Statistical inference on vaccine efficacy under covariate-adaptive randomization
Open AccessIn clinical trials and causal inference studies, the primary goal of the experiment is to compare the effectiveness of two treatments. Complete randomization is a commonly used method that assigns patients independently and randomly into two treatment groups, and it is relatively easy to implement. However, this method may result in imbalanced distributions of covariates between the two treatment arms. To address this issue, covariate-adaptive randomization is a popular method in clinical trials with sequentially arriving patients. This method aims to balance treatment assignments across covariates that may have an influence on the response. However, concerns have been raised regarding the validity of statistical inference under covariate-adaptive randomization. Existing theory on tests under covariate-adaptive randomization is limited to estimating the average treatment effect. For example, it is well-known that the usual two-sample t-test of treatment effect is typically conservative. However, many trials do not solely focus on estimating the treatment effect, but other estimators as well. For instance, in a study with binary outcomes, the interested estimator could be the relative risk; in a study with time-to-event outcomes, the interested estimator could be the hazard ratio. In a vaccine study, vaccine efficacy can be estimated based on the relative risk given a fixed time point or based on the hazard ratio if the vaccine efficacy changes over time.The purpose of this study is twofold. Firstly, we have observed that although the covariate-adaptive randomization procedure has been utilized in vaccine trials, the validity of classical statistical methods for analyzing relative risk or vaccine efficacy after such randomization remains uncertain. In most cases, practitioners tend to adopt a conventional test to compare relative risk or vaccine efficacy, which is controversial since tests derived under complete randomization may not be valid under covariate-adaptive randomization schemes. Thus, it is critical to investigate the validity of classical statistical methods for analyzing relative risk or vaccine efficacy under covariate-adaptive randomization. Moreover, the issue of conservativeness also arises under survival analysis. Based on numerous reports, vaccine efficacy tends to decline over time, and the statistical inference of time-dependent vaccine efficacy also needs to be adjusted under covariate-adaptive randomization. Therefore, we aim to investigate the inferential properties of time-dependent vaccine efficacy under covariate-adaptive randomization using appropriate statistical methods.In addition, we observed that the statistical theories available for covariate-adaptive randomization primarily apply to linear models or generalized linear models with equal allocation ratios. While random assignment to treatment arms on a 1:1 basis (equal allocation ratio) has been standard practice, many 2-armed clinical trials now employ unequal allocation schemes due to ethical and economic advantages. Thus, we aim to investigate the inferential properties of unequal ratio tests for treatment effects in Chapter 2.In Chapter 3, our focus is on the evaluation of vaccine efficacy at a particular point. Typically, in such cases, researchers employ the logistic regression model to estimate the relative risk and vaccine efficacy. We provide a theoretical framework to study the asymptotic distribution of test statistics under covariate-adaptive randomization with omitted covariates. Our findings reveal that the hypothesis test on relative risk is conservative when the working model is misspecified. To address this issue, we propose an adjusted method for test statistics based on the asymptotic result, ensuring that the corrected test remains valid under covariate-adaptive randomization. Our approach offers a practical solution to the issue of conservative hypothesis tests on relative risk under covariate-adaptive randomization with omitted covariates.In Chapter 4, our focus is on the evaluation of vaccine efficacy as a time-dependent function. The duration of protection offered by a vaccine is uncertain, and vaccine efficacy typically declines over time. We consider the time-dependent Cox model to estimate hazard ratio and vaccine efficacy over time.Our findings reveal that when some important covariates are omitted, the estimator of vaccine efficacy can be biased, and the test statistics can introduce a very large Type-I error. Furthermore, when the model is mis-specified, we propose a stratified estimator of vaccine efficacy and construct a valid test statistics. Our study demonstrates that the stratified estimator is consistent, and the stratified test could attain the correct Type-I error under complete randomization and covariate-adaptive randomization. Our numerical studies confirm the theoretical findings and demonstrate the effectiveness of the proposed adjustment method. Overall, our approach offers a practical solution to the issue of bias in vaccine efficacy estimation and test statistics under time-dependent Cox models with omitted covariates.
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