Kuznetsova Olga M, Tymofyeyev Yevgen
Late Development Statistics, Merck Sharp & Dohme Corp., Rahway, NJ 07065-0900, USA.
Pharm Stat. 2013 Mar-Apr;12(2):82-91. doi: 10.1002/pst.1556. Epub 2013 Feb 14.
Proschan, Brittain, and Kammerman made a very interesting observation that for some examples of the unequal allocation minimization, the mean of the unconditional randomization distribution is shifted away from 0. Kuznetsova and Tymofyeyev linked this phenomenon to the variations in the allocation ratio from allocation to allocation in the examples considered in the paper by Proschan et al. and advocated the use of unequal allocation procedures that preserve the allocation ratio at every step. In this paper, we show that the shift phenomenon extends to very common settings: using conditional randomization test in a study with equal allocation. This phenomenon has the same cause: variations in the allocation ratio among the allocation sequences in the conditional reference set, not previously noted. We consider two kinds of conditional randomization tests. The first kind is the often used randomization test that conditions on the treatment group totals; we describe the variations in the conditional allocation ratio with this test on examples of permuted block randomization and biased coin randomization. The second kind is the randomization test proposed by Zheng and Zelen for a multicenter trial with permuted block central allocation that conditions on the within-center treatment totals. On the basis of the sequence of conditional allocation ratios, we derive the value of the shift in the conditional randomization distribution for specific vector of responses and the expected value of the shift when responses are independent identically distributed random variables. We discuss the asymptotic behavior of the shift for the two types of tests.
普罗尚、布里顿和卡默曼做了一个非常有趣的观察，即对于一些不等分配最小化的例子，无条件随机化分布的均值偏离了0。库兹涅佐娃和季莫菲耶夫将这一现象与普罗尚等人论文中所考虑例子中各次分配的分配比例变化联系起来，并主张使用在每一步都保持分配比例的不等分配程序。在本文中，我们表明这种偏移现象扩展到了非常常见的情形：在等额分配的研究中使用条件随机化检验。这种现象有着相同的原因：条件参考集中分配序列之间的分配比例变化，这一点此前未被注意到。我们考虑两种条件随机化检验。第一种是常用的基于治疗组总数进行条件设定的随机化检验；我们通过置换区组随机化和偏倚硬币随机化的例子描述了这种检验下条件分配比例的变化。第二种是郑和平提出的针对具有置换区组中心分配的多中心试验、基于中心内治疗总数进行条件设定的随机化检验。基于条件分配比例序列，我们针对特定的反应向量推导了条件随机化分布中的偏移值，以及当反应是独立同分布随机变量时偏移的期望值。我们讨论了这两种检验下偏移的渐近行为。
