On multistage experiments with restrictions in the randomization of treatments

This paper considers multistage experiments which involve, in each stage, two-level factors whose levels are hard to change. Because of such factors, in each stage, there are restrictions in the randomization of runs leading to two types of experimental units, called whole plots and split plots. As a result, there are two types of random errors, in each stage, that need to be taken into account when modeling the response variable. It is assumed that a linear mixed effects model is appropriate for analyzing observations, and that the method of generalized least squares estimation is used to obtain estimators for the fixed effects in the model. It is also assumed that the model matrix of the fixed effects is based on a general two-level fractional factorial design. The goal of this paper is to provide an analytic form of the covariance matrix of the generalized least squares estimators of the fixed factorial effects in the model, that is useful for evaluating designs. This form shows how any confounding of (fixed) effects with the whole plots, associated with the different stages, affects the variances of their generalized least squares estimators. Some special cases of this form, which correspond to a model matrix based on either a two-level regular factorial or a two-level full factorial design, are also discussed. Results can be extended to multistage experiments with randomization restrictions of the runs, in each stage, with a model matrix based on a general multilevel fractional factorial design.