent vanishes and the required sample size explodes.</p> <p>A worked calibration to published summary statistics from a pretrial release study makes the frontier concrete, reporting the exact common-effect identified intervals under varying budget assumptions and the exact finite-sample separation quantities under a simulated calibrated risk screen proxy.</p> |
| title | HIDDEN-SUBGROUP CAUSAL EFFECTS: EXACT SIGN NONIDENTIFICATION AND RECOVERY WITH CALIBRATED PROXIES |
| topic | Simpson's paradox causal inference causal identification statistics Simpson reversal latent stratum model binary mixture model nonidentifiability partial identification identification frontier confounding capacity marginal risk difference within-group contrast subgroup effect treatment effect heterogeneity unmeasured confounding proxy variable channel matrix mixture model identifiability sign certification aggregation paradox contingency table binary treatment binary outcome epidemiology methods |
| url | https://doi.org/10.5281/zenodo.19420323 |