By Alessandro Baldi Antognini, Alessandra Giovagnoli
Adaptive experimental designs are sequential strategies for amassing info. In every one step, the observer can use all the details accrued to figure out even if to forestall or keep it up with the examine. This ebook specializes in lately constructed tools in adaptive randomization for therapy allocation. It illustrates designs that take note of earlier therapy allocations in basic terms, discusses methods that use prior info, and offers compromise options that care for moral issues.
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Extra info for Adaptive Designs for Sequential Treatment Allocation
A + σB Starting with an equal number n0 of allocations of A and B, let σ ˆAi and σ ˆBi be the current MLEs of σA and σB at step i ≥ 2n0 ; at the next step, we assign the treatment A with probability equal to the estimated target: Pr δi+1 = 1 | δ (i) , Y (i) = σ ˆAi . σ ˆAi + σ ˆBi In the rest of the book, this type of design will be called a Sequential Maximum Likelihood design. s. s. so the design is asymptotically optimal. Furthermore, let µ ˆAn and µ ˆBn denote the MLEs of µA and µB after n steps, then as n → ∞ σB 2 0 σ 1 + √ µ ˆAn − µA A 0 σA n →d N ; 2 A 0 µ ˆBn − µB 0 σB 1 + σσB and also √ n [(ˆ µAn − µ ˆBn ) − (µA − µB )] →d N 0, (σA + σB )2 .
35) where 0 < tj < 1 for all j = 1, . . , v and the vector t = (t1 , . . s. s. s. for all j = 1, . . , v; • M(θ | πn ) → Σ−1 = diag • √ tv t1 V ar[Yi |δi1 =1] , . . s. n(θˆn − θ) →d N (0 ; Σ). 35) is true. Moreover, it gives the asymptotic properties of the estimators of the quantities of interest θ. 17), and does not take into account the nuisance parameters, but sometimes it is necessary to estimate the nuisances as well; this point will find a further warrant in Chapter 3. 3 Asymptotic optimality of a sequential design In order to simplify the notation, in this section we do not explicitly state the possible dependence of the targets on the unknown parameters.
Normal model M(µ1 , . . 22) . ,v 4. Exponential model M(λ1 , . . 24) where the expected value is taken with respect to the distribution of the random vector πn of treatment proportions. 23) above, πjn is replaced by E[πjn ] (for j = 1, . . , v). 5 Inference: Conditional on the design or unconditional? When the experiment is sequential, the potentially random nature of the experimental design may pose some problems for the statistical analysis of the experimental data. The question of the correct inferential paradigm after a sequential experiment was first posed by Silvey (1980).