Optimal designs for mixed-effects models with random nested factors

The main objective of this research is to obtain experimental designs for estimating both fixed effects and variance components, in the presence of nested random effects. This is an important applied area in industrial experimental designs, including, but not limited to, robust product or process design. For this type of problem, a general class of experimental designs for mixed-effects models with random nested factors, called assembled designs, is introduced. The assembled designs are crossed factor designs with a nested design placed at each treatment combination of the crossed factor design.; A mixed-effects model approach to the analysis of data from assembled designs is theoretically sound, analytically flexible, and computationally feasible. This approach offers experimental designs for the joint estimation of location (fixed) effects, dispersion effects, and variance components, thereby avoiding possible biases introduced by considering them separately. That unbalanced data and fractional designs can be easily accommodated is very useful, since these are often needed to reduce the experimental effort.; For the estimation of fixed effects and two variance components, theorems establishing conditions for existence and uniqueness of D-optimal assembled designs and algorithms for finding these designs are presented. Specifically, balancing the observations as much as possible among all the branches of the nested design results in D-optimal designs for maximum likelihood estimation, except when a single sample is taken from a large number of batches. It is assumed that the data are approximately normally distributed and that the variance ratio is at least one.; If dispersion effects are to be estimated, then assembled designs within a fixed budget are compared. A heuristic algorithm for finding a nearly D-optimal assembled design with two variance components for a given budget is provided. A plot, called a precision plot, is introduced to help determine the expected effectiveness of the chosen design to identify effects on variance.; Ready to use computer programs for the presented experimental design procedures and analysis technique are included. This research also provides the practitioner with clear guidelines about the best design available for their needs, which is extremely useful and very needed in practice.