Balancing covariates in multi-arm trials via adaptive randomization

Abstract

Multi-arm trials are common in medical and health research for comparing the efficacy of competing drugs and interventions, among other applications. While ensuring covariate balance is a critical issue for comparative studies to be successful, classical multi-arm trials often fail to balance covariates among multi-treatments. An adaptive randomization via Mahalanobis distance for multi-arm trials is proposed to improve the covariate balance and thus the quality of the subsequent treatment effect estimation. The investigation scope includes the implementation of the proposed method and also its theoretical properties. Both theoretical and numerical results demonstrate the proposed method can attain desirable covariate balance, and thus improving the subsequent estimation efficiency. Compared with other competing methods, the computational cost of the proposed method is also favorable. An illustrative real case analysis of the efficacy of different doses of Canagliflozin, a treatment for patients with type 2 diabetes, also proves that the proposed method has broad applicability.

Introduction

Multi-arm randomized trials are commonly used in medical and health research (Linden et al., 2016). In multi-arm trials, multi-valued treatments may include more than two discrete conditions (such as comparing the efficacy of competing drugs or interventions) or multiple levels of one treatment (such as various doses of a particular drug). Wason and Trippa (2014) note that multi-arm trials usually provide efficiency gains over separate trials when several experimental treatments are available for testing. Chataway et al. (2020) develop the multiple sclerosis-secondary progressive multi-arm randomized trial in which three treatments for multiple sclerosis patients are compared with a common control group. Palmcrantz et al. (2021) conduct a multi-arm randomized trial to explore the impact of training with an electromechanically assisted gait training system when integrated with conventional rehabilitation focused on gait and mobility.

Our study is motivated by an epidemiological trial comparing the efficacy and safety of Canagliflozin, an SGLT2 inhibitor, with Glimepiride for treating subjects with type 2 diabetes (Cefalu et al., 2013). In this clinical trial, participants are randomly assigned to one of three treatment groups to receive 100 mg or 300 mg of Canagliflozin or Glimepiride (up-titrated to 6 mg or 8 mg per day) orally once daily. The random assignment is carried out by a computer-generated random sequence via an interactive voice or web response system. One important issue with this experiment is the covariate imbalance among multi-treatments caused by adopting complete randomization (CR). This can be seen in Table 1, which lists five covariates in this experiment and shows the proportion of imbalanced bias among the treatment groups, calculated as the sum of absolute values of the percentage bias between each treatment group and the average level over three arms. It is clear that the substantial covariate imbalance among treatment groups exists.

The covariate imbalance among treatment groups has a negative impact on the subsequent analysis. Yang et al. (2020) note that the bias in the treatment effect estimation is proportional to the degree of covariate imbalance at the baseline. According to Li et al. (2020), any multi-arm trial can, if improperly designed, provide misleading information on the basis of the baseline covariates distributions in the treatment groups, potentially leading researchers to draw false conclusions. Indeed, if a substantial covariate imbalance exists, any inferences of the treatment effect will be inaccurate and any subsequent claims would rely on unverifiable assumptions (Zhao et al., 2015; Bugni et al., 2018). Although some ex-post adjustments can cope with such an imbalance, they are far less efficient than achieving the ex-ante balance from the start since additional coefficients must be estimated and further model assumptions are required (Bruhn and McKenzie, 2009). Imbens and Rubin (2015) also suggest that researchers' energies are better spent considering how to prospectively control the randomized assignment rather than make ex-post adjustments. According to a recent review of nearly 300 clinical trials published in 2009 and 2014 (Ciolino et al., 2019), 237 of them used covariate-adaptive randomization. Therefore, randomization designs that minimize the covariate imbalance are essential in multi-arm trials.

There are many randomization methods of balancing discrete or continuous covariates (Rosenberger and Sverdlov, 2008; Kuznetsova and Johnson, 2017). Li et al. (2020) note that the covariate balance in multi-arm trials tends to be more important since including more treatment groups in the experiment even raises the chance of observing imbalanced covariates. To balance the covariates in multi-arm trials, Kuznetsova and Tymofyeyev (2012) introduce the allocation ratio preserving biased coin minimization (BCM) design. BCM discretizes the continuous covariates so as to measure the covariate imbalance which is often less efficient and changes the nature of the continuous covariates. Xu and Kalbfleisch (2013) propose a balanced match weighted design which aims to select the assignment that yields the best covariate balance. They worry about the high computational cost of obtaining an approximate optimal solution under such a design, especially as the sample size and the number of covariates increases. Branson et al. (2016) propose the rerandomization (RR) method for multi-arm trials and the asymptotic properties of this experimental design framework are studied by Li et al. (2020). The major computational concern for RR is how long it takes to find the acceptable randomization, which is closely related to the selection of the threshold. Despite recent improvements in computational power, finding the most efficient randomization method with covariate balance for multi-arm trials remains a challenging research task.

In this paper, we present a new method namely the adaptive randomization via Mahalanobis distance for multi-arm design (ARMM), for balancing continuous covariates among multi-treatments and thus improving the subsequent estimation efficiency. The proposed method takes the covariates of units into account sequentially and allocates units adaptively since in many trials units are not all available for simultaneous assignment of treatments but rather arrive sequentially. In the proposed method, we adjust the probability of assignment according to the covariates of current units and the existing covariate imbalance of the previously allocated units. The proposed method is investigated both theoretically and numerically.

We contribute to the literature in the following aspects. We design an easy-to-implement covariate-adaptive randomization method specifically for the continuous covariates in multi-arm trials, whereas most of the existing methods are either designed for discrete covariates or lack theoretical justification. For our method, we establish its theoretical properties in randomization and treatment effect estimation. In particular, we demonstrate its optimality for estimation precision, i.e., the estimated treatment effect under the proposed method attains the minimum variance asymptotically. In addition, compared with other competing methods, the computational cost of the proposed method is also favorable.

The remainder of this paper is organized as follows. We introduce the proposed method in Section 2 and investigate its theoretical properties in Section 3. The numerical studies verifying the theoretical properties are shown in Section 4. We present a real data analysis to demonstrate its superiority in Section 5. Finally, we conclude with a discussion in Section 6.