Statistical Inference for Covariate-Adaptive Randomization Procedures

Comparison between asymptotic and re-randomisation tests under non-proportional hazards in a randomised controlled trial using the minimisation method

BACKGROUND

Pocock-Simon's minimisation method has been widely used to balance treatment assignments across prognostic factors in randomised controlled trials (RCTs). Previous studies focusing on the survival outcomes have demonstrated that the conservativeness of asymptotic tests without adjusting for stratification factors, as well as the inflated type I error rate of adjusted asymptotic tests conducted in a small sample of patients, can be relaxed using re-randomisation tests. Although several RCTs using minimisation have suggested the presence of non-proportional hazards (non-PH) effects, the application of re-randomisation tests has been limited to the log-rank test and Cox PH models, which may result in diminished statistical power when confronted with non-PH scenarios. To address this issue, we proposed two re-randomisation tests based on a maximum combination of weighted log-rank tests (MaxCombo test) and the difference in restricted mean survival time (dRMST) up to a fixed time point τ , both of which can be extended to adjust for randomisation stratification factors.

METHODS

We compared the performance of asymptotic and re-randomisation tests using the MaxCombo test, dRMST, log-rank test, and Cox PH models, assuming various non-PH situations for RCTs using minimisation, with total sample sizes of 50, 100, and 500 at a 1:1 allocation ratio. We mainly considered null, and alternative scenarios featuring delayed, crossing, and diminishing treatment effects.

RESULTS

Across all examined null scenarios, re-randomisation tests maintained the type I error rates at the nominal level. Conversely, unadjusted asymptotic tests indicated excessive conservatism, while adjusted asymptotic tests in both the Cox PH models and dRMST indicated inflated type I error rates for total sample sizes of 50. The stratified MaxCombo-based re-randomisation test consistently exhibited robust power across all examined scenarios.

CONCLUSIONS

The re-randomisation test is a useful alternative in non-PH situations for RCTs with minimisation using the stratified MaxCombo test, suggesting its robust power in various scenarios.

Network and covariate adjusted response-adaptive design for binary response

Randomization is a distinguishing feature of clinical trials for unbiased assessment of treatment efficacy. With a growing demand for more flexible and efficient randomization schemes and motivated by the idea of adaptive design, in this article we propose the network and covariate adjusted response-adaptive (NCARA) design that can concurrently manage three challenges: (1) maximizing benefits of a trial by assigning more patients to the superior treatment group randomly; (2) balancing social network ties across treatment arms to eliminate potential network interference; and (3) ensuring balance of important covariates, such as age, gender, and other potential confounders. We conduct simulation with different network structures and a variety of parameter settings. It is observed that the NCARA design outperforms four alternative randomization designs in solving the above-mentioned problems and has comparable power and type I error for detecting true difference between treatment groups. In addition, we conduct real data analysis to implement the new design in two clinical trials. Compared to equal randomization (the original design utilized in the trials), the NCARA design slightly increases power, largely increases the percentage of patients assigned to the better-performing group, and significantly improves network and covariate balances. It is also noted that the advantages of the NCARA design are augmented when the sample size is small and the level of network interference is high. In summary, the proposed NCARA design assists researchers in conducting clinical trials with high-quality and high-efficiency.

Minimization in randomized clinical trials

In randomized trials, comparability of the treatment groups is ensured through allocation of treatments using a mechanism that involves some random element, thus controlling for confounding of the treatment effect. Completely random allocation ensures comparability between the treatment groups for all known and unknown prognostic factors. For a specific trial, however, imbalances in prognostic factors among the treatment groups may occur. Although accidental bias can be avoided in the presence of such imbalances by stratifying the analysis, most trialists, regulatory agencies, and other stakeholders prefer a balanced distribution of prognostic factors across the treatment groups. Some procedures attempt to achieve balance in baseline covariates, by stratifying the allocation for these covariates, or by dynamically adapting the allocation using covariate information during the trial (covariate-adaptive procedures). In this Tutorial, the performance of minimization, a popular covariate-adaptive procedure, is compared with two other commonly used procedures, completely random allocation and stratified blocked designs. Using individual patient data of 2 clinical trials (in advanced ovarian cancer and age-related macular degeneration), the procedures are compared in terms of operating characteristics (using asymptotic and randomization tests), predictability of treatment allocation, and achieved balance. Fifty actual trials of various sizes that applied minimization for treatment allocation are used to investigate the achieved balance. Implementation issues of minimization are described. Minimization procedures are useful in all trials but especially when (1) many major prognostic factors are known, (2) many centers of different sizes accrue patients, or (3) the trial sample size is moderate.

