High-dimensional imbalance of additional covariates under IE-CAR

Derive the asymptotic behavior of the imbalance of additional covariates that are not included in the feature map under imbalance-efficient covariate-adaptive randomization procedures when the feature-map dimension diverges with the sample size, using high-dimensional methods beyond the low-dimensional Markov-chain analysis.

Background

The paper studies imbalance-efficient covariate-adaptive randomization (IE-CAR) and imbalance-robust covariate-adaptive randomization (IR-CAR) when the covariate dimension or feature-map dimension diverges with the sample size. Its main high-dimensional IE-CAR results concern the overall imbalance of the specified covariate features, whereas additional covariates not incorporated into the feature map may behave differently.

In low-dimensional settings, the authors indicate that the asymptotic behavior of additional-covariate imbalance can be analyzed through positive recurrence, invariant measures, ergodic laws of large numbers, and Poisson equations for an associated Markov chain. They state that these tools are less effective in the high-dimensional regime and leave the corresponding asymptotic analysis unresolved.

References

Under low-dimensional settings, the asymptotic behavior of the imbalance of additional covariates can be derived using classical tools from Markov chain theory, including the establishment of invariant measures via drift conditions, ergodic laws of large numbers, and the analysis of the associated Poisson equation. However, in high-dimensional settings, these tools become less effective, making the analysis of the asymptotic behavior of the imbalance of additional covariates considerably more challenging and we leave this problem for future research.

— Theoretical Properties of Covariate-Adaptive Randomization with a Diverging Number of Covariates  (2608.13442 - Tao et al., 13 Aug 2026) in Section 3, subsection “Theoretical Properties on Imbalance Measure” under IE-CAR

Second, when $q=\Omega(n)$, we show that existing CAR procedures do not perform well. Whether there exists a procedure that can achieve satisfactory performance in this regime remains an open problem.

— Theoretical Properties of Covariate-Adaptive Randomization with a Diverging Number of Covariates  (2608.13442 - Tao et al., 13 Aug 2026) in Section 8, Conclusion

Third, proposed a general non-Markovian CAR framework to address the shift problem under unequal allocation, which was further extended to covariate-adjusted response-adaptive settings by . Establishing theoretical properties for these two procedures under high-dimensional settings remains an important research problem.

— Theoretical Properties of Covariate-Adaptive Randomization with a Diverging Number of Covariates  (2608.13442 - Tao et al., 13 Aug 2026) in Section 8, Conclusion

Several directions remain open for future research. First, this work focused on equal allocation; extending the framework to unequal allocation ratios would increase its applicability. Second, our asymptotic results assume fixed covariate dimension and fixed strata counts, but in practice these may grow with sample size. Understanding non-asymptotic performance in high-dimensional or diverging-strata regimes is an important next step. Third, as discussed in Section \ref{sec:model robust}, the extension of model-robust inference after CAR to incorporate continuous variables merit further study. Finally, extending the framework to other outcome types, e.g., binary/count with generalized linear models \citep{zhai2026validtestmultiarmtrials} or survival data could broaden its practical relevance.

— Discretization in covariate-adaptive randomization: gains and losses  (2609.11012 - Zhao et al., 10 Sep 2026) in Section 10, Conclusion, final paragraph