Simpson Reversal as Exact Latent-Stratum Nonidentifiability and Recovery — Sharp Identification Frontiers in Binary Mixture Models
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| Sprache: | Englisch |
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2026
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| _version_ | 1866902074516570112 |
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>This paper develops a complete exact identification theory for subgroup effects in binary latent-stratum models. The setting is a binary treatment, binary outcome, and unobserved binary group indicator, where the observable data consists only of the marginal 2x2 table. The paper proves the nonidentifiability result in its strongest form: for every interior observable law with nonzero marginal association, the fiber of latent two-group decompositions contains both an all-positive and an all-negative subgroup-effect model, so no function of the observable table can determine the sign of subgroup contrasts.</p> <p>The paper then derives the exact algebraic decomposition of the marginal risk difference for an arbitrary finite latent stratum: the marginal contrast equals the treatment-weighted average of within-group contrasts plus a confounding-capacity term equal to the covariance between treatment propensity and baseline risk divided by the marginal treatment variance. For two groups this yields an explicit bound on the gap between the marginal association and the weighted average subgroup effect.</p> <p>Under a common subgroup effect assumption and primitive quantitative bounds on propensity spread and baseline risk spread, the exact identified set for the common subgroup effect is proved to be a closed interval centered at the marginal risk difference with radius equal to the confounding-capacity constant. The sign of the common subgroup effect is identified if and only if the marginal risk difference exceeds this radius.</p> <p>The paper then solves the fully heterogeneous two-parameter problem. Dropping the common-effect assumption, the exact identified set of the pair of subgroup contrasts in the two-dimensional plane is characterized as a union of explicit parallelograms, each indexed by the treated posterior weight and the posterior treatment-control gap, with the admissible posterior-gap region derived exactly from the propensity bound via a quadratic inequality. The auxiliary coordinates are then eliminated in two stages. First, an exact two-variable semialgebraic kernel in the treated posterior weight and the baseline risk difference is derived by algebraic substitution, reducing the four-dimensional parameterization to two variables. Second, cylindrical algebraic decomposition via the Tarski-Seidenberg theorem eliminates the remaining two auxiliary variables entirely, producing a direct semialgebraic description of the identified set in the plane of subgroup contrasts with no auxiliary coordinates. A scalar uniform sign certificate under a bounded effect-oscillation budget follows as a corollary.</p> <p>Finally, an exact proxy identifiability theorem is proved: if a binary proxy for the latent stratum is observed with a known channel matrix and is conditionally independent of treatment and outcome given the stratum, then the full latent parameter vector is exactly identified if and only if the proxy channel matrix is invertible, and identification fails precisely when the channel is singular. All results are proved in full.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19361706 |
| 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 a complete exact identification theory for subgroup effects in binary latent-stratum models. The setting is a binary treatment, binary outcome, and unobserved binary group indicator, where the observable data consists only of the marginal 2x2 table. The paper proves the nonidentifiability result in its strongest form: for every interior observable law with nonzero marginal association, the fiber of latent two-group decompositions contains both an all-positive and an all-negative subgroup-effect model, so no function of the observable table can determine the sign of subgroup contrasts.</p> <p>The paper then derives the exact algebraic decomposition of the marginal risk difference for an arbitrary finite latent stratum: the marginal contrast equals the treatment-weighted average of within-group contrasts plus a confounding-capacity term equal to the covariance between treatment propensity and baseline risk divided by the marginal treatment variance. For two groups this yields an explicit bound on the gap between the marginal association and the weighted average subgroup effect.</p> <p>Under a common subgroup effect assumption and primitive quantitative bounds on propensity spread and baseline risk spread, the exact identified set for the common subgroup effect is proved to be a closed interval centered at the marginal risk difference with radius equal to the confounding-capacity constant. The sign of the common subgroup effect is identified if and only if the marginal risk difference exceeds this radius.</p> <p>The paper then solves the fully heterogeneous two-parameter problem. Dropping the common-effect assumption, the exact identified set of the pair of subgroup contrasts in the two-dimensional plane is characterized as a union of explicit parallelograms, each indexed by the treated posterior weight and the posterior treatment-control gap, with the admissible posterior-gap region derived exactly from the propensity bound via a quadratic inequality. The auxiliary coordinates are then eliminated in two stages. First, an exact two-variable semialgebraic kernel in the treated posterior weight and the baseline risk difference is derived by algebraic substitution, reducing the four-dimensional parameterization to two variables. Second, cylindrical algebraic decomposition via the Tarski-Seidenberg theorem eliminates the remaining two auxiliary variables entirely, producing a direct semialgebraic description of the identified set in the plane of subgroup contrasts with no auxiliary coordinates. A scalar uniform sign certificate under a bounded effect-oscillation budget follows as a corollary.</p> <p>Finally, an exact proxy identifiability theorem is proved: if a binary proxy for the latent stratum is observed with a known channel matrix and is conditionally independent of treatment and outcome given the stratum, then the full latent parameter vector is exactly identified if and only if the proxy channel matrix is invertible, and identification fails precisely when the channel is singular. All results are proved in full.</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.19361706 |