The covariate-adaptive randomization method was proposed for clinical trials a long time ago but very little theoretical work has been done for statistical inference associated with it. Practical users often apply test procedures available for simple randomization, which is controversial since procedures valid under simple randomization may not be valid under other randomization. We provide some theoretical results for testing hypotheses after covariateadaptive randomization. We show that one way to obtain a valid test procedure is to use a correct model between outcomes and covariates, including those used in randomization. We also show that the simple two sample t-test, without using any covariate, is conservative under covariate-adaptive biased coin randomization in terms of its type I error, and that a valid bootstrap t-test can be constructed. The power of several tests are examined theoretically as well as empirically. Our study provides a guidance for applications and sheds light on further research in this area.

Some covariate-adaptive randomization methods have been used in clinical trials for a long time, but little theoretical work has been done about testing hypotheses under covariate-adaptive randomization until Shao et al. (2010) who provided a theory with detailed discussion for responses under linear models. In this article, we establish some asymptotic results for covariate-adaptive biased coin randomization under generalized linear models with possibly unknown link functions. We show that the simple t-test without using any covariate is conservative under covariate-adaptive biased coin randomization in terms of its Type I error rate, and that a valid test using the bootstrap can be constructed. This bootstrap test, utilizing covariates in the randomization scheme, is shown to be asymptotically as efficient as Wald's test correctly using covariates in the analysis. Thus, the efficiency loss due to not using covariates in the analysis can be recovered by utilizing covariates in covariate-adaptive biased coin randomization. Our theory is illustrated with two most popular types of discrete outcomes, binary responses and event counts under the Poisson model, and exponentially distributed continuous responses. We also show that an alternative simple test without using any covariate under the Poisson model has an inflated Type I error rate under simple randomization, but is valid under covariate-adaptive biased coin randomization. Effects on the validity of tests due to model misspecification is also discussed. Simulation studies about the Type I errors and powers of several tests are presented for both discrete and continuous responses.

Formal inference in randomized clinical trials is based on controlling the type I error rate associated with a single pre-specified statistic. The deficiency of using just one method of analysis is that it depends on assumptions that may not be met. For robust inference, we propose pre-specifying multiple test statistics and relying on the minimum p-value for testing the null hypothesis of no treatment effect. The null hypothesis associated with the various test statistics is that the treatment groups are indistinguishable. The critical value for hypothesis testing comes from permutation distributions. Rejection of the null hypothesis when the smallest p-value is less than the critical value controls the type I error rate at its designated value. Even if one of the candidate test statistics has low power, the adverse effect on the power of the minimum p-value statistic is not much. Its use is illustrated with examples. We conclude that it is better to rely on the minimum p-value rather than a single statistic particularly when that single statistic is the logrank test, because of the cost and complexity of many survival trials.

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