Proschan, Brittain and Kammerman [45] described a real-life study [57] that used a version of unequal allocation minimization designed by the authors of the study. In this study, the unconditional randomization test pre-specified as the primary analysis resulted in a p-value above 0.05, while the analysis of variance resulted in a much smaller, statistically significantp-value. The au- thors [57] demonstrated that the unconditional re-randomization distribution in their study was not centered at zero, which was the argument in favor of the analysis of variance (ANOVA) as the more trusted analytical approach.
Proschan, Brittain and Kammerman [45] considered more general exam- ples of unequal allocation minimization that, like the algorithm used in [57], did not preserve the unconditional allocation ratio at every step. They pointed out that in all these examples the unconditional re-randomization distribution of thet-test statistics was not centered at zero. This, they showed, led to a low power of the unconditional randomization test as well as difficulties in interpreting study results.
Kuznetsova and Tymofyeyev [30] showed that this problem is common to all non-ARP allocation procedures, fixed as well as covariate–adaptive. In contrast, for ARP allocation procedures, the mean of the unconditional re- randomization test statistics is asymptotically zero.
Kuznetsova and Tymofyeyev [30] derived the value of the shift in the un- conditional re-randomization distribution in a two-group study whereN sub- jects are allocated in aC1:C2 (C1< C2) ratio (or with probabilitiesρ1 and ρ2= 1−ρ1) to treatment groupsT1andT2. Following the notations presented in the introductions, they called
ei=πi2/ρ2−πi1/ρ1
the excess probability to allocate treatment T2 at the ith allocation. This measure of the extent to which the unconditional probability to allocateT2at theith allocation deviates from the targeted probability is not a probability measure: −1/ρ1 < ei <1/ρ2. When ei= 0, the unconditional probability to allocateT2 at theith allocation is equal to the targeted probability:πi2=ρ2; whenei>0,πi2> ρ2; whenei<0,πi2< ρ2. For ARP allocation procedures, ei = 0 for all i = 1, . . . , N. However, for non-ARP allocation procedures, ei varies from allocation to allocation.
They considered responses of the subjects in the treatment groupsT1(Con- trol) and T2 (Active Treatment) to be normally distributed with the means à1< à2 (the normality of the responses is not essential). Suppose the vector of responsesY ={Yi, i= 1, . . . , N}is observed in the study. Such study data are commonly analyzed using an ANOVA (ANCOVA) model where treatment group is one of the factors. The differenceà2−à1is estimated from the model.
To perform an unconditional re-randomization test [46], the difference in the treatment meansD(ω,Y) that corresponds to the randomization sequence ω is derived for each ω from Ω. The responses of the study subjects are considered fixed, while the treatment assignments follow those inω. The re- randomization test p-value (one-sided) is defined as the overall probability of the set of the randomization sequences ω for which the difference in the treatment group meansD(ω,Y) exceeds or is equal to the differenceD(ω0,Y) observed in the study.
In a study that follows an equal allocation randomization symmetric with respect to treatmentsT1andT2, the distribution ofD(ω,Y) is symmetric with mean zero. Indeed, as noted by Proschan, Brittain and Kammerman [45], for everyωfromΩ, there is a mirror allocation sequence where treatmentsT1and T2are switched places that has the same probability to occur asω. Thus, the differences in the treatment group means corresponding to the two sequences have the same absolute values but opposite signs. This, however, is not the case for unequal allocation.
Kuznetsova and Tymofyeyev [30] showed that for large N, the expecta- tion of the difference in the treatment means (the shift of the unconditional randomization distribution) is approximately
Eω∈ΩD(ω,Y) = 1 N
N
X
i=1
Yiei (8.2)
and thus, in general it is not equal to 0 when the allocation ratio is not preserved at every allocation. For an ARP procedureei = 0, and therefore,
the expectation of the difference in treatment meansD(ω,Y) is approximately zero for largeN.
