📊 statistics

Asymptotic inference with flexible covariate adjustment under rerandomization and stratified rerandomization

This paper establishes the asymptotic theory for a broad class of covariate-adjusted estimators, including M-estimators and data-adaptive machine learning methods, under rerandomization and stratified rerandomization, demonstrating that while these designs preserve asymptotic linearity, they may induce non-Gaussian distributions unless covariates are appropriately adjusted to restore asymptotic normality and efficiency.

Original authors: Bingkai Wang, Fan Li

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bingkai Wang, Fan Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are organizing a massive cooking competition. You have two teams: Team A (using a new secret recipe) and Team B (using the old standard recipe). Your goal is to see which recipe makes better food.

To make the test fair, you need to ensure the teams are evenly matched before they start cooking. If Team A gets all the professional chefs and Team B gets all the beginners, you can't tell if the recipe is better or if it's just the chefs.

The Problem: The "Coin Flip" isn't Perfect

Usually, scientists use Simple Randomization, which is like flipping a coin for every person to decide their team. While this is fair on average, sometimes you get unlucky. You might flip heads 10 times in a row, putting all the "professional chefs" on Team A. This is called imbalance.

The Solution: "Rerandomization" (The Strict Judge)

To fix this, the authors introduce a method called Rerandomization.
Imagine a strict judge who watches every coin flip.

The judge flips the coins.
The judge checks the teams. "Oh, Team A has too many pros and Team B has too many beginners. That's not fair."
The judge throws away that entire set of assignments and starts over.
They keep flipping and checking until they find a set of teams that looks perfectly balanced.

This ensures the starting line is fair. But here is the big question the paper answers: Does this strict judging change how we calculate the results at the end?

The Old Way vs. The New Way

In the past, statisticians knew that if you used simple math (like a straight line) to adjust for the differences between teams, the strict judge (rerandomization) didn't really matter for the final calculation. You could just pretend you flipped coins normally.

However, modern science uses much more complex, flexible tools (like Machine Learning or Generalized Linear Models) to analyze the data. These tools are like "smart algorithms" that can find complex patterns in the data, not just straight lines.

The big unknown was: If you use these fancy, flexible tools, does the strict judge (rerandomization) mess up the math? Does it make the results weird or unpredictable?

The Paper's Discovery: "The Magic of Adjustment"

The authors, Wang and Li, did the heavy mathematical lifting to prove two main things:

1. The "Shape" of the Results Changes (But only if you ignore the judge)
If you use a fancy tool but forget to tell it about the strict judge's rules, the math gets weird. The results don't follow the standard "bell curve" (the normal distribution) that statisticians love. It's like trying to measure a round object with a square ruler; the numbers get distorted.

2. The Fix: Just Tell the Tool About the Judge
The paper proves that if you simply include the variables the judge used (the "baseline covariates" like age, skill level, etc.) into your fancy analysis tool, everything becomes normal again.

The Analogy: Imagine you are explaining the cooking results to a reporter. If you say, "We used a strict judge to pick the teams, and here are the variables the judge looked at," the reporter can understand the math perfectly.
The Result: Once you adjust for those variables, the fancy tools work exactly as they would have if you had just flipped coins. The "strict judge" becomes invisible to the final calculation.

What About "Stratified" Rerandomization?

Sometimes, you don't just want to balance the whole group; you want to balance specific subgroups first (e.g., ensuring every team has an equal mix of men and women, then balancing their cooking skills). This is called Stratified Rerandomization.

The authors show that the same magic works here too. If you tell your fancy analysis tool about both the subgroups (strata) and the variables the judge balanced, the math stays clean and normal.

The "Super-Tool" (Machine Learning)

The paper also looks at Machine Learning (AI) models, which are the most flexible tools of all. These tools can learn from the data without you telling them exactly what to look for.

The Finding: Even with these super-smart AI tools, if you make sure the AI "sees" the variables the judge balanced, the AI remains perfectly efficient. It finds the true answer as fast as possible, and the math holds up.

The Bottom Line for Everyday People

If you are running an experiment (like a medical trial or a policy test) and you use a strict method to ensure the groups are balanced at the start:

Don't panic: You don't need to throw away your complex analysis tools.
Just be honest: When you run your analysis, make sure you include the specific factors you used to balance the groups.
The result: Your results will be accurate, your confidence intervals will be correct, and you can trust the numbers just as much as if you had used simple coin flips.

The paper essentially gives a "green light" to using the most advanced, flexible statistical tools in experiments that use strict balancing, as long as you remember to tell the math about the balancing rules.

Technical Summary: Asymptotic Inference with Flexible Covariate Adjustment under Rerandomization and Stratified Rerandomization

Problem Statement
Rerandomization is a restricted randomization procedure designed to control baseline covariate imbalance by discarding treatment allocation schemes that fail to meet a pre-specified balance criterion. While the asymptotic properties of unadjusted and linearly-adjusted estimators under rerandomization have been extensively studied within a finite-population design-based framework, it remains unclear whether these results generalize to the broader class of M-estimators. This class includes flexible covariate adjustment strategies such as g-computation with generalized linear models, doubly-robust estimators for missing data, mixed-effects models for cluster-randomized trials, and efficient estimators utilizing data-adaptive machine learners. Existing literature lacks formal asymptotic theory for these estimators under rerandomization, creating a gap in understanding how to conduct valid inference when flexible adjustment is combined with restricted randomization.

