Simpson Reversal as Exact Latent-Stratum Nonidentifiability and Recovery — Sharp Identification Frontiers in Binary Mixture Models
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| Natura: | Recurso digital |
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2026
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>This paper develops an exact causal, extremal, and decision-theoretic frontier for hidden-subgroup effect recovery in binary latent-stratum models under a potential-outcome framework. The setting is binary treatment, binary outcome, latent subgroup, and potential outcomes satisfying consistency and latent ignorability, so the target is the hidden-subgroup average treatment effect profile rather than an observational contrast.</p> <p>The impossibility theorem is proved in its strongest causal form: every interior observable 2x2 law of treatment and outcome admits one latent causal decomposition with all subgroup average treatment effects positive and another with all negative. The scale-robust corollary shows that no aggregate-only statistic — risk difference, risk ratio, odds ratio, logit coefficient, or any other collapse of the observable table — can uniformly certify the uniform sign of the hidden-subgroup causal effects. A minimax lower bound for n-sample sign testing is proved: any randomized test based only on observable treatment-outcome data has worst-case type I plus type II error summing to at least one, with a paired hypothesis pair attaining average error exactly one-half regardless of sample size.</p> <p>For an arbitrary latent variable, an exact causal decomposition of the marginal risk difference is derived, separating the treated-mixture average causal effect from a confounding-capacity term equal to the covariance between treatment propensity and baseline potential-outcome risk divided by the marginal treatment variance. Under oscillation bounds on treatment propensity spread and baseline risk spread, a sharp confounding radius is proved, and the constant one-quarter in the denominator is shown to be attained by an explicit two-point binary extremizer on the central interior strip. For the binary two-group common-effect model, the exact identified interval on the central strip is established, and exact boundary-truncation formulas are derived on the budget-saturating branch where positivity and outcome boundary constraints begin to bind. The full heterogeneous identified set in the plane of subgroup effects is characterized as a union of explicit parallelograms, with a two-variable semialgebraic kernel and cylindrical algebraic decomposition completing the elimination to a direct semialgebraic description.</p> <p>The paper then proves an extremal finite-support reduction theorem for the general latent-state proximal setting: 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. This converts the infinite-dimensional proximal problem into an exact finite-support extremal search. For one binary proxy with a known invertible channel matrix and proxy conditional independence, exact identification of all latent cell vectors and hence all subgroup causal effects is proved; under a singular known channel, identification fails and an explicit converse is given for the unknown-channel case. Under epsilon-level proxy misspecification or channel calibration error, recovery error scales at rate kappa(M) times epsilon, where kappa(M) is the condition number of the channel matrix.</p> <p>Finally, the paper formalizes the information hierarchy via Blackwell experiment comparison. The augmented experiment with a binary proxy strictly Blackwell-dominates the aggregate experiment for the latent-sign decision problem: on a paired least-favorable hypothesis pair, every aggregate-only metric is equivalent to the null experiment, while a single calibrated noisy proxy bit yields strictly smaller Bayes risk.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19376121 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Simpson Reversal as Exact Latent-Stratum Nonidentifiability and Recovery — Sharp Identification Frontiers in Binary Mixture Models 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 causal, extremal, and decision-theoretic frontier for hidden-subgroup effect recovery in binary latent-stratum models under a potential-outcome framework. The setting is binary treatment, binary outcome, latent subgroup, and potential outcomes satisfying consistency and latent ignorability, so the target is the hidden-subgroup average treatment effect profile rather than an observational contrast.</p> <p>The impossibility theorem is proved in its strongest causal form: every interior observable 2x2 law of treatment and outcome admits one latent causal decomposition with all subgroup average treatment effects positive and another with all negative. The scale-robust corollary shows that no aggregate-only statistic — risk difference, risk ratio, odds ratio, logit coefficient, or any other collapse of the observable table — can uniformly certify the uniform sign of the hidden-subgroup causal effects. A minimax lower bound for n-sample sign testing is proved: any randomized test based only on observable treatment-outcome data has worst-case type I plus type II error summing to at least one, with a paired hypothesis pair attaining average error exactly one-half regardless of sample size.</p> <p>For an arbitrary latent variable, an exact causal decomposition of the marginal risk difference is derived, separating the treated-mixture average causal effect from a confounding-capacity term equal to the covariance between treatment propensity and baseline potential-outcome risk divided by the marginal treatment variance. Under oscillation bounds on treatment propensity spread and baseline risk spread, a sharp confounding radius is proved, and the constant one-quarter in the denominator is shown to be attained by an explicit two-point binary extremizer on the central interior strip. For the binary two-group common-effect model, the exact identified interval on the central strip is established, and exact boundary-truncation formulas are derived on the budget-saturating branch where positivity and outcome boundary constraints begin to bind. The full heterogeneous identified set in the plane of subgroup effects is characterized as a union of explicit parallelograms, with a two-variable semialgebraic kernel and cylindrical algebraic decomposition completing the elimination to a direct semialgebraic description.</p> <p>The paper then proves an extremal finite-support reduction theorem for the general latent-state proximal setting: 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. This converts the infinite-dimensional proximal problem into an exact finite-support extremal search. For one binary proxy with a known invertible channel matrix and proxy conditional independence, exact identification of all latent cell vectors and hence all subgroup causal effects is proved; under a singular known channel, identification fails and an explicit converse is given for the unknown-channel case. Under epsilon-level proxy misspecification or channel calibration error, recovery error scales at rate kappa(M) times epsilon, where kappa(M) is the condition number of the channel matrix.</p> <p>Finally, the paper formalizes the information hierarchy via Blackwell experiment comparison. The augmented experiment with a binary proxy strictly Blackwell-dominates the aggregate experiment for the latent-sign decision problem: on a paired least-favorable hypothesis pair, every aggregate-only metric is equivalent to the null experiment, while a single calibrated noisy proxy bit yields strictly smaller Bayes risk.</p> |
| title | Simpson Reversal as Exact Latent-Stratum Nonidentifiability and Recovery — Sharp Identification Frontiers in Binary Mixture Models |
| 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.19376121 |