Optimal Bayesian design applied to logistic regression experiments

Abstract

A traditional way to design a binary response experiment is to design the experiment to be most efficient for a best guess of the parameter values. A design which is optimal for a best guess however may not be efficient for parameter values close to that best guess. We propose designs which formally account for the prior uncertainty in the parameter values. A design for a situation where the best guess has substantial uncertainty attatched to it is very different from a design for a situation where approximate values of the parameters are known.We derive a general theory for concave design criteria for non-linear models and then apply the theory to logistic regression. Two numerical algorithms are described, an algorithm using the Nelder and Mead version of the simplex algorithm, which requires that the number of design points be specified, and an algorithm like the Wynn-Federov algorithm for linear design. The Wynn-Federov algorithm is much the slower of the two. Designs found by the Nelder-Mead algorithm are examined for a range of prior distributions and a range of criteria. The theoretical results are used to verify that the designs are indeed optimal~ A general finding is that as the uncertainty in the prior distribution increases so does the number of support points in the optimal design.

Some robust design strategies for percentile estimation in binary response models

“…Turning to the comparison of minimal c p -efficiencies, things change dramatically, especially for large parameter regions. We consider, as a representative example, the optimal designs in the logit model for estimating the percentile x p with p = 0.9 and assume that the experimenter is able to specify the parameter region Θ = [−1, 1] × [1,2]. The averaged efficiency of the Bayesian optimal design is given by the value b(ξ * B ) = 0.7283, indicating high c p -efficiencies for this design at most points in the parameter region Θ, whereas the integrated efficiency of the corresponding maximin optimal design is given by b(ξ * M ) = 0.5826, which is of course smaller than the result for the Bayesian design, but still in acceptable limits.…”

“…In the present context, the Fisher information for the parameter ϑ depends on the unknown value of ϑ itself, which complicates the determination of optimal designs substantially. Much effort has been devoted to the problem of finding good designs for the estimation of the parameter ϑ in the binary response model [see, e.g., Chaloner and Larntz (1989), Sitter and Wu (1993) among many others]. The goal of this article is to provide designs ξ, which are on the one hand efficient for estimating the 100pth percentile x p and on the other hand robust against misspecifications of the model assumptions.…”

“…This leads to so-called Bayesian optimality criteria [see e.g. Chaloner and Larntz (1989)]. As an alternative for the construction of robust designs, we propose a maximin approach based on c-efficiencies, which only requires the specification of a certain range for the unknown parameter.…”

Optimal Designs for Regression Models With a Constant Coefficient of Variation

“…Particular attention is paid to the case of normally distributed responses, but the methodology can easily be transferred to other distributions.We discuss locally optimal designs in the sense of Chernoff (1953) and robust design strategies as introduced in Chaloner and Larntz (1989) and Dette (1997). Section 3 is devoted to the characterization of optimal designs by equivalence theorems in the sense of Kiefer and Wolfowitz (1960).…”

Robust designs for binary data: applications of simulated annealing

“…There are three main approaches: sequential experimentation [5,6], maximin designs [7] and Bayesian designs [8]. More recent work [9] has used computer intensive optimisation methods to find designs for several variables which are robust to misspecification of the parameter values.…”