By Design: Exchange Algorithms to Construct Exact Model-Robust and Multiresponse Experimental Designs
Open Access
- Author:
- Smucker, Byran Jay
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 08, 2010
- Committee Members:
- Enrique Del Castillo And James Rosenberger, Dissertation Advisor/Co-Advisor
Enrique Del Castillo, Committee Chair/Co-Chair
James Landis Rosenberger, Committee Chair/Co-Chair
Rana Arnold, Committee Member
Bing Li, Committee Member
Susan H Xu, Committee Member - Keywords:
- split-plot design
optimal design
maximin - Abstract:
- Optimal experimental design procedures, utilizing criteria such as $mathcal{D}$-optimality, are often used under nonstandard experimental conditions such as constrained design spaces, and produce designs with desirable variance properties. However, to implement these methods the form of the regression function must be known emph{a priori}, an often unrealistic assumption. Model-robust designs are those which, from our perspective, are robust for a set of specified possible models. In this dissertation, we present new model-robust exchange algorithms for exact experimental designs which improve upon current, practical model-robust methodology. We also extend these ideas to experiments with multiple responses and split-plot structures, settings for which few or no flexible, practicable model-robust procedures exist. We first develop a model-robust technique which, when the possible models are nested, is $mathcal{D}$-optimal with respect to an associated multiresponse model. In addition to providing a justification for the procedure, this motivates a generalization of the modified Fedorov exchange algorithm which is used to construct exact model-robust designs. We give several examples and compare our designs with two model-robust procedures in the literature. For a given set of models, the aforementioned algorithm tends to produce designs which have higher $mathcal{D}$-efficiencies for some models and lower $mathcal{D}$-efficiencies for others. To mitigate this unbalancedness, we develop a model-robust maximin exchange algorithm which maximizes the lowest efficiency over the set of models and consequently produces designs for which there is worst-case protection. Furthermore, we present a generalization of this technique which allows the user to express varying levels of interest in each model, often resulting in a design suggestive of these differences. Some asymptotic properties of this criterion are explored, including a condition which guarantees complete balance in terms of (generalized) efficiencies. We also show that even if this condition is not satisfied, this balance will be achieved in some subset of at least two models for nontrivial cases. We give several examples illustrating the procedure. Since many, if not most, experiments have multiple responses, we extend our methodology to such designs. In addition to the problem of unknown model forms, which in this case is exacerbated by the fact that there are multiple such forms to specify, the response covariance matrix is generally unknown at the design stage as well. We present an exchange algorithm for multiresponse $mathcal{D}$-optimal designs, using generalizations of matrix-updating formulae to serve as its computational engine. However, this procedure requires knowledge of the model forms, so we develop an expanded multiresponse model which allows each response to accommodate a set of possible models. The optimal design with respect to this larger model constitutes a design robust to these sets. We find, as has been noted before, that the covariance matrix is generally of little import, and it is much less consequential than the unknown model forms. We use several examples to compare the model-robust designs to designs optimal for the largest assumed model (i.e. usual practice). Finally, we consider model-robust split-plot designs using the maximin approach. Split-plot experiments are appropriate when some factors are difficult or expensive to change relative to other factors. They require two levels of randomization which induces an error structure that renders ordinary least squares analysis incorrect in general. The design of such experiments has garnered much attention over the last twenty years, and has spawned work in split-plot $mathcal{D}$-optimal designs. However, as in the case of completely randomized experiments, these procedures rely on the assumption that the form of the model relating the factors to the response is correctly specified. We relax that assumption, again by allowing the experimenter to specify a set of model forms, and use the maximin criterion to produce designs that have high $mathcal{D}$-efficiencies for each of the models in the set. Furthermore, a generalization allows the experimenter to exert some control over the efficiencies by specifying a level of interest in each model. We demonstrate the procedure with two examples.