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Design and Analysis of Screening Experiments Assuming Effect Sparsity University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2008 Design and Analysis of Screening Experiments Assuming Effect Sparsity David Joseph Edwards University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Business Administration, Management, and Operations Commons Recommended Citation Edwards, David Joseph, "Design and Analysis of Screening Experiments Assuming Effect Sparsity. " PhD diss., University of Tennessee, 2008. https://trace.tennessee.edu/utk_graddiss/435 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by David Joseph Edwards entitled "Design and Analysis of Screening Experiments Assuming Effect Sparsity." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Business Administration. Robert W. Mee, Major Professor We have read this dissertation and recommend its acceptance: Mary G. Leitnaker, Russell L. Zaretzki, Myong K. Jeong Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by David Joseph Edwards entitled “Design and Analysis of Screening Experiments Assuming Effect Sparsity”. I have examined the final electronic copy of this dissertation for form and content and rec- ommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Business Administration. Robert W. Mee, Major Professor We have read this dissertation and recommend its acceptance: Mary G. Leitnaker Russell L. Zaretzki Myong K. Jeong Accepted for the Council: Carolyn R. Hodges, Vice Provost and Dean of the Graduate School (Original Signatures are on file with official student records.) Design and Analysis of Screening Experiments Assuming Effect Sparsity A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville David Joseph Edwards August 2008 Copyright c 2008 by David Joseph Edwards. All rights reserved. ii Dedication This dissertation is dedicated in honor of: Elizabeth, Joseph, LaVerne, Kevin, Arlett, and Lillie andinmemoryof: Wesley and Christine. iii Acknowledgments I would like to thank my advisor, Dr. Robert Mee, for the many hours you have spent teaching me how to be a better researcher and educator. Thank you for your guidance, patience, persistence, and attention to details. You have helped to provide considerable growth and insight for my professional career in academia. Thank you for always keeping your door open. Thanks also to the other members of my dissertation committee. I would like to thank Dr. Mary Leitnaker for teaching me that there is more to industrial statistics than just a set of methods. Your sense for the needs of practitioners has inspired me to always think of them when developing new methods. Thanks to Dr. Russell Zaretzki for being both a great teacher and colleague. I have enjoyed our conversations on topics ranging from research, work, to life in general. Thanks also to Dr. Myong Jeong for bringing your thoughts and insights based on your expertise in both statistics and industrial engineering. Thanks to my office mates, Artin Armagan and Maria Weese, for their friendship. Without all of the conversations, encouragements, and laughs, I would have been done with this dissertation six months ago. Finally, I want to thank all of the faculty and staff in the Department of Statistics, Operations, and Management Science for your help and support over the years. This has truly been an adventure. iv Abstract Many initial experiments for industrial and engineering applications employ screening designs to determine which of possibly many factors are significant. These screening designs are usually a highly fractionated factorial or a Plackett-Burman design that focus on main effects and provide limited information for interactions. To help sim- plify the analysis of these experiments, it is customary to assume that only a few of the effects are actually important; this assumption is known as ‘effect sparsity’. This dissertation will explore both design and analysis aspects of screening experiments assuming effect sparsity. In 1989, Russell Lenth proposed a method for analyzing unreplicated factorials that has become popular due to its simplicity and satisfactory power relative to alternative methods. We propose and illustrate the use of p-values, estimated by simulation, for Lenth t-statistics. This approach is recommended for its versatility. Whereas tabulated critical values are restricted to the case of uncorrelated