The shift (8.2) depends on the sequence of observed responses. Kuznetsova and Tymofyeyev [30] also showed that for largeN the expectation of the shift over the normally distributed responses is approximately
(à2−à1)ρ1ρ2
N
N
X
i=1
e2i. (8.3)
This shows that the expected shift in the distribution of the difference in means is in the direction of the treatment effect (à2−à1) as was noted by Proschan, Brittain and Kammerman [45]. This shift lowers the power of the randomization test compared to the power of the ANOVA. In the example of the 60-patient study with 1 : 2 BCM allocation (pH(1)= 0.8) that balances on the treatment totals whereà1= 0 andà2= 1, the power of the unconditional randomization test was 0.785 compared to 0.903 power of the ANOVA [30].
It also follows from (8.3) that the expected shift is greater when the fluc- tuations in the unconditional allocation ratio (and therefore, the sum of ei
squares) are greater. This matches another observation of Proschan, Brittain and Kammerman [45] that the shift is greater when the probability of allocat- ing the preferred treatment is set to a higher value. Indeed, as we had noted earlier for BCM, a higher probability of allocating the preferred treatment leads to higher magnitude of the fluctuations in the excess probability.
All these considerations point toward using the allocation procedures that preserve the allocation ratio at every allocation.
For unequal allocation ARP procedures the shift, while asymptotically zero, is typically very small for studies of moderate size as was demonstrated by Kuznetsova and Tymofyeyev [30] for the ARP unequal allocation versions of biased coin randomization and minimization.
For many unequal allocation ARP procedures, attempts to investigate the shift in the randomization distribution for small sample sizes are complicated by the fact that for two randomization sequences that assign all subjects to the same treatment, the difference in the treatment means is not defined. In a study of small size such sequences cannot be ignored as their probabilities might not be negligible, in particular, for the sequence with all allocations to the larger group. For example, in a 10-subject study with 1 : 2 complete randomization to treatmentsT1 andT2, the probability of having all 10 sub- jects assigned to T2 is 0.017. It can be shown that in a study with unequal complete randomization, the mean value of the difference in treatment means across all other randomization sequences (that is, sequences that have subjects randomized to both treatments) is zero.
It is easy to show that for permuted block randomization the shift is zero if the study enrolled complete blocks of subjects, that is if N =m×(C1+ C2). However, this is not the case in a study for which the last block on the randomization schedule is left incomplete. For example, in a study with
N =C1+C2+ 1, the shift in the randomization distribution of the treatment differences is
Eω∈ΩD(ω,Y) = (ρ1−ρ2) (C1+ 1)(C2+ 1)
PC1+C2
i=1 Yi
C1+C2 −YC1+C2+1
!
. (8.4) Kaiser [19] developed a theory for randomization-based estimation in a linear model with unit-treatment additivity for an arbitrary randomization proce- dure including dynamic randomization. He showed that the treatment effect estimator from the randomization perspective is biased when the allocation ratio varies from allocation to allocation and recommended to avoid unequal allocation procedures with this property. Han, Yu and McEntegart [14] took a different approach—they allowed variations in the allocation ratio but pro- posed a weighted randomization test that adjusts for such variations.
While for the unconditional randomization test the shift exists only for studies with unequal allocation (with the rare exception of very uncom- mon non-symmetric equal allocation procedures), for the conditional ran- domization test the shift exists for studies with equal allocation. Indeed, as Kuznetsova and Tymofyeyev [31] noted, a conditional randomization test in a study with equal allocation is associated with variations in the allocation ratio in the conditional reference set. Indeed, when unequal group totalsN1andN2
are achieved in a study and the randomization test conditions on the group totals, the allocation ratio in the set of sequences with observed group totals varies from allocation to allocation. This happens to all allocation procedures, with the exception of complete randomization and the random allocation rule [46] for which all permutations ofN1treatment assignments to the first group and N2 treatment assignments to the second group are equally likely. Vari- ations in the conditional allocation ratio lead to the shift in the conditional randomization distribution.
Kaiser [19] noted that conditional inference will typically lead to bias in the randomization-based estimator of the treatment effect for unequal alloca- tion, but not for equal allocation. In contrast to Kuznetsova and Tymofyeyev [31], his conditional reference set included sequences with the group totals of (N1, N2) as well as symmetric sequences with group totals of (N2, N1), thus eliminating variations in the allocation ratio. This kind of conditional randomization test is not common in clinical trial practice.