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Graphical Tools, Incorporating Cost and Optimizing Central Composite Designs for Split- Plot Response Surface Methodology Experiments

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Graphical Tools, Incorporating Cost and Optimizing Central Composite Designs for Split- Plot Response Surface Methodology Experiments GRAPHICAL TOOLS, INCORPORATING COST AND OPTIMIZING CENTRAL COMPOSITE DESIGNS FOR SPLIT- PLOT RESPONSE SURFACE METHODOLOGY EXPERIMENTS by Li Liang Dissertation submitted to the faculty of the Virginia Polytechnic Institute and State University in fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics C.M. Anderson-Cook, Co-Chair T.J. Robinson, Co-Chair E.P. Smith G.G. Vining K. Ye March 28, 2005 Blacksburg, Virginia Keywords: 3-dimensional variance dispersion graph, fraction of design space, cost penalized evaluation, multiple criteria, optimal factorial levels, central composite structure GRAPHICAL TOOLS, INCORPORATING COST AND OPTIMIZING CENTRAL COMPOSITE DESIGNS FOR SPLIT- PLOT RESPONSE SURFACE METHODOLOGY EXPERIMENTS Li Liang (Abstract) In many industrial experiments, completely randomized designs (CRDs) are impractical due to restrictions on randomization, or the existence of one or more hard-to-change factors. Under these situations, split-plot experiments are more realistic. The two separate randomizations in split-plot experiments lead to different error structure from in CRDs, and hence this affects not only response modeling but also the choice of design. In this dissertation, two graphical tools, three-dimensional variance dispersion graphs (3-D VDGs) and fractions of design space (FDS) plots are adapted for split-plot designs (SPDs). They are used for examining and comparing different variations of central composite designs (CCDs) with standard, V- and G-optimal factorial levels. The graphical tools are shown to be informative for evaluating and developing strategies for improving the prediction performance of SPDs. The overall cost of a SPD involves two types of experiment units, and often each individual whole plot is more expensive than individual subplot and measurement. Therefore, considering only the total number of observations is likely not the best way to reflect the cost of split-plot experiments. In this dissertation, cost formulation involving the weighted sum of the number of whole plots and the total number of observations is discussed and the three cost adjusted optimality criteria are proposed. The effects of considering different cost scenarios on the choice of design are shown in two examples. Often in practice it is difficult for the experimenter to select only one aspect to find the optimal design. A realistic strategy is to select a design with good balance for multiple estimation and prediction criteria. Variations of the CCDs with the best cost- adjusted performance for estimation and prediction are studied for the combination of D-, G- and V-optimality criteria and each individual criterion. Dedicated To my husband, my son and my family ii i ACKNOWLEDGEMENTS I would like to wholeheartedly thank my advisor Dr. Christine Anderson-Cook for her invaluable guidance and advice. Her encouragement, support, and thoughtfulness as a mentor and a friend are highly appreciated. I would also like to thank my co-advisor Dr. Timothy Robinson for his valuable advice and help throughout this research. I would also like to thank my committee members: Dr. Keying Ye, Dr. Geoffrey Vining, and Dr. Eric Smith for their helpful comments and suggestions. I also like to thank the professors in Statistics department for enhancing my knowledge with their excellent teaching. I would like to thank the friends at Blacksburg, with whom I spend great time in the past five years. Thanks go to Xin Zhong, Bo Jin, Ayca Ozol-Godfrey, J. D. Williams, and many others. I would like to express my sincere gratitude and deep appreciation to my beloved parents for their love, encouragement, and believing in me over all these years. I also thank my brothers Yong and Long, and my sister Yan for their love and support. I would like to deeply thank my husband Xiaopeng for his love, support, and patience, without which this work would not have been done. Thanks also go to my son Eric for the great happiness he has brought to my life. iv TABLE OF CONTENTS List of Tables............................................................................................................................................ viii List of Figures...............................................................................................................................................x Chapter 1: Introduction ............................................................................................................................ 1 1.1 Split-Plot Designs ........................................................................................................................ 2 1.1.1 Split-Plot Designs in Industry .......................................................................................... 2 1.1.2 Model and Analysis ........................................................................................................... 4 1.1.3 Optimal Split-Plot Designs ............................................................................................ 12 1.2 Design Optimality Criteria ........................................................................................................ 13 1.2.1 D-optimality....................................................................................................................... 13 1.2.2 G-optimality ...................................................................................................................... 14 1.2.3 V-optimality ..................................................................................................................... 16 1.2.4 The Role of the Total Number of Runs, N, in Optimality Criteria ....................... 16 1.2.5 Desirability Function for Combining Multiple Criteria ............................................ 17 1.3 Graphical Tools .......................................................................................................................... 19 1.3.1 Variance Dispersion Graph (VDG).............................................................................. 20 1.3.2 Three-Dimensional VDG (3-D VDG) ....................................................................... 21 1.3.3 Quantile Dispersion Graph (QDG) ............................................................................ 22 1.3.4 Fractional Design Space (FDS) ..................................................................................... 22 Chapter 2: Three-Dimensional Variance Dispersion Graphs (3-D VDGs) ................................ 24 Abstract .............................................................................................................................................. 24 2.1 Split-Plot Designs ...................................................................................................................... 24 2.2 Model and analysis ..................................................................................................................... 25 2.3 Alphabetic Optimality Criteria for Comparing Competing Designs ............................... 28 2.4 Variance Dispersion Graphs (VDGs) ................................................................................... 30 2.5 Three-Dimensional VDGs (3-D VDGs) for Split-Plot Designs ...................................... 31 2.6 Example 1 ................................................................................................................................... 35 2.6.1 V-optimal CCD ............................................................................................................... 39 2.6.2 Robustness of the V-optimal CCDs to Changes of the Optimal Factorial Levels ................................................................................................................................ 41 v 2.6.3 Robustness of the V-optimal CCDs to Changes in the Variance Component Ratio .................................................................................................................................. 42 2.6.4 G-optimal CCDs .............................................................................................................. 44 2.7 Example 2 ................................................................................................................................... 46 2.8 Example 3 ................................................................................................................................... 49 2.9 Conclusions ................................................................................................................................. 52 Chapter 3: Fraction of Design Space (FDS) Plots ............................................................................. 54 Abstract .............................................................................................................................................. 54 3.1 Introduction................................................................................................................................. 54 3.2 FDS Plots for Split-Plot Designs ............................................................................................ 59 3.3 Example 1 ................................................................................................................................... 63 3.4 Example 2 ................................................................................................................................... 69 3.5 Conclusions ................................................................................................................................
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