Shift in re‐randomization distribution with conditional randomization test

Abstract

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.

Generalized Efron's biased coin design and its theoretical properties

“…For the latter, Proshcan et al [22] discovered that with two treatment arms and an unequal allocation ratio, the null distribution of the test statistic may be not centered at zero, which can lead to power loss compared with the population-based test. Similar discoveries were noted by Kuznetsova and Tymofyeyev [17], when the allocation ratio is equal but the randomization test is conditional. Kuznetsova and Tymofyeyev [16] further found that the shift from zero can be mitigated for large sample sizes if, at each step of allocation, a randomization procedure is able to keep the mean ratio the same as the target ratio.…”

Adaptive randomization for balancing over covariates

“…Accordingly, it is subject to a serious loss of power compared to normal theory under a population model. Kuznetsova and Tymofyeyev find that such shift even exists in equal allocation when a conditional randomization test is applied. Kuznetsova and Tymofyeyev explain that a shift occurs when a randomization method does not preserve the expected allocation ratio at every step, where the expectation is taken with respect to the distribution of the randomization sequences.…”

Considerations in the paper by Proschan, Brittain, and Kammerman are not an argument against minimization. In response to Vance W Berger ‘Minimization: not all it’s cracked up to be’, Clin Trials 2011; 8: 443

“…In this case, the shift in randomization distribution converges to 0 as the sample size increases and is typically very small in studies of moderate size. Of note, when a conditional randomization test is used in studies with equal allocation, a small shift in conditional randomization distribution will be observed [8] with many common randomization procedures, including permuted blocks [9], biased coin randomization [10], or a maximal procedure [4].…”