Robust integer -valued designs for generalized linear model

We investigate the construction of designs for misspecified generalized linear models. Possible misspecification of generalized linear models includes linear predictor misspecification, use of an incorrect link function and an inadequate variance function. In this work we assume a finite design space and our interest is in the construction of integer-valued designs over the finite design space. The advantage of our integer-valued construction over the approximate design approach is that our designs are exact and thus readily implementable.;We begin with the problem of designing for models with a misspecified linear predictor. We adopt the average mean squared error of predictions over the design space as the loss function. The complicated dependence of the loss function on the unknown contamination function renders the problem of designing for generalized linear models with misspecified linear predictor not easily amenable to the minimax treatment which has been successful in the context of linear models. We propose a new criterion for robust designs termed "minave" designs. Using the average (over the design space) mean squared error of predictions as the loss function, the minave design is the design that minimizes the average mean squared error of predictions over the specified contamination neighbourhood. This averaging was carried out using a procedure based on a singular value decomposition of the design matrix.;We give a holistic treatment to the problem of designing for misspecified generalized linear models by using the same approach for constructing designs when the link function is possibly misspecified and when there is overdispersion. The general approach is to derive the mean squared error of predictions-based criterion in each case of model misspecification. Having established the design criterion based on relevant statistical consideration we obtain the integer-valued designs such that the criterion is minimized. In all cases the problem is a nonlinear integer optimization problem. We employ the simulated annealing algorithm to solve the resulting optimization problem.