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Covariate Shift Corrected Conditional Randomization Test
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Proposes csPCR, a covariate-shift-corrected Pearson chi-squared conditional randomization test that uses importance weights and control variates to test conditional independence in a target population using source data.
Problem
Testing conditional independence X ⊥ Y | Z in a target population when only labeled source data is available and covariates (X, Z, V) follow different distributions across populations (covariate shift), which invalidates standard CRT.
Approach
Extends the Pearson chi-squared CRT (PCR) by weighting label counts with density ratios e(X,Z,V)=P_T/P_S and reducing variance of weighted label sums via a weighted-regression control variate; the null distribution becomes a weighted sum of chi-squares.
Key results
- In simulation with beta=1.4, power is 0.33 for IS, 0.44 for csPCR, and 0.8 for csPCR(pe).
- With gamma=1, power is 0.4 for IS, 0.62 for csPCR, and 0.86 for csPCR(pe).
- On the COVID-19 data, csPCR p=0.025 and csPCR(pe) p=0.032 are significant, while IS gives p=0.663.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use csPCR instead of IS | Conditional independence testing under covariate shift · Simulation, β = 1.4 | power: 0.33 -> 0.44 | Helped |
| Add control variates for power enhancement (ablation) | csPCR · Simulation, β = 1.4 | power: 0.44 -> 0.8 | Helped |
| Use csPCR instead of IS | Conditional independence testing under covariate shift · Simulation, γ = 1 | power: 0.4 -> 0.62 | Helped |
| Add control variates for power enhancement (ablation) | csPCR · Simulation, γ = 1 | power: 0.62 -> 0.86 | Helped |
| Add control variates for power enhancement (ablation) | csPCR · Simulation, nonlinear indirect effect as θ approaches 1 | power | No clear effect |
| Use csPCR instead of IS | Conditional independence testing under covariate shift · COVID-19 dataset | p-value: 0.663 vs 0.025 | Mixed |
| Increase L (ablation) | csPCR · Simulation, true density ratio | Type-I error rate: 0.05125 (L=2) -> 0.02425 (L=20) | Mixed |
| Increase L (ablation) | csPCR · Simulation, estimated density ratio | Type-I error rate: 0.04620 (L=2) -> 0.02175 (L=20) | Mixed |
Surprising findings
- When the V|X,Z model is misspecified (theta near 1), the power-enhanced version's advantage over csPCR disappears.
Limitations
- Relies on the model-X assumption that the joint distribution of (X, Z, V) is known or well estimated.
- Power-enhancement benefit vanishes under misspecified V|X,Z model and inaccurate density ratio.
- Power analysis is only empirical; authors note theoretical power analysis is future work.
Experimental setup
Datasets: Simulated data Duke University Health System COVID-19 admissions (N=3,057)
Benchmarks: Simulated covariate-shift CI testing (Type-I error and power) COVID-19 90-day mortality dataset
Metrics: Type-I error rate power p-value
Baselines: IS method (importance-resampling CRT) PCR
Builds on: Conditional Randomization Test Pearson Chi-squared Conditional Randomization (PCR) model-X framework importance sampling CRT
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