An approximate Bayesian up-down method for estimating a percentage point on a dose-response curve

While the up-down method for estimating a percentage point on a dose-response curve has received considerable attention, a general Bayesian solution to the up-down design and estimation has never been presented, probably due to its computational complexity both in design and use. This paper presents a theoretical approach for up-down experimental designs with unknown location and slope parameters, and a practical approach for their use. The simplex method is used to find the optimal starting dose level and step sizes that minimize the expected root mean square error for a fixed number of observations and a reduced number of step sizes. The Bayesian estimate is then approximated by a polynomial formula. The coefficients of the formula are also chosen using simplex minimization. Two example solutions are given with uniform-uniform and normal-gamma joint prior distributions, showing that the simplifying assumptions make the method far easier to use with only a marginal increase in expected root mean square error. We show how to adapt these prior distributions to a wide range of frequently encountered applications.