Randomization in studies with unequal allocation

Published on September 29, 2016   24 min
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Hi. I'm Olga Kuznetsova. I'm a Senior Principal Scientist at Merck & Company. I will talk about randomization in studies with unequal allocation.
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This presentation is largely based on our joint works with Dr. Yevgen Tymofyeyev.
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Unequal allocation where the treatment groups are planned to have unequal sizes is often used in clinical trials. Among common reason unequal allocation are power considerations, better enrollment in a trail with higher allocation to the experimental group. Increased exposure to the experimental treatment or reduced cost. Unequal allocation is often used in adaptive design studies list two or more treatment times. Dose-ranging or multi-stage studies, studies with treatment selection design, sample size re-estimation or response adaptive randomization.
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Numerous equal allocation procedures are developed to meet different design needs. Robustness with respect to the accidental bias due to a time trend, reduced potential for selection bias, balance in important covariates. This algorithms are usually symmetric with respect to the treatment arms. As a result all patients, regardless of their place in the allocation sequence, are allocated with the same unconditional allocation ratio. For example, in a study with equal allocation to two treatments, the patients allocated first, second, third and so on, all have probability of ½ to be allocated to the first treatment.
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Kuznetsova and Tymofyeyev called procedures that preserve the unconditional allocation ratio, at every step. Allocation ratio preserving or ARP procedures. While common for equal allocation procedures, the ARP property does not necessarily hold for existent unequal allocation procedures.
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For example, in 1:2 allocation to treatments A and B, following biased coin minimization by Han et al. The unconditional probability of a allocation depicted on figure one varies with other phenomena. Instead of being 1/3 for all patients, it varies from approximately 0.1 to approximately 0.7 depending on the allocation number.
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Many existing unequal allocation procedures lack the ARP property. Non ARP procedures include, the urn design described by a Rosenberger and Lachin. Expansion of the maximal procedure by Salama et al. Biased coin randomization and minimization expansion by Han et al. Doubly adaptive biased coin designed procedure by Hu and Zhang, applied to fixed unequal allocation, as described by Sverdlov and Zhang. Minimum quadratic distance constrained balance randomization by Titterington. Adaptation of Biased Coin Randomization by Frane. Generalized method for adaptive randomization by Russel et al. And generalized multidimensional dynamic allocation method by Lebowitsch et al.
3:20
Variations in the unconditional allocation ratio are problematic. Awareness of such variations introduces potential for selection and evaluation bias, even in double blind studies. For example, if it is known that subject randomized second, has higher than average chance to receive active treatment. A subject with better prognosis can be selected for randomization in the second slot in the randomization sequence. Such variations can also lead to an accidental bias. In particular, in multi-center studies with randomization stratified by center.
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Such variations can also lead to randomization test problems. Proschan et al, considered examples of minimization with variations in the allocation ratio and showed that the distribution of the randomization test statistics is shifted away from zero. This lowered the power of the randomization test. Kuznetsova and Tymofyeyev showed that this problem exists for any non-ARP allocation procedure, either dynamic or fixed. They also derived the value of the shift from the sequence of the unconditional allocation ratios. Kaiser demonstrated that such variations cause a treatment effect estimator to be biased from a randomization perspective. To avoid these problems, equal allocation procedures should be expanded to unequal allocation, in a way that preserves the unconditional allocation ratio at every allocation.
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Any equal allocation procedure, symmetric with respect to treatment arms, can be expanded to an unequal allocation ARP procedure, following the mapping principle as described by Kuznetsova and Tymofyeyev. To generate an unequal allocation procedure the K treatment groups G1 to GK in C1 to CK allocation ratio, one can do the following. Let us denote by S is the sum of C 1 through CK. First, an equal allocation of S fake treatment arms F1 to Fs is executed, following the equal allocation procedure. There the groups of fake treatment arms are mapped to the actual treatment arms. Specifically, the first C1 fake treatments arms are mapped to treatment G1. The next C2 fake treatment arms are mapped to treatment G2 and so on. And finally the last CK fake treatment arms are mapped to treatment GK. Due to symmetry with respect to fake treatment arms such procedure provides equal allocation to fake treatment arms and that's C1 to CK unconditional allocation ratio to actual treatment groups at every allocation.
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Randomization in studies with unequal allocation

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