Testing for treatment effect in covariate-adaptive randomized trials with generalized linear models and omitted covariates

Abstract

Concerns have been expressed over the validity of statistical inference under covariate-adaptive randomization despite the extensive use in clinical trials. In the literature, the inferential properties under covariate-adaptive randomization have been mainly studied for continuous responses; in particular, it is well known that the usual two-sample t-test for treatment effect is typically conservative. This phenomenon of invalid tests has also been found for generalized linear models without adjusting for the covariates and are sometimes more worrisome due to inflated Type I error. The purpose of this study is to examine the unadjusted test for treatment effect under generalized linear models and covariate-adaptive randomization. For a large class of covariate-adaptive randomization methods, we obtain the asymptotic distribution of the test statistic under the null hypothesis and derive the conditions under which the test is conservative, valid, or anti-conservative. Several commonly used generalized linear models, such as logistic regression and Poisson regression, are discussed in detail. An adjustment method is also proposed to achieve a valid size based on the asymptotic results. Numerical studies confirm the theoretical findings and demonstrate the effectiveness of the proposed adjustment method.

Consistent covariances estimation for stratum imbalances under marginal design for covariate adaptive randomization

“…The Monte Carlo approach could be viewed as a parametric bootstrap, and since Z is discrete, if pn is the empirical pmf, it is also a non-parametric bootstrap [19] REMARK. As mentioned before, an consistent estimator for Σ ∞ (p 0 ) is required to develop asymptotically valid statistical tests for various models; see, e.g., [14,29,11]. In simulation studies in Section 3, we consider adjusted tests using the proposed estimator, ΣB n ( pn ), for survival analysis.…”

“…For data collected under covariate adaptive randomization, due to correlated treatment assignments, adjustments are required for inferential procedures which work under the simple randomization [20,28,29,8,16,14]. Specifically, denote by Z 1 , .…”

“…Finally, denote by Σ n the covariance matrix of the within-stratum imbalances n −1/2 S n , where S n = (S n (z) : z ∈ Z), and by Σ ∞ the large-n limit of Σ n if it exists. A consistent estimator for the limiting covariance matrix Σ ∞ is needed to develop valid robust tests for treatment effects, e.g., under generalized linear models with omitted variables [14] or with survival data under possibly mis-specified models [29,11].…”

Consistent covariances estimation for stratum imbalances under minimization method for covariate‐adaptive randomization

“…Instead of balanced allocation within each stratum, it aims for marginal balance for the stratification factors and is thus also known as the marginal method. The popularity of the minimization method is noted in Taves (2010) that it has been used over 500 times during 1989-2008. For data collected under covariate-adaptive randomization, due to correlated treatment assignments, adjustments are required for inferential procedures which are developed for the simple randomization (Bugni et al, 2018(Bugni et al, , 2019Bugni & Gao, 2023;Hu & Zhang, 2020;Li et al, 2021;Liu et al, 2023;Ma et al, 2020;Shao et al, 2010;Xu et al, 2016;Ye et al, 2022;Ye & Shao, 2020). Specifically, denote by Z 1 ,…, Z n a sequence of discrete vectors that are used in covariate-adaptive randomization.…”

“…For data collected under covariate‐adaptive randomization, due to correlated treatment assignments, adjustments are required for inferential procedures which are developed for the simple randomization (Bugni et al, 2018, 2019; Bugni & Gao, 2023; Hu & Zhang, 2020; Li et al, 2021; Liu et al, 2023; Ma et al, 2020; Shao et al, 2010; Xu et al, 2016; Ye et al, 2022; Ye & Shao, 2020). Specifically, denote by a sequence of discrete vectors that are used in covariate‐adaptive randomization.…”