Bayesian design of experiment for quantal responses: What is promised versus what is delivered

This article considers a design problem in quantal response analysis, where an experimenter must choose a set of dose levels and number of independent observations to take at these levels, subject to some total sample size, in order to minimize the expected or predicted posterior variance of some characteristics ø of the tolerance distribution F_θ, with unknown parameters θ. An exact solution to this problem is demonstrated when ø is the unknown LD50 of the one parameter logistic tolerance distribution, under the restriction that an equal number of observations are taken at each of a set of equally spaced levels. The solution is based on a combination of simulated outcomes and Monte Carlo integration to evaluate the predicted variance. The numerical results are compared to those obtained previously by asymptotic approximations in Tsutakawa (1972), (J. Amer. Statist. Assoc. 67 584–590). The wide variability in the simulated posterior variance suggests that the expected posterior variance alone is not a good criterion for design selection.