Equivariance and Invariance for Optimal Designs in Generalized Linear Models Exemplified by a Class of Gamma Models

The main intention of the present work is to outline the concept of equivariance and invariance in the design of experiments for generalized linear models and to demonstrate its usefulness. In contrast with linear models, pairs of transformations have to be employed for generalized linear models. These transformations act simultaneously on the experimental settings and on the location parameters in the linear component. Then, the concept of equivariance provides a tool to transfer locally optimal designs from one experimental region to another when the nominal values of the parameters are changed accordingly. The stronger concept of invariance requires a whole group of equivariant transformations. It can be used to characterize optimal designs which reflect the symmetries resulting from the group actions. The general concepts are illustrated by models with gamma distributed response and a canonical link. There, for a given transformation of the experimental settings, the transformation of the parameters is not unique and may be chosen to be nonlinear in order to fully exploit the model structure. In this case, we can derive invariant maximin efficient designs for the D- and the IMSE-criterion.

Optimal Designs for Comparing Regression Curves: Dependence Within and Between Groups

We consider the problem of designing experiments for the comparison of two regression curves describing the relation between a predictor and a response in two groups, where the data between and within the group may be dependent. In order to derive efficient designs we use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in a corresponding continuous model. It is demonstrated that in general simultaneous estimation using the data from both groups yields more precise results than estimation of the parameters separately in the two groups. Using the BLUE from simultaneous estimation, we then construct an efficient linear estimator for finite sample size by minimizing the mean squared error between the optimal solution in the continuous model and its discrete approximation with respect to the weights (of the linear estimator). Finally, the optimal design points are determined by minimizing the maximal width of a simultaneous confidence band for the difference of the two regression functions. The advantages of the new approach are illustrated by means of a simulation study, where it is shown that the use of the optimal designs yields substantially narrower confidence bands than the application of uniform designs.