Sequential Optimal Designs with Unknown Link Function

Optimal designs provide a very efficient way to maximize the amount of information gained in an experiment. For linear models the information is a factor of the covariates, but for non-linear models maximizing the information becomes more difficult because of the relationship of information with the unknown parameter. To efficiently utilize the information gained, we consider sequential optimal designs. The work in this paper deals with sequential designs for binary regression and Poisson regression models, where the link functions are from a location scale family. In both cases, we assume the functional form of the link function, F, to be known, such as logistic or probit. That leads to the unknown parameters α and β and quantiles of the curve being of primary interest. We provide an empirical study to investigate the robustness of several sequential designs for binary regression models under different stimulus-response curves for two binaryregression applications: Phase I clinical trial and reliability sensitivity testing. The analysis involves studying the properties of the design measure and the parameter estimates. We propose nonparametric estimation of the stimulus-response curve via isotonic smoothing spline estimator under monotonicity constraint and incorporate the estimator into the sequential allocation scheme. This makes the procedure more robust against incorrect specification of the stimulus-response curve and at the same time maintains the objective of efficient estimation.We extend both the study and the proposed estimator to the Poisson regression model. We provide some theoretical results for the ISS sequential design algorithm and formulate the semiparametric Bayesian sequential optimal design approach for the binary regression model, which involves placing a prior distribution on the using mixtures of Dirichlet process priors.