Abstract
This paper deals with the statistical properties of a response adaptive design, described in terms of a two colors urn model, targeting prespecified asymptotic allocations. Results on the rate of divergence of number of patients assigned to each treatment are proved as well as on the asymptotic behavior of the urn composition. Suitable statistics are introduced and studied to test the hypothesis on treatments’ difference.
Highlights
The superior treatment, and (b) increasing the allocation of units to the superior treatment
We have completed the study of asymptotic statistical properties of the MRRU design, a response adaptive design, expressed in term of a randomly reinforced urn model, able to target asymptotically any prespecified allocation
There are a lot of interesting open problems whose solution could help in the research on optimal randomized adaptive designs; in particular, further studies based on changing the values of the parameters δ and η can contribute to explore the possibilities offered by the MRRU design
Summary
Consider a clinical trial with two competitive treatments, say R and W. . .) of Bernoulli random variables, with Xn representing the color of the ball sampled from the urn at time n, and a process (Z, D) = ((Zn, Dn), n = 0, 1, 2, . Since the urn proportion Zn−1 represents the conditional probability of assign the subject n to treatment R, this result shows that the target allocation depends on which is the superior treatment. Notice that in a RRU model the sequence Dn/n converges almost surely to the mean of the superior treatment. = = = + a→.s. 0 where the almost surely convergence to zero of the last terms can be proved with the same arguments used to prove Proposition 2.1 This result implies (2.8) due to the fact that n i=1.
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