Typically, when the treatment groups need to be balanced in known predic- tors of the response, permuted block randomization [61] stratified by baseline factors is employed in the study. However, when the study is small and the number of predictors is large, the stratification cells become small and might not even contain a single permuted block. Small strata can also arise when one of the factors (typically, the study center) has a large number of levels.
When the strata are very small, stratified randomization fails to produce the desired balance in baseline covariates [56].
In this case, balance can be achieved with dynamic allocation procedures [1, 2, 5, 16, 38, 41, 42, 44, 50, 53, 55, 58, 62]. The most popular and most discussed covariate–adaptive procedure isminimization introduced by Taves [55] and Pocock and Simon [44]. The version by Pocock and Simon explicitly adds a random element at every allocation through the use of a biased coin, while in the Taves version, random allocation occurs only when assignments to either treatment would result in equal imbalance. The Pocock and Simon version is preferred by the regulators [7, 11, 18], since it lessens the potential for selection bias in open-label studies.
Pocock and Simon [44] describe the minimization for an equal allocation to K ≥ 2 treatment groups Tk, k = 1, . . . , K in the following way. When a subject arrives for randomization, the subject’s sequence of covariates is recorded. For each covariate (for example, gender), the pre-specified marginal imbalance function (for example, the range) is derived across all subjects with the same level of the covariate as the new subject (for example, male), for Kdifferent scenarios assuming the new subject is assigned treatmentTk. The total imbalanceT otk that would arise ifTk is assigned is commonly calculated as a sum of the marginal imbalances across all levels of the covariates of the new subject. The treatmentsTk’s are then ranked in the total imbalanceT otk. The treatment with the rankj is then selected for the new subject with the probabilitypj, where the probabilitiesp1≥p2≥. . .≥pK (p1+p2+. . .+pk= 1) are pre-specified. Thus, the treatment that would result in the smallest imbalance (thepreferred treatment) is assigned with the highest probability.
When several treatment groups lead to the same total imbalance, establishing a random order among them is one of the ways to handle ties.
Typically, the preferred treatment is assigned with high probability p1
(for example, p1 = 0.8 or 0.9) and the rest of the probabilities are set to be equal: pj = (1−p1)/(K−1). Thus, the allocation procedure consists of two steps: 1) finding the preferred treatment and 2) assigning the pre- ferred treatment with probabilityp1 or one of the remaining treatments with probabilitypj.
8.2.1 How Variations in the Allocation Ratio Arise
As the two-step algorithm above is symmetric with respect to K treatment groups, the unconditional allocation ratio is the same (1 : 1 : . . . : 1) for all subjects, regardless of their place in the allocation sequence.
This, however, is not the case for many versions of minimization expansion to unequal allocation described in the literature [13, 35, 45, 48, 57]. In a study withC1:C2: . . . :CK allocation to treatment groups Tk,k= 1, . . . , K, the marginal imbalance within the levelmof the covariatef is commonly defined as the range of theKratios of the treatment totalsNkf m,k= 1, . . . , K, among the subjects with the covariate levelmdivided by the respective allocation ra- tioCk. For example, for unequal allocation inC1:C2(C1< C2) ratio (or with probabilitiesρ1andρ2= 1−ρ1) to treatment groupsT1andT2, marginal im- balance across males is defined asM Imale=|N2,male/ρ2−N1,male/ρ1|, where N2,male is the number of males allocated toT2 andN1,male is the number of males allocated toT1. With that, the two-step algorithm described above for the equal allocation results in variations in the unconditional allocation ratio from allocation to allocation.
