Robust designs for multivariate generalized linear models with misspecification in the linear predictor: A quantile dispersion graphs approach

The purpose of this article is to evaluate and compare designs for multivariate generalized linear models using quantile dispersion graphs when the linear predictor is misspecified. The uncertainty in the linear predictor is represented by an unknown function. The comparison of the designs are based on a scalar-valued function of the mean square error of prediction (MSEP) matrix, which incorporates both variance and bias of the prediction caused by the misspecification in the linear predictor. For a given design, quantiles of the largest eigenvalue of the MSEP matrix are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor and the unknown function assumed to be the cause of model misspecification. If initial data is available, the unknown function is estimated using multivariate parametric kriging. To address the dependence of the quantiles on the unknown parameters, a 100(1-α)% confidence region of the parameters is computed and used as a parameter space. A numerical example based on multinomial response models is presented to illustrate the methodology.