A New Family of Covariate-Adjusted Response Adaptive Designs and their Asymptotic Properties

Response-adaptive designs for clinical trials incorporate sequentially accruing response data into future allocation probabilities. A major objective of response-adaptive designs in clinical trials is to minimize the number of patients that is assigned to the inferior treatment to a degree that still generates useful statistical inferences. The preliminary idea of response adaptive randomization can be traced back to Thompson (1933) and Robbins (1952). A lot of response-adaptive designs have been proposed in literature (e.g., Rosenberger andLachin 2002, Hu andRosenberger, 2006). Much recent work has focused on proposing better randomized adaptive designs. The three main components for evaluating a response-adaptive design are allocation proportion, efficiency (power), and variability. The issue of efficiency or power was discussed by Hu and Rosenberger (2003), who showed that the efficiency is a decreasing function of the variability induced by the randomization procedure for any given allocation proportion. Hu, Rosenberger and Zhang (2006) showed that there is an asymptotic lower bound on the variability of response-adaptive designs. A response-adaptive design that attains this lower bound will be said to be first order efficient. More recently, Hu, Zhang and He (2008) proposed a new family of efficient randomized adaptive designs that can adapt to any desired allocation proportion. But all these studies are limit to the designs that do not incorporate covariates.

In many clinical trials (Pocock andSimon, 1975, Taves, 1974), covariate information is available and has a strong influence on the responses of patients. For instance, the efficacy of a hypertensive drug is related to a patient’s initial blood pressure and cholesterol level, whereas the effectiveness of a cancer treatment may depend on whether the patient is a smoker or a non-smoker. Covariate-adaptive designs have been proposed to balance covariates among treatment groups (see Pocock and Simon, 1975, Taves, 1974and Zelen, 1974). Hu and Rosenberger (2006) defined a covariate-adjusted response-adaptive (CARA) design as a design that incorporate sequentially history information of accruing response data and covariate as well as the observed covariate information of the incoming patient into future allocation probabilities.

In a CARA design, the assignment of a treatment depends on the history information and the covariate of the incoming patient. This generates a certain level of technical complexity for studying the properties of the design. Zhang, et al (2007) got a limit success on CARA designs by proposing a class of CARA designs that allow a wide spectrum of applications to very general statistical models and obtaining the asymptotic properties to provide a statistical basis for inferences after using this kind of designs. However, the CARA designs in Zhang, et al (2007) often have high variabilities and therefore are not efficient (Hu and Rosenberger, 2003). The major purpose of this paper is to study the variability and efficiency of CARA designs and to propose a new family of CARA designs with small variabilities.

The paper is organized as follows. In Section 2, the Fisher information and the best asymptotic variability are derived for a CARA design with any given target allocation proportion. We will find that the Fisher information and the variability depend on the distribution of each individual response, the target function and the distribution of the covariate. In Section 3, we propose a new CARA design that can adapt to target any allocation function and in which a parameter can be tuned such that the asymptotic variability approaches to the best one. The design proposed by Zhang, et al (2007) is a special case of this new design and has the largest variability in all this kind of designs. The new design is also an extension of the doubly adaptive biased coin design (BDCD) proposed by Eisele and Woodroofe (1995) and Hu and Zhang (2004a). The technical proofs are put on the Appendix.

2 Variability and efficiency of CARA designs

General framework of CARA designs.

Given a clinical trial with K treatments. Supposing that a patient with a covariate vector ξ is assigned to treatment k, k = 1, . . . , K, and the observed response is Y k , assume that the response Y k has a conditional distribution f k (y k |θ k , ξ) for given the covariate ξ. Here θ k , k = 1, . . . , K, are unknown parameters, and Θ k ⊂ R d is the parameter space of θ k .

In an adaptive design, we let X 1 , X 2 , … be the sequence of random treatment assignments. For the m-th subject, X m = (X m,1 , . . . , X m,K ) represents the assignment of treatment such that if the m-th subject is allocated to treatment k, then all elements in X m are 0 except for the k-th component, X m,k , which is 1. Suppose that {Y m,k , k = 1, . . . , K, m = 1, 2 . . .} denote the responses such that Y m,k is the response of the m-th subject to treatment k, k = 1, . . . , K. In practical

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