Design of blocked and split-plot experiments for fixed-effects and variance-components estimation

Many industrial experiments, such as block experiments and split-plot experiments, involve one or more restrictions on the randomization. In these experiments the observations are obtained in groups. A key difference between blocked and split-plot experiments is that there are two sorts of factors in split-plot experiments. Some factors are held constant for all the observations within a group or whole plot, whereas others are reset independently for each individual observation. The former factors are called whole-plot factors, whereas the latter are referred to as sub-plot factors. Often, the levels of the whole-plot factors are, in some sense, hard to change, while the levels of the sub-plot factors are easy to change. D-optimal designs, which guarantee efficient estimation of the fixed effects of the statistical model that is appropriate given the random block or split-plot structure, have been constructed in the literature by various authors. However, in general, model estimation for block and split-plot designs requires the use of generalized least squares and the estimation of two variance components. We propose a new Bayesian optimal design criterion which does not just focus on fixed-effects estimation but also on variance-component estimation. A novel feature of the criterion is that it incorporates prior information about the variance components through log-normal or beta prior distributions. Finally, we also present an algorithm for generating efficient designs based on the new criterion. We implement several lesser-known quadrature approaches for the numerical approximation of the new optimal design criterion. We demonstrate the practical usefulness of our work by generating optimal designs for several real-life experimental scenarios.

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very good explantion, i eliminated many doubts thanks