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Resolution of Simpson's paradox via the common cause principle
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Re-estimates Simpson's paradox frequency under Dirichlet priors and proves that, for binary variables and a binary common cause (or screening variable), the fine-grained (conditioned on B) association is the correct one, with a Gaussian analogue.
Problem
Simpson's paradox leaves decision-makers unsure whether to use marginal or conditional-on-lurking-variable association; prior resolutions rely on assumed time-ordered causal graphs that often don't fit real examples. The paper asks how frequent the paradox is under unbiased data generation and how i
Approach
Assume an unobserved minimal common cause C (or screening variable) that makes A and B conditionally independent given C; then the sign of association conditioned on C matches the fine-grained option conditioned on B when C is binary, and for Gaussian variables with a scalar cause, the conditional covariance given B determines the decision.
Key results
- Under the non-informative Dirichlet prior with alpha=1/8, Simpson's paradox frequency is about 4.29% for 3 binary variables, versus about 1.66% (homogeneous) and 2.67% (Jeffreys).
- Theorem 1: for binary A1, A2, B and binary common cause C, any such C gives the same association sign as conditioning on B, so the paradox is resolved in the fine-grained direction.
- Theorem 2: in the minimal 3-scalar Gaussian setup with scalar common cause, the sign of the conditional covariance given B is the one to follow.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the unbiased Dirichlet choice α₁=...=α₈=1/8 instead of the homogeneous prior α=1 | Estimating binary Simpson's paradox frequency · Binary Simpson's paradox | paradox frequency: ≈ 1.66% -> ≈ 4.29 % ± 0.001% | Helped |
| Use the unbiased Dirichlet choice α₁=...=α₈=1/8 instead of the Jeffreys prior α=0.5 | Estimating binary Simpson's paradox frequency · Binary Simpson's paradox | paradox frequency: ≈ 2.67% -> ≈ 4.29 % ± 0.001% | Helped |
| Condition on the proposed binary common cause rather than marginalize over it | Binary Simpson's paradox · Binary variables A₁, A₂, B, and C | direction of association | No clear effect |
| Assume a binary genetic common cause of smoking and age, and use the fine-grained association | Smoking and surviving · Smoking and surviving | death rate association: 0.1820 > 0.1206; 0.8056 > 0.7829 | Hurt |
| Assume a binary healthcare-system common cause and use the fine-grained option | COVID-19 Italy vs China · COVID-19 Italy vs China | death rate association: 0.0608 < 0.0760; 0.0507 > 0.04900; 0.150 > 0.135 | Mixed |
| Use a minimal scalar common cause in the Gaussian formulation | Continuous Simpson's paradox · Three scalar Gaussian variables and scalar cause | conditional covariance sign | No clear effect |
| Use a two-component rather than minimal scalar common cause | Gaussian Simpson's paradox · Continuous Gaussian variables | conditional covariance sign | Mixed |
| Use a tertiary rather than binary common cause | Binary Simpson's paradox · Binary Simpson's paradox with non-binary C | direction of association | Mixed |
Surprising findings
- For non-binary (tertiary) common causes, all three options of the paradox become possible, so prior knowledge of C is needed.
- The COVID-19 example under a binary healthcare-quality common cause reverses the conclusion of the original authors (China did worse than Italy).
Limitations
- Resolution holds only for minimal (binary or scalar) common causes; non-minimal causes allow all options.
- The common cause C is hypothesized, not observed or inferred, and the examples' plausibility is argued qualitatively.
- Numerical examples for non-binary causes are stated to be available only upon request.
Experimental setup
Datasets: smoking and survival data (Tables from cited survey) COVID-19 Italy vs China death rate data Lindley-Novick medical treatment and plant yield examples synthetic Dirichlet-sampled probabilities (N=10^7)
Metrics: frequency of Simpson's paradox under Dirichlet priors sign of conditional association given C
Baselines: homogeneous Dirichlet prior frequency Jeffreys prior frequency Barigelli-Scozzafava retrodiction criterion Rudas criterion
Builds on: Barigelli and Scozzafava retrodiction proposal Pearl back-door criterion Lindley and Novick exchangeability Dirichlet non-informative prior
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