By Jeremy D. Finn

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B) Gauss–Hermite quadrature. 40) is derived from the asymptotic approximation of li by a quadratic function of b in a small neighborhood of bˆ i as λmin (−l¨i (bˆ i |σ , α , β ) → ∞, where λmin (·) denotes the minimum eigenvalue of a symmetric matrix. Therefore, such formula may produce significant approximation error for Li if the corresponding λmin (−l¨i (bˆ i |σ , α , β )) is not sufficiently large. One way to reduce the possible approximation error is to compute Li by using an adaptive Gauss–Hermite quadrature rule, as in Liu and Pierce (1994).

In this case the MTD has to be redefined as η= max{λ ∈ Λ : Fθ (λ ) ≤ p} if Fθ (λi ) ≤ p for some i, λ1 otherwise. 28). Yin and Yuan (2009) have proposed a Bayesian model averaging design based on the monotone dose– toxicity relationship. Chapter 3 Sequential Testing Theory and Stochastic Optimization Over Time The first seven sections of this chapter give an overview of the theory of fully sequential tests, starting with simple hypotheses involving likelihood ratio statistics and then extending the theory to composite hypotheses via generalized likelihood ratio (GLR) statistics.

Noting that the nonparametric approach in stochastic approximation seems too ambitious for moderate sample sizes, Wu (1985) proposed to use a parametric modification of the stochastic approximation scheme in Sect. 1, taking xk+1 to be the pth quantile of Fθˆ , where θˆ k is the MLE of θ based on the doses and responses k of the first k patients. O’Quigley et al. (1990) proposed a similar design but from a Bayesian point of view, called the CRM, that estimates the MTD at each stage by the posterior mean of θ with respect to a chosen prior distribution.

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