estimates, we illustrate the use of p-values for both orthogonal and nonorthogonal designs. For cases where there is limited replication, we suggest computing t-statistics and p-values using an estimator that combines the pure error mean square with a modified Lenth’s pseudo standard error. Supersaturated designs (SSDs) are designs that examine more factors than runs available. SSDs were introduced to handle situations in which a large number of fac- tors are of interest but runs are expensive or time-consuming. We begin by assessing v the null model performance of SSDs when using all-subsets and forward selection regression. The propensity for model selection criteria to overfit is highlighted. We subsequently propose a strategy for analyzing SSDs that combines all-subsets regres- sion and permutation tests. The methods are illustrated for several examples. In contrast to the usual sequential nature of response surface methods (RSM), recent literature has proposed both screening and response surface exploration using only one three-level design. This approach is named “one-step RSM”. We discuss and illustrate two shortcomings of the current one-step RSM designs and analysis. Subsequently, we propose a new class of three-level designs and an analysis strategy unique to these designs that will address these shortcomings and aid the user in being appropriately advised as to factor importance. We illustrate the designs and analysis with simulated and real data. vi Contents 1 Introduction 1 2 Empirically Determined p-Values for Lenth t-Statistics 10 2.1IntroductionandMotivation....................... 10 2.2SimulationstoComputep-values.................... 14 2.2.1 Example 1: An Unreplicated Orthogonal Design . ...... 16 2.3TheImpactofCorrelatedParameterEstimates............ 17 2.3.1 Example 2: A 24 withOneObservationMissing........ 20 2.3.2 Example3:RetchshaffnerDesignforSevenFactors...... 22 2.4PureErrorDegreesofFreedom..................... 25 2.4.1 Example 4: A Fractional Factorial with Center Point Replication 32 2.5Discussion................................. 35 3 Supersaturated Designs: Are Our Results Significant? 38 3.1IntroductionandMotivation....................... 38 3.2AStrategyforAnalyzingSupersaturatedDesigns........... 50 3.2.1 More on All-Subsets vs. Forward Selection ........... 50 3.2.2 PermutationTests........................ 55 3.2.3 ProposedAnalysisStrategy................... 58 3.3Examples................................. 62 3.3.1 Williams Data ........................... 62 3.3.2 ABayesianD-OptimalSSD................... 73 3.3.3 AIDSData............................ 78 3.4Discussion................................. 81 4 A New Class of Designs for Factor Screening and Response Surface Exploration 85 4.1IntroductionandMotivation....................... 85 4.2 New Three-Level Designs for Screening and Response Surface Explo- ration................................... 96 4.2.1 Motivation............................. 96 4.2.2 Resolution III Subsets of Four Runs (Fractional Box-Behnken Designs).............................. 102 vii 4.2.3 OtherDesigns........................... 119 4.3AnalysisStrategy............................. 121 4.3.1 AnalysisApproach1....................... 125 4.3.2 AnalysisApproach2....................... 153 4.4Discussion................................. 159 5 Concluding Remarks 164 Bibliography 171 Vita 179 viii List of Tables 2.1ExamplefromHicksandTurner[40].................. 16 2.2 Example 1: Lenth t Statistics and Comparison of P-value Approxima- tions.................................... 17 2.3 α = 0.05 Critical Values and IER for m Equicorrelated Contrasts . 19 2.4 Example2-a24 withonemissingobservation............. 21 2.5Example2EstimatesandP-valueComparisons............ 21 2.6 Example 3 - Retchshaffner’s Nonorthogonal Design .......... 23 2.7 Example 3 Estimates and Empirical P-values for Saturated Model . 24 ∗ ∗ 2.8 Power Simulation [m =7,π =2.3¯/(2.3+¯ df PE), ω =2.3¯/(2.3+5¯ df PE)] 27 ∗ ∗ 2.9 Power Simulation [m = 15,π =5/(5 + df PE), ω =5/(5 + 5df PE)] . 28 ∗ ∗ 2.10 Power Simulation [m = 31,π =10.3¯/(10.3+¯ df PE), ω =10.3¯/(10.3+¯ 5df PE)]................................... 29 2.11 Power Simulation with Effects of Various Sizes [π∗ =(m/3)/((m/3) + ∗ df PE), ω =(m/3)/((m/3) + 5df PE)].................. 31 2.12 Example 4- 25−1 Designwith3CenterPointRuns..........
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