Allocation That Result in Non-ARP Procedures
Most covariate–adaptive procedures were developed following one of the two major approaches: Efron’s biased coin randomization [10] or Zelen’s method
[61], sometimes used in combination. In the simplest case of an equal allocation to two arms, the procedures based on the biased coin approach [13, 44] identify the preferred treatment arm that would lead to the lowest imbalance and allocate the patient using the coin biased toward the preferred arm. Biased coin provides an element or randomness, thus reducing the selection bias in open-label single center trials; it also enriches the set of allocation sequences that can be generated by the allocation procedure [22].
Other dynamic allocation procedures—procedures based on the modified Zelen’s approach and hierarchical dynamic balancing procedures [16, 53] were developed along the lines of Zelen’s [61] method that sets a threshold for allowed imbalance. In this section we will describe na¨ıve expansions of these dynamic methods that result in non-ARP procedures.
8.4.1 Na¨ıve Expansion of Modified Zelen’s Approach in a Multi-Center Study
Zelen [61] proposed a method for equal allocation to two treatment groups that provides balance in a baseline factor (most commonly center). The al- location sequence follows a pre-generated randomization schedule as long as the imbalance in treatment assignments within a factor level does not exceed the pre-specified threshold. When the next assignment on the randomization schedule would lead to an imbalance above the allowed threshold, it is skipped and the first assignment to an opposite treatment is used instead. Here a pre- generated randomization schedule provides randomness in treatment assign- ments.
The approach was modified and expanded to equal allocation to K ≥2 treatment armsTk,k= 1, . . . , K, for use in multi-center trials by McEntegart [39]. In a modified Zelen’s approach, the gaps in the allocation schedule are filled by assigning subjects to the first unused entry on the randomization schedule that complies with the imbalance threshold. This modification en- sures good balance in treatment assignments even in a small sample, as most blocks on the randomization schedule are filled by the end of allocation. For equal allocation toK≥2 treatment arms, the within-center imbalance is de- fined as the range in the treatment totals at the center. Morrissey, McEntegart and Lang [41] showed that equal allocation toK≥2 treatment arms using a modified Zelen’s approach provides good within-center and across-study bal- ance in treatment assignments even in a moderate size study.
Modified Zelen’s approach can be stratified by factors other than center.
To this end, a separate schedule is prepared for each stratum formed by a combination of levels of the factors other than center. A new patient is al- located to the first unassigned treatment on the allocation schedule for their stratum available at their center. This is a very useful feature in studies with a large number of small centers (especially those with several treatment arms) where within-center balance is required in addition to balance in other factors.
Conventional randomization stratified by all factors including center will not
work in this case as each center will be broken into several strata with few or, possibly, no subjects.
When in a study with equal modified Zelen’s approach allocation toK≥2 arms, the maximum allowed range in treatment assignments is set to 1, the al- location sequence at any given center becomes a sequence of permuted blocks with the block sizeK. The random permutations within the blocks are not pre- defined in advance as is the case with stratified by center permuted block allo- cation but instead are determined dynamically by the randomization schedule and the order in which patients from different sites enter the study.
This might suggest the way to expand the modified Zelen’s approach to unequal C1:C2: . . . :CK allocation to K ≥ 2 arms by dynamically filling the entire permuted block of C1+C2+. . .+CK allocations at the center before moving on to fill in the second block (as in Frane [12]). Specifically, the subject would be allocated to the first unused treatment assignment on the schedule that falls within the remaining allocations at the unfilled per- muted block on the dynamically formed center-specific schedule. Such a na¨ıve approach, however, presents a problem, as the allocation ratio will depend on the subject’s order of allocation within their center, signifying a non-ARP expansion.
Kuznetsova and Tymofyeyev [28] considered an example of the na¨ıve ex- pansion of the modified Zelen’s approach to 1 : 2 allocation to Control and Active treatments in a 240-subject study with 80 small centers. The center sizes ranged from 1 to 13; the average center size was 3 patients. The authors examined the percentage of Control allocations by order of allocation within the center through simulations. The distribution of the center sizes was ran- domly generated and fixed through all 500 simulations. The order of subjects’
arrival for randomization was also randomly generated and fixed through all simulations.