Regression-based multiple treatment effect estimation under covariate-adaptive randomization

Covariate-adaptive randomization methods are widely used in clinical trials to balance baseline covariates. Recent studies have shown the validity of using regression-based estimators for treatment effects without imposing functional form requirements on the true data generation model. These studies have had limitations in certain scenarios; for example, in the case of multiple treatment groups, these studies did not consider additional covariates or assumed that the allocation ratios were the same across strata. To address these limitations, we develop a stratum-common estimator and a stratum-specific estimator under multiple treatments. We derive the asymptotic behaviors of these estimators and propose consistent nonparametric estimators for asymptotic variances. To determine their efficiency, we compare the estimators with the stratified difference-in-means estimator as the benchmark. We find that the stratum-specific estimator guarantees efficiency gains, regardless of whether the allocation ratios across strata are the same or different. Our conclusions were also validated by simulation studies and a real clinical trial example.

Rematching on-the-fly: Sequential matched randomization and a case for covariate-adjusted randomization

Covariate-adjusted randomization (CAR) can reduce the risk of covariate imbalance and, when accounted for in analysis, increase the power of a trial. Despite CAR advances, stratified randomization remains the most common CAR method. Matched randomization (MR) randomizes treatment assignment within optimally identified matched pairs based on covariates and a distance matrix. When participants enroll sequentially, sequentially matched randomization (SMR) randomizes within matches found "on-the-fly" to meet a pre-specified matching threshold. However, pre-specifying the ideal threshold can be challenging and SMR yields less-optimal matches than MR. We extend SMR to allow multiple participants to be randomized simultaneously, to use a dynamic threshold, and to allow matches to break and rematch if a better match later enrolls (sequential rematched randomization; SRR). In simplified settings and a real-world application, we assess whether these extensions improve covariate balance, estimator/study efficiency, and optimality of matches. We investigate whether adjusting for more covariates can be detrimental upon covariate balance and efficiency as is the case of traditional stratified randomization. As secondary objectives, we use the case study to assess how SMR schemes compare side-by-side with common and related CAR schemes and whether adjusting for covariates in the design can be as powerful as adjusting for covariates in a parametric model. We find each SMR extension, individually and collectively, to improve covariate balance, estimator efficiency, study power, and quality of matches. We provide a case-study where CAR schemes with randomization-based inference can be as and more powerful than non-CAR schemes with parametric adjustment for covariates.

Randomizing a clinical trial in neuro-degenerative disease

The paper studies randomization rules for a sequential two-treatment, two-site clinical trial in Parkinson's disease. An important feature is that we have values of responses and five potential prognostic factors from a sample of 144 patients similar to those to be enrolled in the trial. Analysis of this sample provides a model for trial analysis. The comparison of allocation rules is made by simulation yielding measures of loss due to imbalance and of potential bias. A major novelty of the paper is the use of this sample, via a two-stage algorithm, to provide an empirical distribution of covariates for the simulation; sampling of a correlated multivariate normal distribution is followed by transformation to variables following the empirical marginal distributions. Six allocation rules are evaluated. The paper concludes with some comments on general aspects of the evaluation of such rules and provides a recommendation for two allocation rules, one for each site, depending on the target number of patients to be enrolled.

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

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.

Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomization

BACKGROUND

Classical Brownian motion (BM) has been commonly used in monitoring clinical trials including those with covariate adaptive randomization (CAR). Independent increment property is commonly assumed in the sequential monitoring process of the clinical trials with CAR designs. However, in reality, correlation may exist in the error terms of the underlying model, resulting in dependent increment in the sequential monitoring process.

METHODS

We conducted simulations for estimating the Hurst exponent to evaluate the stochastic property in the covariate adaptive randomized clinical trials under two scenarios: 1. CAR designs with independent and identically distributed error terms. 2. CAR designs with correlated error terms. The theoretical properties of covariate adaptive randomized clinical trials with correlated error structure were investigated. A test statistic including the covariance pattern of the error terms was proposed.

CONCLUSION

In our study, the sequential test statistics under CAR procedure is shown to be asymptotically Brownian motion when the error structure is correctly specified. Further, Brownian motion is a special case of fractional Brownian motion when Hurst exponent equals to 0.5. Our simulations are consistent with the theoretical asymptotic results.

