Regression analysis for covariate‐adaptive randomization: A robust and efficient inference perspective

Abstract

Linear regression is arguably the most fundamental statistical model; however, the validity of its use in randomized clinical trials, despite being common practice, has never been crystal clear, particularly when stratified or covariate-adaptive randomization is used. In this article, we investigate several of the most intuitive and commonly used regression models for estimating and inferring the treatment effect in randomized clinical trials. By allowing the regression model to be arbitrarily misspecified, we demonstrate that all these regression-based estimators robustly estimate the treatment effect, albeit with possibly different efficiency. We also propose consistent non-parametric variance estimators and compare their performances to those of the model-based variance estimators that are readily available in standard statistical software.Based on the results and taking into account both theoretical efficiency and practical feasibility, we make recommendations for the effective use of regression under various scenarios. For equal allocation, it suffices to use the regression adjustment for the stratum covariates and additional baseline covariates, if available, with the usual ordinary-least-squares variance estimator. For unequal allocation, regression with treatment-by-covariate interactions should be used, together with our proposed variance estimators. These recommendations apply to simple and stratified randomization, and minimization, among others. We hope this work helps to clarify and promote the usage of regression in randomized clinical trials.

Handling misclassified stratification variables in the analysis of randomised trials with continuous outcomes

“…First, the stratification variable could have more than two categories and be included in the analysis using random effects, as is generally recommended when stratifying by centre in multicentre trials 26 . Second, unequal treatment allocation (eg, 2:1 randomisation) could be considered, which may necessitate both the inclusion of treatment‐by‐covariate interactions in the model for estimating treatment effects and use of an alternate variance estimator to avoid inflated type I errors 27 . Third, misclassification in balancing variables for other forms of covariate‐adaptive randomisation, such as minimisation, could be investigated in settings where only some errors are discovered, building on previous work by others 14,15 .…”

Regression-Based Multiple Treatment Effect Estimation under Covariate-Adaptive Randomization

“…It was shown that τ[𝑘]𝑎 = Ȳ[𝑘]𝑎 − Ȳ[𝑘]0 is an unbiased and consistent estimator of 𝜏 [𝑘]𝑎 under Assumptions 1-3, and the plug-in estimator τ𝑏𝑚,𝑎 is a consistent estimator of 𝜏 𝑎 (Ma et al, 2022). Let τ𝑏𝑚 = ( τ𝑏𝑚,1 , … , τ𝑏𝑚,𝐴 ) T be the estimator of 𝜏.…”

“…In our formulation, the estimator for 𝜏 𝑎 can be directly obtained. Regression ( 1) is an extension of the third regression in Ma et al (2022) to multiple treatments.…”

“…This approach produces a valid inference by using a working model between responses and covariates, regardless of whether the working model is correct or not. To consider the regression adjustment for baseline covariates in addition to stratification covariates, stratum-common estimators and stratum-specific estimators have been developed, mainly for the case in which the allocation ratios are the same across strata (Liu et al, 2023;Ma et al, 2022;Ye et al, 2022aYe et al, , 2022b. However, little attention has been paid to the case in which the allocation ratios are different across strata, especially when additional baseline covariates are included, although different allocation ratios are commonly used in practice and are more flexible (Angrist et al, 2014;Chong et al, 2016).…”

Covariate Adjustment in Stratified Experiments

“…There has been significant interest in treatment effect estimation under different experimental designs in the recent literature. Some papers studying covariate adjustment under stratified randomization include Bugni et al (2018), Fogarty (2018), Liu and Yang (2020), Lu and Liu (2022), Ma et al (2020), Reluga et al (2022), Wang et al (2021), Ye et al (2022), Zhu et al (2022), and Chang (2023. These works differ from our paper in at least one of the following ways: (1) studying inference on the sample average treatment effect (SATE) rather than the ATE in a super-population, (2) restricting to coarse stratification (stratum size going to infinity), or (3) proving weak efficiency gains but not optimality.…”