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Covariate-Adaptive Randomization in Clinical Trials Without Inflated Variances

Zhang Li-Xin

TL;DR

This work addresses variance inflation in covariate-adaptive randomization (CAR) when balancing covariates via an unequal allocation ratio $\rho:(1-\rho)$. It introduces a new CAR framework using a feature map $\boldsymbol{\phi}(\boldsymbol{X})$, an imbalance vector $\boldsymbol{\Lambda}_n$, and an allocation function $\ell$ with tuning $\gamma \in (0,1)$ to achieve $o_P(n^{1/2})$ balance for specified covariates while keeping the asymptotic variance for any unspecified covariate at or below that under simple randomization, thereby avoiding the shift problem reported in prior work. The paper establishes rigorous moment bounds and CLTs for the covariate imbalances, showing variance inflation is avoided and the estimator variances used in hypothesis testing are well-behaved. An application to testing treatment effects demonstrates that the proposed method yields valid type I error control and offers an adjusted test that preserves size exactly and boosts power when covariate information is available in analysis. Overall, the framework enables robust, efficient inference under CAR with unequal allocation in two-arm trials.

Abstract

Covariate adaptive randomization (CAR) procedures are extensively used to reduce the likelihood of covariate imbalances occurring in clinical trials. In literatures, a lot of CAR procedures have been proposed so that the specified covariates are balanced well between treatments. However, the variance of the imbalance of the unspecified covariates may be inflated comparing to the one under the simple randomization. The inflation of the variance causes the usual test of treatment effects being not valid and adjusting the test being not an easy work. In this paper, we propose a new kind covariate adaptive randomization procedures to balance covariates between two treatments with a ratio $ρ:(1-ρ)$. Under this kind of CAR procedures, the convergence rate of the imbalance of the specified covariates is $o(n^{1/2})$, and at the same time the asymptotic variance of the imbalance of any unspecified (observed or unobserved) covariates does not exceed the one under the simple randomization. The ``shift problem'' found by Liu, Hu, and Ma (2025) will not appear under the new CAR procedures.

Covariate-Adaptive Randomization in Clinical Trials Without Inflated Variances

TL;DR

This work addresses variance inflation in covariate-adaptive randomization (CAR) when balancing covariates via an unequal allocation ratio . It introduces a new CAR framework using a feature map , an imbalance vector , and an allocation function with tuning to achieve balance for specified covariates while keeping the asymptotic variance for any unspecified covariate at or below that under simple randomization, thereby avoiding the shift problem reported in prior work. The paper establishes rigorous moment bounds and CLTs for the covariate imbalances, showing variance inflation is avoided and the estimator variances used in hypothesis testing are well-behaved. An application to testing treatment effects demonstrates that the proposed method yields valid type I error control and offers an adjusted test that preserves size exactly and boosts power when covariate information is available in analysis. Overall, the framework enables robust, efficient inference under CAR with unequal allocation in two-arm trials.

Abstract

Covariate adaptive randomization (CAR) procedures are extensively used to reduce the likelihood of covariate imbalances occurring in clinical trials. In literatures, a lot of CAR procedures have been proposed so that the specified covariates are balanced well between treatments. However, the variance of the imbalance of the unspecified covariates may be inflated comparing to the one under the simple randomization. The inflation of the variance causes the usual test of treatment effects being not valid and adjusting the test being not an easy work. In this paper, we propose a new kind covariate adaptive randomization procedures to balance covariates between two treatments with a ratio . Under this kind of CAR procedures, the convergence rate of the imbalance of the specified covariates is , and at the same time the asymptotic variance of the imbalance of any unspecified (observed or unobserved) covariates does not exceed the one under the simple randomization. The ``shift problem'' found by Liu, Hu, and Ma (2025) will not appear under the new CAR procedures.
Paper Structure (8 sections, 6 theorems, 93 equations)

This paper contains 8 sections, 6 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. New family of CAR procedures
  3. Framework
  4. Theoretical Properties
  5. An application
  6. Proofs of the results
  7. The properties of the CAR procedures
  8. The properties of the test

Key Result

Theorem 2.1

Suppose that Assumptions and are satisfied. Then and if $\textsf{E}[|Z|]<\infty$ and $\gamma> \frac{1}{4}+\frac{1}{4(r_0-1)}$. Moreover, if $r_0\ge 4$, $\gamma>\frac{1}{2}+\frac{1}{4(r_0-1)}$, $\textsf{E} Z^2<\infty$, $\textsf{E} W^2<\infty$ and $\textsf{E}[W]=0$, then where $\vec{\sigma}_Z^2=(1-\rho)\rho\textsf{E}[\widetilde{Z}^2]$, $\widetilde{Z}=Z-\langle\bm\phi(\bm X),\bm x_0\rangle$ and

Theorems & Definitions (10)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Example 2.1
  • Example 2.2
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2