Kuznetsova and Tymofyeyev [30] illustrated this phenomenon with the example of 1 : 2 allocation following Biased Coin Minimization (BCM) intro- duced by Han, Enas and McEntegart [13]. In the latter paper, the authors offered a refinement of the two-step procedure where the probability with which the preferred treatment is assigned differs across the treatment arms depending on their target allocation proportion. They specify the probability pH(1) to assign treatmentT1 when it is the preferred treatment and derive the probabilitypH(k)to assign treatmentTk,k= 1, . . . , K, when it is the preferred treatment frompH(1) and the allocation ratio. This approach approximates the target allocation ratio at the end of study better than the approach where pH(k)are the same for all treatment groupsk= 1, . . . , K [13].
Consider an example of the 1 : 2 BCM allocation to treatment groupsT1
and T2 that balances only on treatment group totals (no covariates) with pH(1) = 0.8 described by Kuznetsova and Tymofyeyev [30]. Figure 8.1 de- picts the unconditional probabilityπi1 to allocate treatmentT1 at allocation iderived iteratively following Han, Enas and McEntegart [13]. AsFigure 8.1 shows, the unconditional probability to allocate T1 at the first allocation is very low (0.1), while the unconditional probability to allocateT1at the second
FIGURE 8.1
Unconditional allocation probability toT1with 1 : 2 BCM allocation [13] that balances only on treatment totals (cf. Kuznetsova and Tymofyeyev [30]).
allocation is very high (0.73); it is low again at the 3rd allocation (0.23). These fluctuations in the unconditional probability to allocateT1converge with time to a periodic cycle of three distinct probabilities, while the overall probability to allocateT1 across all allocated subjects gets close to 1/3 with diminishing fluctuations from allocation to allocation. Hence, the BCM of Han, Enas and McEntegart [13] is not an ARP procedure.
The fluctuations in the unconditional allocation ratio from allocation to allocation are more pronounced when the probability to allocate the preferred treatment is higher. It is easy to show that with this procedure the allocation ratio is held constant only if pH(1) = 1/3 and pH(2) = 2/3, that is, when the biased coin randomization becomes complete randomization.
Variations in the unconditional allocation ratio also exist with covariate–
adaptive BCM. In this case the sequence of unconditional allocation ratios at allocations i= 1,2, . . . depends on the sequence of covariates observed in the trial. Kuznetsova and Tymofyeyev [30] considered an example of a 30- patient study with 1 : 2 covariate–adaptive BCM allocation toT1andT2with pH(1) = 0.8 that balances on gender (male/female) and age (younger/older).
The sequence of covariates observed in the study was generated as a sequence of 30 independent vectors of two independent variables (age and gender) where each level of each variable occurs with probability 1/2. For this sequence of covariates, 10,000 BCM sequences balanced on gender and age were generated.
The unconditional probabilitiesπi1,i = 1, . . . ,30, of assigningT1 at theith allocation were estimated based on simulations.
The dashed line in Figure 8.2that depicts the sequence of unconditional probabilities πi1 demonstrates considerable variations in probability of as- signing treatment T1. For comparison, the solid line in Figure 8.2 presents the probability of assigning treatment T1 (by allocation order) when the se-
FIGURE 8.2
An example of fluctuations in the unconditional allocation probability toT1 with 1 : 2 BCM allocation [13] that balances on two covariates with two levels each (pH(1) = 0.8). Dashed: the unconditional probability based on 10,000 simulations that used fixed sequence of covariates. Solid: average probability over 10,000 simulations that used a different sequence of covariates in each simulation (cf. Kuznetsova and Tymofyeyev [30]).
quence of covariates is random and varies from simulation to simulation. Now the fluctuations are much less pronounced and the probability to assignT1 is close to 1/3 everywhere except for the first couple of allocations.
These observations play a role when evaluating the potential for selection bias that might arise in a study with variations in the allocation ratio.
