Abstract

The Pocock and Simon’s covariate-adaptive randomization (minimization) is employed to dynamically balance the treatment groups in prognostic factors when the number of strata is large. The asymptotic covariance matrix of the within-stratum imbalances plays an important role in the inferential properties of the robust score test and the unstratified logrank test following a covariate-adaptive allocation procedure. However, the explicit form of this asymptotic covariance matrix is known only for case of the equal prevalence of the strata. In this work, based on empirical observations from extensive simulations, the previously unknown explicit form of the asymptotic covariance matrix is provided for the minimization with unequal prevalence and independent covariates. It was determined that the covariance matrix is proportional to the one arising from a simpler probabilistic model intuitively connected to minimization. The coefficient of proportionality V depends on the bias of the biased coin used with minimization. It is proven theoretically that when V ≤1, the maximum eigenvalue of the asymptotic covariance matrix does not exceed 1. This means that the unstratified log-rank test and the robust score test following such minimization is valid or conservative under a misspecified model. These results are supported by simulations of a clinical trial with the treatment groups balanced using minimization on two factors: the region with 5 levels and the risk group with 4 levels.

Key words and phrases: Asymptotic covariance matrix, log-rank test, minimization, Pocock and Simon covariate-adaptive randomization, unequal prevalence of the strata

Information

Preprint No.SS-2025-0213
Manuscript IDSS-2025-0213
Complete AuthorsOlga M Kuznetsova, Victoria P Johnson, Michael Gekhtman
Corresponding AuthorsOlga M Kuznetsova
Emailsolga_kuznetsova@merck.com

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Acknowledgments

The authors are grateful to the anonymous reviewers and the Associate Editor for constructive recommendations that helped substantially improve the paper.

M.G.’s work was supported in part by the Merck & Co., Inc., Rahway, NJ, USA, BARDS grant “Demonstration that Type I error is preserved or reduced with the score test and the log-rank test in studies with Pocock and Simon covariate-adaptive randomization”.

Supplementary Materials

The online Supplementary Material contains tables A1 – A14, the derivation of (3.2), and the proof of Theorem 1.

Supplementary materials are available for download.