On small sample Bayesian inference and sequential design for quantal response curves

Abstract

It is important in many econometric and biological applications to evaluate the effective dose (ED) points in the tails of quantal response curves and the curves according to a sensible criterion. Following Geisser (1971), it seems most sensible to evaluate the posterior mean value function of the response curve, since this also gives predictive probabilities of positive responses across the design region. While plausible, it can be insufficient for small samples to base the calculations upon standard multivariate normal likelihood approximations. Exact determinations, for example, via importance sampling (see Zellner and Rossi, 1984) are needed. Extending Leonard (1982a) it is now also possible to compute the exact posterior distribution of the ED points. These are proposed as “design measures”, since they can be used to sequentially generate further design points. Related procedures (see Leonard, 1982b) yield excellent frequency properties. For example, a total of 40 observations (10 fixed in advance and 30 chosen sequentially) can assess a response curve to 6% accuracy for all design points between the ED60 and ED90 points, with an average of over 90% of the sequentially generated design points falling between the ED60 and ED90 points.