8.2.2 Potential for Selection Bias, Accidental Bias and Observer Bias in a Study That Uses
Unequal Allocation Minimization with Variations in the Allocation Ratio
If the investigators who enroll patients in the study are aware of the variations in the unconditional probability to allocateT1, an opportunity for selection bias and evaluation bias might arise even in a double-blind study. Indeed, if the investigator is aware that certain allocation numbers are associated with a higher than average chance of assigning an experimental treatment, they might use these allocation numbers to allocate patients who they believe would benefit more from the experimental treatment. Thus, the experimental and control group will differ in baseline profile and the selection bias will lead to biased study results. This knowledge can also influence the investigator’s
evaluation of efficacy or safety endpoints as well as treatment decisions, thus giving rise to the evaluation bias.
Variations in the allocation ratio can also lead to an accidental bias if confounded with the time trend in patients’ characteristics. Kuznetsova and Tymofyeyev [30] provide a hypothetical example of a study with 1 : 2 BCM allocation to T1 and T2 where subjects randomized 2nd, 5th, 8th,. . . (that is, when the unconditional probability of T1 allocation is almost double the targeted probability of 1/3 according to Figure 8.1) have a bad prognosis causingT1 group to have more bad prognosis patients compared toT2group.
Although such periodicity in baseline characteristic is possible, it is hardly likely to happen in practice.
A more realistic scenario that leads to an accidental bias was described by Proschan, Brittain and Kammerman [45]. They consider a version of 1 : 2 minimization to Control vs. Treatment that balances on centers only (which is effectively a minimization stratified by center) and where the third patients in their respective centers have lower than 1/3 probability of the Control assignment. They allowed for a time trend within a center such that the first two patients in a center are sick while the third patient is relatively healthy. It is quite common that the patients enrolled early differ in baseline profile from patients enrolled later, and the authors provide possible reasons for that. Since this confounding is repeated across all 200 centers, each with three patients enrolled, a notable accidental bias arises: on average, subjects in the Treatment group are healthier than those in the Control group. As a result, the type I error of theZ-test was also inflated [45].
In a trial with a non-ARP covariate–adaptive randomization, the sequence of unconditional allocation probabilities depends on the sequence of covariates and can be derived through complex calculations or simulations. In a single- center trial or a trial with covariate–adaptive allocation stratified by center, the investigator knows the sequence of covariates of all randomized patients (or all patients at his/her center, respectively) and can potentially calculate the probability to assign T1 at the next allocation (depicted by dashed line inFigure 8.2 example). Thus, he can use this knowledge to introduce the se- lection bias in a double-blind trial. However, the complexity of the required calculations makes it highly unlikely that an investigator would actually per- form the calculations—except, perhaps, for the first few subjects where it is relatively simple. Thus, the potential for selection and evaluation bias in- troduced by the variations in the allocation ratio is likely to remain just a potential when the investigators are concerned. The exception could be an example of a multi-center study with covariate–adaptive allocation stratified by center where the ability to introduce the selection bias in allocation of the first few subjects in each center can lead to a notable selection bias across the study.
However, a statistician who knows the covariates of all patients enrolled in a double-blind study will be able to calculate the sequence of probabilities to allocateT1 and thus enable the review team to introduce evaluation bias.
The more pronounced variations in the unconditional allocation ratio for a fixed sequence of covariates (dashed line inFigure 8.2) compared to less pro- nounced variations averaged across the random sequences of covariates (solid line in Figure 8.2) reflect the investigator’s ability to predict the probability of assigning T1 to the next subject when the investigator knows the full se- quence of covariates (as in a single-center trial) versus when the sequence of covariates is unknown to the investigator (as in a multi-center trial). When the full sequence of covariates is unknown, the fluctuations in the expected probability of theT1 assignment are small and can hardly lead to a selection bias other than in enrollment of the first few patients.
In conclusion, non-ARP expansions of allocation procedures to unequal randomization should be avoided as they can lead to a selection and eval- uation bias even in double-blind studies and can also lead to an accidental bias if confounded with the time trend in baseline characteristics of the study subjects.
Another problem with variations in the allocation ratio identified by Proschan, Brittain and Kammerman [45] is that they lead to a shift in the unconditional randomization distribution and thus lower the power of the unconditional randomization test.