Figure 8.3presents the percentage of Control assignments among subjects allocated first, second, third, and so on in their respective center across all 500 simulations. It shows that the allocation ratio varies with the order of allocation within the center. Indeed, subjects allocated 1st, 3rd, 4th, 6th, 7th, 9th, 10th, 12th, and 13th within their respective center have higher than 1/3 probability of Control allocation, while subjects allocated 2nd, 5th, 8th, and 11th in their respective center have lower than 1/3 probability of Control allocation. The overall fraction of Control allocations remained close to 1/3 in all simulations.
Kuznetsova and Tymofyeyev [29] provided a probabilistic explanation to almost periodic fluctuations in the frequency of Control assignments in a multi-center study with na¨ıve expansion of the modified Zelen’s approach.
They also noted that the unconditional allocation ratio will vary with the or- der of allocation within center if a larger within-center imbalance is allowed, that is, the within-center allocations are allowed to be spread acrossM >1 unequal permuted blocks.
FIGURE 8.3
Percentage of Control treatment assignments by order of allocation in the example of 1 : 2 na¨ıve expansion of modified Zelen’s approach.
8.4.2 Non-ARP Expansions of Allocation Procedures Related to Modified Zelen’s Approach
Modified Zelen’s approach becomes impractical when the number of treatment armsKconsiderably exceeds the center size. In this case, Morrissey, McEnte- gart and Lang [41] propose a dynamic allocation procedure where only partial blocks of supplies are sent to the individual centers. It improves efficiency of the drug management while providing good overall study balance and reason- able within-center balance in treatment assignments.
The procedure is similar to modified Zelen’s approach. First, a random- ization sequence is generated to allocate subjects. Then a separate schedule is generated for drug supplies. This drug supplies schedule is cut in short seg- ments (smaller than the block size) that are then distributed across the sites.
When the site approaches the point where drug resupplies should be shipped, the first unused segment on the drug schedule determines the contents of the shipment. The next segment becomes available for randomization at the site only when the previous segment is completely filled. The subjects are allocated to the first unused treatment on the randomization schedule that is available for randomization at the site. Similar to modified Zelen’s approach, dynamic allocation with partial block supplies can be stratified by factors other than center.
Kuznetsova and Tymofyeyev [29] showed that if this procedure is expanded to unequal allocation by using unequal allocation permuted block schedules in place of equal allocation schedules, the unconditional allocation ratio will vary with the order of allocation within a center.
Another rich class of covariate–adaptive randomization procedures consists of hierarchical dynamic allocation procedures [16, 53]. In these procedures, typically defined for studies with equal allocation to two treatment arms, the hierarchy among the factors is established according to their importance.
The allowed imbalance thresholds are set for each of the factors; thresholds typically increase for the factors lower in hierarchy. When a new subject enters the study, the existing imbalances within the subject’s levels of the factors are examined in the order of the hierarchy. If none of the imbalances exceeds the threshold set for that factor, the patient is allocated at random. However, if, when moving down the list of factors, the imbalance above the allowed threshold is encountered, the patient is immediately allocated to the treatment that will decrease the imbalance in that factor. Sometimes, a biased coin element is added to the hierarchical balancing scheme to allow exceeding pre- specified thresholds with small probability [53]. Hierarchical procedures in studies with more than two treatment arms require more involved allocation rules [32].
Modified Zelen’s approach can be incorporated in a hierarchical allocation scheme by using center as the top factor in the hierarchy. Alternatively, within- center balancing through modified Zelen’s approach can be incorporated in a minimization-type procedure that balances on several other factors [1, 42].
The minimization-type procedures use an overall imbalance function that is to be minimized by the treatment assigned to the new subject within the constraint on a within-center imbalance.
In studies with small centers and equal allocation to a large number of arms, the dynamic allocation based on partial block supplies sent to the centers can also be incorporated into a hierarchical or minimization-type scheme to balance on other factors.
When a hierarchical procedure is na¨ıvely expanded to unequal allocation by establishing thresholds on the differences of observed allocation proportions (for example, |N2,male/ρ2−N1,male/ρ1| in a two-group study that balances on gender), the allocation ratio varies from allocation to allocation. Na¨ıve expansions of allocation procedures that incorporate within-center balancing through modified Zelen’s approach or dynamic allocation with partial block supplies sent to the centers share the same problem.