Methodology
The authors adopt a super-population sampling-based framework, assuming independent and identically distributed (i.i.d.) draws from an unknown distribution with finite second moments. The study covers two primary randomization procedures:

Rerandomization: Treatment assignments are regenerated until the Mahalanobis distance (or a generalized distance) of the imbalance statistic on a subset of covariates ( $X_r$ ) falls below a threshold $t$ .
Stratified Rerandomization: A combination of stratified randomization (to ensure exact balance on categorical strata) and rerandomization (to control imbalance on continuous covariates).

The paper develops asymptotic theory for M-estimators, defined as solutions to estimating equations $\sum \psi(O_i; \theta) = 0$ , and efficient estimators based on the efficient influence function (EIF) using double machine learning (DML) with cross-fitting. The analysis addresses the correlation among observations induced by the rejection sampling inherent in rerandomization.

Key Contributions and Results

Asymptotic Linearity and Influence Function Invariance:
The authors prove that for any M-estimator satisfying standard regularity conditions, the asymptotic linearity and the influence function remain identical under simple randomization and rerandomization. Consequently, if an estimator is consistent under simple randomization, it remains consistent under rerandomization. This result obviates the need to re-derive influence functions for M-estimators under rerandomization.
Non-Gaussian Asymptotic Distribution:
Unlike simple randomization, which yields asymptotic normality, rerandomization generally leads to a non-Gaussian asymptotic distribution. The limiting distribution is shown to be a linear combination of a standard normal variable and a truncated normal variable:
$\sqrt{n}(\hat{\Delta} - \Delta) \xrightarrow{d} \sqrt{V} \left( \sqrt{1-R^2} z + \sqrt{R^2} r_{q,t} \right)$
where $z \sim N(0,1)$ , $r_{q,t}$ is a truncated normal variable, $V$ is the asymptotic variance under simple randomization, and $R^2$ measures the alignment between the estimator's influence function and the rerandomized covariates. This distribution is symmetric and bell-shaped but has a variance reduced by a factor dependent on $R^2$ and the balance threshold $t$ .
Conditions for Asymptotic Normality (Ignorability of Rerandomization):
The paper identifies specific conditions under which $R^2 = 0$ , rendering the asymptotic distribution normal and the rerandomization procedure asymptotically ignorable during the analysis stage. For common estimators (ANCOVA, logistic regression g-computation, DR-WLS, and mixed-ANCOVA), asymptotic normality is achieved if:
- The treatment allocation is balanced ( $\pi=0.5$ ) and the estimand is on the difference scale;
- The outcome regression model includes treatment-by-covariate interaction terms for the rerandomization variables; or
- The outcome regression model is correctly specified.
  Crucially, if the analysis model adjusts for the rerandomization variables (and strata variables in the stratified case), standard inference procedures based on asymptotic normality remain valid.
Extension to Stratified Rerandomization:
Parallel results are established for stratified rerandomization. The asymptotic distribution follows a similar form, with adjustments to the variance component to account for variance reduction from stratification. The condition for asymptotic normality requires adjusting for both the rerandomization variables and the stratum indicators.
Efficient Estimators with Machine Learning:
The authors extend these findings to efficient estimators based on the efficient influence function and data-adaptive machine learners (DML). They prove that as long as the machine learning models adjust for the rerandomization variables, the estimators remain semiparametrically efficient. The asymptotic distribution follows the same non-Gaussian form unless the adjustment set includes the rerandomization variables, in which case $R^2=0$ and normality is restored. The paper demonstrates that rerandomization does not lower the classical semiparametric efficiency bound but can reduce the variance of estimators that are not orthogonal to prognostic covariates.

Significance and Claims
The paper claims to provide the first formal development of asymptotic theory for a general class of covariate-adjusted estimators under rerandomization and stratified rerandomization. Its primary significance lies in:

Clarifying Inference Procedures: It demonstrates that for many commonly used estimators, if the analysis appropriately adjusts for the variables used in the randomization design, standard inference (assuming simple randomization) is asymptotically valid. This simplifies the practical application of rerandomization.
Bridging Design and Analysis: The results connect the design-based benefits of rerandomization (improved balance) with the analysis-based benefits of flexible covariate adjustment, showing that the two are complementary but that the efficiency gains from rerandomization may be fully captured by the adjustment in large samples.
Validating Machine Learning Approaches: The work supports the use of data-adaptive machine learners in rerandomized experiments, proving their efficiency optimality and providing the necessary theoretical justification for their asymptotic behavior.

The authors note that while their results are derived under a super-population framework, they serve as the sampling-based counterparts to earlier design-based results for linear estimators. They acknowledge that in finite samples, particularly with small sample sizes or model misspecification, rerandomization may still offer precision gains even if the asymptotic variance is theoretically unchanged, a phenomenon observed in their simulations and re-analysis of a cluster-randomized experiment.