Multi-arm covariate-adaptive randomization

Simultaneously investigating multiple treatments in a single study achieves considerable efficiency in contrast to the traditional two-arm trials. Balancing treatment allocation for influential covariates has become increasingly important in today's clinical trials. The multi-arm covariate-adaptive randomized clinical trial is one of the most powerful tools to incorporate covariate information and multiple treatments in a single study. Pocock and Simon's procedure has been extended to the multi-arm case. However, the theoretical properties of multi-arm covariate-adaptive randomization have remained largely elusive for decades. In this paper, we propose a general framework for multi-arm covariate-adaptive designs which also includes the two-arm case, and establish the corresponding theory under widely satisfied conditions. The theoretical results provide new insights into the balance properties of covariate-adaptive randomization procedures and make foundations for most existing statistical inferences under two-arm covariate-adaptive randomization. Furthermore, these open a door to study the theoretical properties of statistical inferences for clinical trials based on multi-arm covariate-adaptive randomization procedures.

Incorporating covariates information in adaptive clinical trials for precision medicine

Precision medicine is the systematic use of information that pertains to an individual patient to select or optimize that patient's preventative and therapeutic care. Recent studies have classified biomarkers into predictive and prognostic biomarkers based on their roles in clinical studies. To design a clinical trial for precision medicine, predictive biomarkers and prognostic biomarkers should both be included. In statistical analysis, biomarkers are mathematically treated as covariates. We first classify covariates into predictive and prognostic covariates according to their roles. We then provide a brief review of recent advances in adaptive designs that incorporate covariates. However, the literature includes no designs that incorporate both prognostic covariates and predictive covariates simultaneously. In this paper, we propose a new family of covariate-adjusted response-adaptive (CARA) designs that incorporate both prognostic and predictive covariates and the responses. It is important to note that the predictive biomarkers and prognostic biomarkers play different roles in the new designs. The advantages of the proposed methods are demonstrated via numerical studies, and some further statistical issues are also discussed.

k-Resolution sequential randomization procedure to improve covariates balance in a randomized experiment

Balancing allocation of assigning units to two treatment groups to minimize the allocation differences is important in biomedical research. The complete randomization, rerandomization, and pairwise sequential randomization (PSR) procedures can be employed to balance the allocation. However, the first two do not allow a large number of covariates. In this article, we generalize the PSR procedure and propose a k-resolution sequential randomization (k-RSR) procedure by minimizing the Mahalanobis distance between both groups with equal group size. The proposed method can be used to achieve adequate balance and obtain a reasonable estimate of treatment effect. Compared to PSR, k-RSR is more likely to achieve the optimal value theoretically. Extensive simulation studies are conducted to show the superiorities of k-RSR and applications to the clinical synthetic data and GAW16 data further illustrate the methods.

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

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.

Sample size re-estimation for covariate-adaptive randomized clinical trials

Covariate-adaptive randomization (CAR) procedures have been developed in clinical trials to mitigate the imbalance of treatments among covariates. In recent years, an increasing number of trials have started to use CAR for the advantages in statistical efficiency and enhancing credibility. At the same time, sample size re-estimation (SSR) has become a common technique in industry to reduce time and cost while maintaining a good probability of success. Despite the widespread popularity of combining CAR designs with SSR, few researchers have investigated this combination theoretically. More importantly, the existing statistical inference must be adjusted to protect the desired type I error rate when a model that omits some covariates is used. In this article, we give a framework for the application of SSR in CAR trials and study the underlying theoretical properties. We give the adjusted test statistic and derive the sample size calculation formula under the CAR setting. We can tackle the difficulties caused by the adaptive features in CAR and prove the asymptotic independence between stages. Numerical studies are conducted under multiple parameter settings and scenarios that are commonly encountered in practice. The results show that all advantages of CAR and SSR can be preserved and further improved in terms of power and sample size.

The impact of misclassification on covariate-adaptive randomized clinical trials

Covariate-adaptive randomization (CAR) is widely used in clinical trials to balance treatment allocation over covariates. Over the past decade, significant progress has been made on the theoretical properties of covariate-adaptive design and associated inference. However, most results are established under the assumption that the covariates are correctly measured. In practice, measurement error is inevitable, resulting in misclassification for discrete covariates. When covariate misclassification is present in a clinical trial conducted using CAR, the impact is twofold: it impairs the intended covariate balance, and raises concerns over the validity of test procedures. In this paper, we consider the impact of misclassification on covariate-adaptive randomized trials from the perspectives of both design and inference. We derive the asymptotic normality, and thereby the convergence rate, of the imbalance of the true covariates for a general family of covariate-adaptive randomization methods, and show that a superior covariate balance can still be attained compared to complete randomization. We also show that the two sample t-test is conservative, with a reduced Type I error, but that this can be corrected using a bootstrap method. Moreover, if the misclassified covariates are adjusted in the model used for analysis, the test maintains its nominal Type I error, with an increased power. Our results support the use of covariate-adaptive randomization in clinical trials, even when the covariates are subject to misclassification.