Research map / Probabilistic and scientific modeling / Treatment effects
Conditional Outcome Equivalence: A Quantile Alternative to CATE
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Introduces the conditional quantile comparator (CQC), a response-space estimand matching equivalent quantiles of treated and untreated outcomes, and a doubly robust DR-learner estimator with finite-sample bounds, validated on simulations and employment and colon cancer data.
Problem
Estimating conditional quantile treatment effects (CQTE) is hard because it requires marginal conditional quantile functions, which may be less smooth than the treatment effect itself; existing doubly robust CQTE estimators need reciprocal conditional density nuisances that are hard to estimate.
Approach
Define the conditional quantile comparator, the treated outcome at the same conditional quantile as a given untreated outcome, and estimate it via a CCDF contrasting function using a DR pseudo-outcome regression (treated as a CATE problem), followed by isotonic projection and inversion. Its smoothness can be high even when the CCDFs and CQTE are not.
Key results
- In simulations with fixed 2n=1000, the proposed DR CQC estimator attains lower mean absolute error than Separate, IPW and CQTE-based baselines and is only marginally worse than the oracle DR estimator, with the CQTE approach degrading as the frequency gamma increases.
- On the employment data, the CQC reveals that participants around age 23 and aged 32-37 benefit most, and the lowest income quantile shows the least change.
- On the colon cancer data, the CQC suggests the treatment mainly increases the proportion of patients with no event during the trial rather than delaying events.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the doubly robust CQC estimator instead of separate CCDF estimation | CQC estimation with Nadaraya-Watson regression · Simulated experiments | mean absolute error | Helped |
| use the doubly robust CQC estimator instead of the IPW pseudo-outcome estimator | CQC estimation with Nadaraya-Watson regression · Simulated experiments | mean absolute error | Helped |
| use the doubly robust CQC estimator instead of the CQTE estimator | CQC estimation · Simulated experiments as nuisance smoothness decreases | mean absolute error | Helped |
| replace estimated nuisance quantities with their true values in the pseudo-outcome | Doubly robust CQC estimator · Simulated experiments | mean absolute error | Hurt |
| increase sample size | CQC estimators · Simulated experiments | mean absolute error | Helped |
| increase nuisance-parameter frequency | Doubly robust CQC estimator · Simulated experiments | mean absolute error | Hurt |
| make the CCDF contrasting function constant as well as the CQC (ablation) | CQC estimation · Simulated experiment with uniform responses | mean absolute error | No clear effect |
| use the DR CQC estimator in a 10-dimensional setting | CQC estimation · 10-dimensional simulated example | mean absolute error | Helped |
Surprising findings
- The CQTE estimator's error plateaus with increasing sample size, which the authors attribute to instability in estimating the reciprocal of the conditional density.
- Making both the CQC and the CCDF contrasting function constant gave no material improvement in estimator performance.
Limitations
- Theory requires weight decay assumptions (Assumption b) and does not directly bound smoothness of the CQC itself; smoothness of CQC is only linked to the contrasting function.
- Estimation requires learning the contrasting function and inverting it, which is less direct and computationally streamlined than a direct estimator.
- The framework is limited to deterministic treatment effects and lacks individual-level interpretation.
Experimental setup
Datasets: simulated data employment programme dataset colon cancer trial data (survival R package)
Benchmarks: simulated Gaussian CCDF experiments employment programme dataset colon cancer trial data
Metrics: mean absolute error 95% confidence intervals
Baselines: Separate CCDF estimator IPW pseudo-outcome estimator CQTE estimator of prior work Oracle DR estimator
Builds on: DR-learner R-learner CQTE doubly robust estimator Nadaraya-Watson regression PAVA isotonic regression
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