Nonparametric Tests under Restricted Treatment-Assignment Rules

Abstract

Abstract This article considers comparative clinical trials with R (R ≥ 2) treatments and allocation designs that aim to balance the assignment of treatments while controlling selection bias. A general form of these restricted designs was presented by Smith (1984a). His class of designs encompasses most of the other designs, including Efron's (1971) biased coin design (BCD). Widespread use of these designs is impeded by the lack of knowledge concerning the proper statistical analyses of the resulting data. In particular, little is known about the small-sample randomization distributions of test statistics. Thus current approaches rely on large-sample approximations, simulation techniques, and distributional results derived under classical schemes such as the complete randomization design. Clearly, none of these approaches is exact. A recursion procedure is presented for computing the exact conditional randomization distribution of a wide class of test statistics under these restricted designs. This procedure, which allows for a systematic and more efficient (compared to complete “brute force” enumeration) method of obtaining randomization distributions, enables exact significance tests of the hypothesis that the R treatments are equivalent. To amplify, let {T j = (T j1, …, T jR–1)′: j = 1, …, n} be the sequence of treatment-assignment vectors, with Tji equal to 1 or 0 according to whether patient j (j = 1, …, n) is or is not assigned treatment i (i = 1, …, R – 1), respectively. Let x = (x 1, …, xn )′ be the vector of observed patient responses (e.g., time to death, time to relapse, etc.) and a = (a 1, …, an )′ be the vector of scores (e.g., ranks, Gehan scores, Savage scores, log-rank scores, etc.) associated with x. Then, if T 1 + ··· + T n = m is the observed terminal allocation imbalance, the procedure given allows one to obtain the exact randomization distribution of the statistic S n = (T 1 …, T n )a, conditional on m. The observed value of S n is then compared with the exact distribution and a decision is made on whether at least two of the R treatments are different. The procedure is demonstrated on some clinical trial data of Mehta, Patel, and Wei (1986). Special attention is given to Efron's two-treatment BCD. Smythe and Wei (1983) pointed out that under the BCD, the unconditional randomization distribution of Sn is not asymptotically normal. Whether conditional asymptotic normality holds, however, remains an open question. The recursion is used to obtain empirical results that provide strong evidence that conditional asymptotic normality does not hold under the BCD. Exact distributional results, conditional on the terminal imbalance, are also obtained for the treatment-assignment variables. These results, of interest in their own right, illuminate difficulties with a large-sample approximation suggested by Efron in the presence of allocation imbalance.