Exact Properties of Restricted Randomization Procedures

EXACT PROPERTIES OF RESTRICTED RANDOMIZATION PROCEDURES Hui Shao, PhD George Mason University, 2015 Dissertation Director: Dr. William F. Rosenberger Restricted randomization is employed in clinical trials to achieve balanced or near balanced treatment assignments, but is associated with a loss of randomness. For example, the permuted block design, probably the most commonly used restricted randomization procedure, randomizes patients block by block and achieves perfect balance within each block. However, there is at least one predictable treatment assignment within each block. To avoid deterministic treatment allocation in the permuted block design, many clinical trialists prefer randomizing the block sizes as well. While such a procedure is rarely formalized, it is generally assumed that the design will be less predictable. In this research, we first formalize the random block design by assuming a discrete uniform distribution for block size. We obtain its distributional properties such as the joint distribution of the block size and position within the block. With this distribution theory, we can analyze properties of prediction and balance. We then explore properties of restricted randomization procedures in clinical trials, including the permuted block design, random block design, Efron’s biased coin design and the big stick design. We investigate the degree of predictability and balancing properties theoretically, and discuss the performance of randomization tests by simulation. Additionally, we conduct graphical comparisons between these randomization procedures with respect to bivariate objectives. The goal of this study is first to provide a statistical understanding of the random block design, and second to provide properties of some commonly used randomization procedures for guidance in designing clinical trials. Our analysis allows clinical trialists to quantitatively assess predictability and balance of a restricted randomization procedure without doing simulations. Chapter 1: Introduction and Literature Review 1.1 Randomization in Clinical Trials Participants in a clinical trial are randomly assigned to one of two or more treatment groups. Randomization offers at least three major advantages. First, it prevents both the investigators and participants from guessing further treatment assignments. In unmasked studies, investigators may be able to guess the treatment assignment for further patients if they know the past treatment assignments. The investigators may wait to enter a patient into a treatment which they consider to be better suited to that specific patient if they know the next treatment assignment. In this case, a potential bias will be introduced into the trial. Randomization mitigates this type of bias, which is known as selection bias. Selection bias will be described more in Section 1.3. Second, randomization generates comparable treatment groups in both known and unknown covariates with high probability. Any baseline differences in participant characteristics such as gender, age, race and some medical baseline measurements between treatment groups may cause bias. These covariates are known in advance and their distribution within each treatment group can be equalized by certain randomization techniques, for example, stratified randomization and covariate-adaptive randomization. Randomization also helps increase the probability of comparability with respect to unknown important covariates as well (Cornfield, 2012). Third, randomization provides an assumption-free statistical test of the treatment effect. These tests are known as randomization tests, originally proposed by Fisher (Fisher, 1935). Randomization tests are performed simply on the basis that participants were randomly assigned to treatment groups. Proper randomization ensures that statistically significant differences between treatment groups are indeed due to an actual treatment effect. 1 Hu and Rosenberger (2006) described five classes of randomization procedures: complete randomization, restricted randomization, covariate-adaptive randomization, responseadaptive randomization, and covariate-adjusted response-adaptive randomization. Let Tj be a random variable that is 1 if the jth patient is assigned to treatment A and 0 if the jth patient is assigned to treatment B. Then T1, . , Tn be a sequence of random treatment assignments. In complete randomization, a balanced coin is used to determine the assignment. The assignment Tj is Bernoulli distributed with P (Tj = 1) = 1/2 and T1, . , Tn are independent. Complete randomization has the largest degree of randomness but also a large probability of producing imbalanced treatment groups and a small probability of serious imbalance. Restricted randomization is used to achieve equal or approximately equal assignments among treatment groups. In a restricted randomization procedure, the jth patient is assigned to either treatment A or B with an allocation probability based on the prior treatment assignment. The allocation probability is defined as φj = P (Tj = 1|T1, . . . , Tj−1). Covariate-adaptive randomization is used to assign patients to minimize the imbalances of certain known important covariates within treatment groups. Response-adaptive randomization is employed when it is desirable to allow more patients to be assigned to the superior treatment. For example, 50-50 allocation may be unethical if the outcome of the control group is serious disease or high probability of death. In response-adaptive randomization, the treatment assignments depend on the responses of the previous patients. The aim of covariate-adjusted response-adaptive randomization is to allocate greater numbers of patients to the superior treatment according to the patients’ covariate values, achieving high statistical efficiency and power in comparing treatment effects. In a covariate-adjusted response-adaptive randomization procedure, the allocation probabilities depend on the previous patients’ responses and covariates as well as the incoming patient’s covariates. A statistician always faces the question of selecting an appropriate randomization procedure when designing a randomized clinical trial. The goal of this research is to provide properties of some commonly used restricted randomization procedures that are useful in