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, 97 (1), 109-121

Stochastic Approximation With Virtual Observations for Dose-Finding on Discrete Levels


Stochastic Approximation With Virtual Observations for Dose-Finding on Discrete Levels

Ying Kuen Cheung et al. Biometrika.


Phase I clinical studies are experiments in which a new drug is administered to humans to determine the maximum dose that causes toxicity with a target probability. Phase I dose-finding is often formulated as a quantile estimation problem. For studies with a biological endpoint, it is common to define toxicity by dichotomizing the continuous biomarker expression. In this article, we propose a novel variant of the Robbins-Monro stochastic approximation that utilizes the continuous measurements for quantile estimation. The Robbins-Monro method has seldom seen clinical applications, because it does not perform well for quantile estimation with binary data and it works with a continuum of doses that are generally not available in practice. To address these issues, we formulate the dose-finding problem as root-finding for the mean of a continuous variable, for which the stochastic approximation procedure is efficient. To accommodate the use of discrete doses, we introduce the idea of virtual observation that is defined on a continuous dosage range. Our proposed method inherits the convergence properties of the stochastic approximation algorithm and its computational simplicity. Simulations based on real trial data show that our proposed method improves accuracy compared with the continual re-assessment method and produces results robust to model misspecification.


Fig. 1.
Fig. 1.
Model fits for the NeuSTART data. (a) Fitted dose-toxicity curve by the continual reassessment method (dashed) and the isotonic fit (solid); the target probability is also indicated (thicker solid line). (b) Liver function outcomes. Each observation is indicated (o), as well as the isotonic fit (solid) and the toxicity threshold (dashed).
Fig. 2.
Fig. 2.
(a) and (c) The proportion of selecting ν1 vs. σ (θ) by the virtual observation recursion assuming normal noise. (b) and (d) The proportion of selecting an acceptable dose by our method. Our method was run with β = 0.15 (dot-dashed), 0.29 (solid), 0.45 (dotted) under normal noise in (a) and (b), and with β = 0.29 and noise generated from the logistic (solid), t5 (dotted) and Gumbel (dot-dashed) distributions in (c) and (d). The selection probabilities of the continual re-assessment method (heavier solid) and the nonparametric optimal design (heavier dashed) are indicated.

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