Validity of Tests under Covariate-Adaptive Biased Coin Randomization and Generalized Linear Models

Abstract

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.

The impact of covariate misclassification using generalized linear regression under covariate–adaptive randomization

“…11 . With the quasi-Poisson regression models, the estimated variability of the outcome based on the unadjusted model then becomes $\phi Efalse(Y_{ij}false)$, where $\phi$is the estimated over-dispersion parameter and is greater than 1 if the data are over-dispersed.…”

“…9 However, under covariate-adaptive randomization, the intended type I error probability can be maintained only when the model is correctly specified, which means that all covariates included in the randomization procedure are also included in the analytic model. 10,11 …”

Adaptive randomization for balancing over covariates

“…Moreover, when adjusting for covariates, the power under covariate‐adaptive randomization is slightly higher than under complete randomization. Shao and Yu further investigate the above problem for discrete responses (binary or count) and exponential continuous responses, and find similar conclusions. One intriguing exception is that under complete randomization the analogous simple t ‐test for data from a Possion regression model leads to inflated type I error rate, whereas under covariate‐adaptive randomization the test is valid so that bootstrap estimator of variance is not needed.…”

Validity and power of minimization algorithm in longitudinal analysis of clinical trials

“…It was also shown that with the treatment assignment being generated through minimization, the simple two-sample t-test, without adjusting any covariate, was conservative in terms of its type I error [14]. Shao and Yu then extended their work to prove that in generalized linear model with even unknown link functions; those conclusions drawn in simple two-sample t-tests still held true [15]. …”