Bayesian Design for Dose-Response Curves with Penalized Risk

This paper considers some Bayesian design problems in quantal response analysis. An experimenter must choose a set of dose levels and number of independent observations to take at these levels, subject to a total sample size, in order to estimate some characteristic phi, e.g., ED50, of a tolerance distribution F theta, where theta is the vector of unknown parameters. It is shown for the logistic model that the sampling variability of the posterior variance of phi is such that the predicted posterior variance alone is an undesirable criterion for design selection. A family of penalty functions is introduced that penalizes any excess in the posterior variance over the expected or predicted variance and protects against unexpected outcomes. The goal is to find a design that avoids experimental results with little information, at the expense of a small sacrifice in the Bayes risk. Numerical results indicate that the chance of an extreme posterior variance can be reduced by sacrificing a small amount of posterior risk.