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Design-based theory for causal inference from adaptive experiments

Xinran Li, Anqi Zhao

Abstract

Adaptive designs dynamically update treatment probabilities using information accumulated during the experiment. Existing theory for causal inference from adaptive experiments primarily assumes the superpopulation framework with independent and identically distributed units, and may not apply when the distribution of units evolves over time. This paper makes two contributions. First, we extend the literature to the finite-population framework, which allows for possibly nonexchangeable units, and establish the design-based theory for causal inference under general adaptive designs using inverse-propensity-weighted (IPW) and augmented IPW (AIPW) estimators. Our theory accommodates nonexchangeable units, both nonconverging and vanishing treatment probabilities, and nonconverging outcome estimators, thereby justifying inference using AIPW estimators with black-box outcome models that integrate advances from machine learning methods. To alleviate the conservativeness inherent in variance estimation under finite-population inference, we also introduce a covariance estimator for the AIPW estimator that becomes sharp when the residuals from the adaptive regression of potential outcomes on covariates are additive across units. Our framework encompasses widely used adaptive designs, such as multi-armed bandits, covariate-adaptive randomization, and sequential rerandomization, advancing the design-based theory for causal inference in these specific settings. Second, as a methodological contribution, we propose an adaptive covariate adjustment approach for analyzing even nonadaptive designs. The martingale structure induced by adaptive adjustment enables valid inference with black-box outcome estimators that would otherwise require strong assumptions under standard nonadaptive analysis.

Design-based theory for causal inference from adaptive experiments

Abstract

Adaptive designs dynamically update treatment probabilities using information accumulated during the experiment. Existing theory for causal inference from adaptive experiments primarily assumes the superpopulation framework with independent and identically distributed units, and may not apply when the distribution of units evolves over time. This paper makes two contributions. First, we extend the literature to the finite-population framework, which allows for possibly nonexchangeable units, and establish the design-based theory for causal inference under general adaptive designs using inverse-propensity-weighted (IPW) and augmented IPW (AIPW) estimators. Our theory accommodates nonexchangeable units, both nonconverging and vanishing treatment probabilities, and nonconverging outcome estimators, thereby justifying inference using AIPW estimators with black-box outcome models that integrate advances from machine learning methods. To alleviate the conservativeness inherent in variance estimation under finite-population inference, we also introduce a covariance estimator for the AIPW estimator that becomes sharp when the residuals from the adaptive regression of potential outcomes on covariates are additive across units. Our framework encompasses widely used adaptive designs, such as multi-armed bandits, covariate-adaptive randomization, and sequential rerandomization, advancing the design-based theory for causal inference in these specific settings. Second, as a methodological contribution, we propose an adaptive covariate adjustment approach for analyzing even nonadaptive designs. The martingale structure induced by adaptive adjustment enables valid inference with black-box outcome estimators that would otherwise require strong assumptions under standard nonadaptive analysis.
Paper Structure (38 sections, 26 theorems, 155 equations, 4 figures, 1 table)

This paper contains 38 sections, 26 theorems, 155 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting
  3. Adaptive randomization
  4. Potential outcomes and treatment effects
  5. Estimation by inverse propensity weighting
  6. Design-based theory of the IPW estimator
  7. Sampling properties
  8. Variance estimation and confidence sets
  9. Design-based theory of the AIPW estimator
  10. Sampling properties
  11. Variance estimation and confidence sets
  12. Adaptive analysis of nonadaptive designs
  13. Extension to block adaptive designs
  14. Treatment assignment mechanism
  15. Inference using the IPW and AIPW estimators
  16. ...and 23 more sections

Key Result

Theorem 1

Assume the adaptive randomization in Definition def:ad. Then $\mathbb{E}(\hat{\tau}_{\textup{ipw}, C}) =\tau_C$ and $\text{cov} (\hat{\tau}_{\textup{ipw}, C}) = T^{-1} CV_{\textup{ipw}}C^{\top}$ for any fixed matrix $C$ with $K$ columns.

Figures (4)

  • Figure 1: Violin plots of the distributions of $\hat{\tau}_* - \tau$ for each strategy $* = \textup{ipw}, {\textup{sm}}, \textup{aipw}, {\textup{all}}, {\textup{cf}}$ across 1,000 independent replications. The suffixes ".ls" and ".rf" indicate outcome regression using OLS and random forest, respectively.
  • Figure 2: Histograms of the difference-in-means estimator and the IPW estimator under the completely randomized design and the sequential rerandomization design across 1,000 independent replications, along with densities from their Gaussian approximations.
  • Figure 3: Violin plots of the distributions of $\hat{\tau}_{\textup{ipw}, C} - \tau$ across 1,000 independent replications at $\delta = 0.2, 0, -0.2, -0.4$.
  • Figure S1: Violin plots of the distributions of deviations of the estimators from the true values $(\bar{Y}(1), \bar{Y}(2), \tau)$ over 1,000 independent replications. The suffix ".ipw" refers to the standard IPW estimators in \ref{['eq:hyi']}. The suffix ".aw" refers to the adaptively reweighted estimators proposed by hadad2021confidence.

Theorems & Definitions (53)

  • Definition 1: Adaptive randomization
  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • ...and 43 more