Stationary and Convergence Properties of 'Up-and-Down' Methods

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We examine three modifications of the ’up-and-down’ (U&D) median- finding method for binary-response experiments. These modifications - ’biased-coin design’ (BCD), ’k-in-a-row’ (KR) and ’group up-and-down’ (GU&D) - target non-median threshold percentiles. Their stationary and convergence properties are compared theoretically and numerically. KR is found to be superior to BCD in all respects. KR converges faster than GU&D, but the latter has smaller stationary bias. Numerical convergence calculations indicate that sample sizes of 10 or less used in some fields for median estimation are overly optimistic, as are sample sizes of < 30 for finding the 30th or 20th percentiles using any of the three above-mentioned methods. The ’3+3’ method, commonly used for Phase I clinical trials, appears to converge even more slowly. We also conclude that using only 4−6 treatment levels is sub-optimal, due to a convergence-precision tradeoff. Instead, 8 − 10 levels are recommended.

Keywords: Adaptive Staircase; Phase I Clinical Trials; Quantile Estimation; Random Walk experimental design; Sensory Threshold; Up-and-Down;


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