We focused on developing optimality criteria corresponding to multiple inference objectives and combining them in compound criteria allowing for finding compromises between different components, especially in the cases of relatively small experiments.
In the framework of response surface factorial experiments, we take into account the assumption of a potential model misspecification that is expressed in the form of extra polynomial terms that cannot be estimated. Along with obtaining quality estimates of the fitted (primary) model parameters, the contamination arising from the model disturbance is desired to be minimised. In addition, in the case of model uncertainty, the model-independent approach of making inference based on 'pure error' is to be incorporated.
The final criteria developed are the Mean Square Error (MSE) based criteria; their components correspond to maximising the precision of the fitted model estimates, minimising the joint effect of potentially missed terms and minimising the bias of the primary model parameters that might occur due to the model misspecification. The criteria were presented in the determinant- and trace-based forms, and are also adapted for use in blocked experiments. We also provided an alternative way of estimating criteria value for cases where the originally suggested simulations would be too computationally expensive.
An example of a real-life blocked experiment is studied, and we present a set of optimal designs that satisfied the aims of the experimenters and the restrictions of the experimental setup. Finally, we explore the framework of multistratum experiments; together with adaptation of the MSE-based criteria we provide a flexible design construction and analysis scheme.
The criteria and experimental settings presented are accompanied by illustrative examples in order to explore the possible relationship patterns between the criterion components and optimal designs' characteristics, and produce some general practical recommendations.