Cross-Balancing for Data-Informed Design and Efficient Analysis of Observational Studies

TL;DR

Cross-Balancing introduces a two-stage, data-informed design for observational causal inference that leverages outcome information to construct balancing features. By employing sample-splitting, it separates feature-construction error from weight-estimation error, enabling consistent, asymptotically normal, and efficient estimation under mild conditions. The framework covers learned features (prognostic scores) and selected variables (dictionary-based feature selection), with finite-sample bias reduction and multiple robustness properties. Through extensive simulations and an NHANES case study, cross-balancing demonstrates improved estimation and inference while preserving interpretability, and provides practical guidance on integrating outcome information into design without compromising validity.

Abstract

Causal inference starts with a simple idea: compare groups that differ by treatment, not much else. Traditionally, similar groups are constructed using only observed covariates; however, it remains a long-standing challenge to incorporate available outcome data into the study design while preserving valid inference. In this paper, we study the general problem of covariate adjustment, effect estimation, and statistical inference when balancing features are constructed or selected with the aid of outcome information from the data. We propose cross-balancing, a method that uses sample splitting to separate the error in feature construction from the error in weight estimation. Our framework addresses two cases: one where the features are learned functions and one where they are selected from a potentially high-dimensional dictionary. In both cases, we establish mild and general conditions under which cross-balancing produces consistent, asymptotically normal, and efficient estimators. In the learned-function case, cross-balancing achieves finite-sample bias reduction relative to plug-in-type estimators, and is multiply robust when the learned features converge at slow rates. In the variable-selection case, cross-balancing only requires a product condition on how well the selected variables approximate true functions. We illustrate cross-balancing in extensive simulations and an observational study, showing that careful use of outcome information can substantially improve both estimation and inference while maintaining interpretability.

Figures (10)

• Figure 1: Diagram of the cross-balancing process.
• Figure 2: Estimation and inference performance of cross-balancing and other methods using learned features in low-dimensional settings. Each row corresponds to a simulation setting, and each column to a performance metric. (a): estimation bias and root mean squared error (RMSE) for each method. (b): empirical coverage of confidence intervals using the bootstrap (Boot) and Wald-type (CLT) inference. (c): average lengths of both types of confidence intervals.
• Figure 3: Estimation and inference performance of cross-balancing and other methods using learned features in high-dimensional settings. All other details are as described in Figure \ref{['fig:err_lowd']}.
• Figure 4: Performance of cross-balancing for covariate selection from high-dimensional raw features. Each column displays a different performance metric, and each row corresponds to a simulation setting. Within each subplot, results are shown for three estimators combined with three variable selection strategies. The performance of SBW (which does not involve variable selection) is indicated by a dashed horizontal line.
• Figure 5: Performance of cross-balancing and other methods when selecting from cubic spline basis functions across all settings in Section \ref{['subsubsec:simu_varsel_basis']}. Details are otherwise the same as in Figure \ref{['fig:simu_varsel_raw']}.

Theorems & Definitions (21)

• Theorem 3.1: Convergence and consistency
• Proposition 3.2
• Theorem 3.3
• Remark 3.4: Multiple validity with parametric models
• Theorem 4.1
• Theorem 4.2
• Remark 4.3: Finite-sample analysis
• Remark 4.4
• Example 4.5: Linear regression
• Proposition 4.6