This thesis focuses on the construction of optimum designs for the estimation of the variance components in multistage variability models. Variance components are the model parameters that represent the different sources of variability that affect the response of a system.A general and highly detailed way to define the linear mixed effects model is proposed. The extension considers the explicit definition of all the elements needed to construct a model. One key aspect of this formulation is that the random part is stated as a functional that individually determines the form of the design matrices for each random regressor, which gives significant flexibility. Further, the model is strictly divided into the treatment structure and the variability structure. This allows separate definitions of each structure but using the single rationale of combining, with little restrictions, simple design arrangements called factor layouts.To provide flexibility for considering different models, methodology to find and select optimum designs for variance components is presented using MLE and REML estimators and an alternative method known as the dispersion-mean model. Different forms of information matrices for variance components were obtained. This was mainly done for the cases when the information matrix is a function of the ratios of variances. Closed form expressions for balanced designs for random models with 3-stage variability structure, in crossed and nested layouts were found. The nested case was obtained when the information matrix is a function of the variance components. A general expression for the information matrix for the ratios using REML is presented. An approach to using unbalanced models, which requires the use of general formulae, is discussed. Additionally, D-optimality and A-optimality criteria of design optimality are restated for the case of variance components, and a specific version of pseudo-Bayesian criteria is introduced.Algorithms to construct optimum designs for the variance components based on the aforementioned methodologies were defined. These algorithms have been implemented in the R language. The results are communicated using a simple, but highly informative, graphical approach not seen before in this context. The proposed plots convey enough details for the experimenter to make an informed decision about the design to use in practice. An industrial internship allowed some the results herein to be put into practice, although no new research outcomes originated. Nonetheless, this is evidence of the potential for applications. Equally valuable is the experience of providing statistical advice and reporting conclusions to a non statistical audience.
|Date of Award||31 Dec 2014|
- The University of Manchester
|Supervisor||Alexander Donev (Supervisor) & Georgi Boshnakov (Supervisor)|
- variance components multistage optimum design dispersion-mean A-optimality D-optimality optimality