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A Cautionary Tale on Integrating Studies with Disparate Outcome Measures for Causal Inference
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Studies a data integration setting where primary (SOWS) and auxiliary (COWS) studies have disparate, never co-observed outcomes, linking them via three outcome-link assumptions and deriving efficiency bounds, misspecification bias, and finite-sample gains, illustrated on OUD trial data.
Problem
Integrating a primary trial and an auxiliary study that measure different (non-identical) outcome scales for the same construct, with no individual observing both, to improve causal effect estimation efficiency.
Approach
Posits an outcome link Y = alpha(X)W + beta(X) in potential-outcome means and considers three strengths of prior knowledge about alpha and beta (fully known, partially known, unknown), deriving semiparametric efficiency bounds, misspecification bias, and finite-sample gains for each.
Key results
- Known-link estimator (theta_a) has asymptotically smaller variance than primary-only, while partially/unknown-link estimators match the primary-only efficiency bound.
- On the MOUD case study, theta_0 = -0.18 (CI width 1.52), theta_b = 0.42 (CI width 0.65), theta_a = -1.08 (CI width 1.41) with alpha = 0.61.
- Simulations show the unknown-link auxiliary estimator gives finite-sample MSE gains that diminish as n grows.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Integrate auxiliary outcomes when both α and β are known | Causal effect estimation from primary and auxiliary studies · Theoretical efficiency bound | asymptotic variance: 𝕍ᵗʰᵉᵗᵃₐ(X) ≤ 𝕍ᵗʰᵉᵗᵃ₀(X) | Helped |
| Integrate auxiliary outcomes when β is known but α is unknown | Causal effect estimation from primary and auxiliary studies · Theoretical efficiency bound | asymptotic variance: 𝕍ᵗʰᵉᵗᵃᵦ(X) = 𝕍ᵗʰᵉᵗᵃ₀(X) | No clear effect |
| Integrate auxiliary outcomes when both α and β are unknown | Causal effect estimation from primary and auxiliary studies · Theoretical efficiency bound | asymptotic variance: 𝕍ᵗʰᵉᵗᵃ𝒸(X) = 𝕍ᵗʰᵉᵗᵃᵦ(X) = 𝕍ᵗʰᵉᵗᵃ₀(X) | No clear effect |
| Use the two-stage auxiliary outcome regression with unknown α | Primary-only estimator θ̂₀ · Finite-sample estimation | estimation error | Mixed |
| Use the two-stage auxiliary outcome regression with unknown α | Primary-only estimator θ̂₀ · MOUD simulation study | mean squared error | Mixed |
| Integrate POAT COWS using known α | Primary-only estimator θ̂₀ · XBOT, week 4 SOWS | 95% CI width: 1.52 -> 1.41 | Helped |
| Integrate POAT COWS with unknown α | Primary-only estimator θ̂₀ · XBOT, week 4 SOWS | 95% CI width: 1.52 -> 0.65 | Helped |
| Misspecify α in the known-link integration estimator | Known-link integration estimator θ̂ₐ · Causal effect estimation | bias | Hurt |
| Vary α across the stated sensitivity range (ablation) | Known-link integration estimator θ̂ₐ · XBOT and POAT sensitivity analysis | treatment effect estimate: 0.50 to 0.33 | Mixed |
Surprising findings
- Under unknown or partially known link, auxiliary data gives no asymptotic efficiency gain, only finite-sample benefits.
- The known-link estimator's point estimate and significance depend on alpha: for alpha above 0.75 the confidence intervals include zero in sensitivity analysis.
Limitations
- Relies on a linear outcome link Y = alpha(X)W + beta(X) and conditional study exchangeability; the case study assumes alpha=0.61 derived from clinical thresholds with a shared zero point.
- No bridge study where both outcomes are observed; the framework cannot handle treatment or covariate discordance across studies.
Experimental setup
Datasets: XBOT trial (NIDA CTN-0051) POAT study (NIDA CTN-0030) synthetic data
Benchmarks: synthetic simulation MSE XBOT/POAT case study
Metrics: mean squared error point estimate 95% confidence interval width
Baselines: primary-only estimator two-stage auxiliary estimator
Builds on: influence function theory efficient score derivation surrogate outcome data fusion
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