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Randomization Restrictions and the Inadvertent Split Plot in Industrial Randomization restrictions and the inadvertent split plot in industrial experimentation by Derek Frank Webb A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics Montana State University © Copyright by Derek Frank Webb (1999) Abstract: In industrial experimentation there are frequently factors which are costly or time consuming to reset from one run to the next. Given a randomized run order, the experimenter will not reset these hard to change factors when they occur at the same level in consecutive runs. The analysis is then conducted as if full randomization took place when in actuality a split plot structure has been imposed on the model. Experimental designs run with multiple hard to change factors are correctly analyzed as mixed models. Random effects and variance components are added to the models to account for the restrictions on randomization due to the presence of hard to change factors. Various methods of estimating variance components are examined and a particular method is recommended. The mixed model analysis is compared to the traditional analysis. Models with particular blocking structures which result in more cost effective experiments and less variable parameter prediction are recommended for factorial experiments with multiple hard to change factors. Response surface designs are also examined and blocking structures recommended. RANDOMIZATION RESTRICTIONS AND THE INADVERTENT SPLIT PLOT IN INDUSTRIAL EXPERIMENTATION by Derek Frank Webb A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy m Statistics MONTANA STATE UNIVERSITY-BOZEMAN Bozeman, Montana . December 1999 3 > y i t APPROVAL of a thesis submitted by Derek Frank Webb This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. Date " ^IrJjohn Borkowski Chairperson, Graduate Committee Approved for the Major Department //// Date Approved for the College of Graduate Studies Dr. Bruce McLeod Graduate Dean . Ill STATEMENT OF PERMISSION TO USE ■ In presenting this thesis in partial fulfillment, for a doctoral degree at Mon­ tana State University-Bozeman, I agree that the Library shall make it available to borrowers under rules of the'Library. I further agree that copying of this thesis is allowable only for scholarly purposes; consistent with “fair use” as prescribed in the U. S. Copyright Law. Requests for extensive copying or reproduction of this thesis should be referred to University Microfilms International, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the exclusive right to reproduce and distribute copies of the dissertation for sale in and from microform or electronic format, along with the non-exclusive right to reproduce and distribute my abstract in any format in whole or in part.” Signature Date ' I'oi }\\n\i IV ACKNOWLEDGEMENTS I* I would like to thank my advisor Dr. John Borkowski for his guidance in seeing this manuscript through from vision to completion. I would also like to thank Dr. Jim Lucas for his unrelenting energy and dedication to the practical application of statistics in industry. ' Susan, without 5 years of your support and encouragement, this would not have been possible. TABLE OF CONTENTS LIST OF TABLES . ..................................... .............................................. viii Chapter 1 AN OVERVIEW OF RESTRICTIONS ON RANDOMIZATION IN EXPERIMENTAL D E S I G N ............................................................. I 2 THE Lk FACTORIAL EXPERIMENT WITH ONE HARD-TO-CHANGE FACTOR . .............................................. ... 16 f 2.1 The Lk Factorial Experiment ...........................................' .................... 16 2.2 Hard-to-change and Easy-to-change Factors ........................................ 16 2.3 The 23 Factorial Example ....................................................................... 17 2.4 The.Expected Variance-Covariance M a trix ........................................... 22 2.5 Further Variance-Covariance Matrix R e su lts........................................ 23 2.6 Resetting the Hard-To-Change Factor.................................................... 27 3 THE Lk FACTORIAL EXPERIMENT WITH C HARD-TO-CHANGE FACTORS ..................................... 33 3.1 The Standard X and Z Design M atrices............... : - ......................... 33 3.2 The Permutation M atrix .................................................... -.................... 35 3.3 The Expected Variance-Covariance Matrix for c HTC Factors .... 40 4 4 PROPOSED 2k FACTORIAL EXPERIMENTS WITH HARD-TO-CHANGE FACTORS .......................................................... 43 4.1 The Prediction Variance of y ............... ................................................. 43 4.1.1 The Case With One Hard-To-Change F acto r............................ 45 . 4.1.1.1 The 2k Factorial Experiment With a Randomized Run Order . ..................................... 45 v vi 4.1.1.2 The 2k Factorial Experiment With Particular Blocking Structures .................................................... 51 4.1.2 The Case With c Hard-To-Change F a c to rs.......................... 65 4.1.2.1 . The 2k Factorial Experiment With a Randomized Run O r d e r ............................................................................. 65 4.1.2.2 The 2k Factorial Experiment With Particular Blocking Structures .................. 77 -4.1.3 Choosing a Run Order for the 2k Factorial Experiment .... 96 4.1.4 The Most Cost Efficient Run Order for the 2k Factorial Experiment - Extensions and Future Research.......................... 103 5 RESPONSE SURFACE DESIGNS ...........................................................108 5.1 The Expected Variance-Covariance Matrix for One HTC Factor . 109 5.2 The Expected Variance-Covariance Matrix for c HTC Factors .... 114 5.3 Blocking With the Central Composite Design ..................................... 121 5.4 Blocking With the Box-Behnken Design ..................... ' ....................... 128 5.5 Related Research and Future Work . .................................................. 135 6 THE MIXED M O D E L ................................................................................... 137 6.1 Mixed Model-Related D efinitions.......................................................... 137 6.2 The Mixed M odel...................................................................................... 138 6.3 Variance Component Estimation.............................................................. 139 6.3.1 Balanced Data Variance Component Estim ation....................... 140 6.3.1.1 The ANOVA Method .............................................. 140 6.3.2 Unbalanced Data Variance Component Estimation ............... 144 6.3.2.1 Henderson’s Methods.................................................... 144 6.3.2.2 Maximum Likelihood and Restricted Maximum Likelihood........................ 147 6.3.3 Conclusions and Recommendations . 149 vii 7 EXAMPLES . ... ............... •............. ................................. 151 7.1 A Box-Behnken design .................. .................................................... 151 . 7.2 A Second Box-Behnken design................................................. •............. I57 • 7.3 A Factorial Experiment.............................................. : ........................... 163 BIBLIOGRAPHY ........................ ............ .....................................173 LIST OF TABLES I Run orders for the 23 factorial experiment which result in distinct blocking structures. •.............................................................. 26 2 The distribution of the ZZ' matrix for the 23 factorial experiment with X i HTC............... ............................................................. 27 3 The expected number of resets of one HTC factor for a random run o rd e r............... .................................................... 31 4 The coefficients a and bi for various 2k factorial experiments . 49 5 Multipliers of of and a2 for the maximum expected prediction variance for various 2k factorial experiments........................................ 50 6 The maximum prediction variance for the 22 factorial experiment with X i being HTC.................................................................................. 60 7 The maximum prediction variance for the 23 factorial experiment with X i being HTC.................................................................................. ' 60 8 The maximum prediction variance for the 24 factorial experiment with X i being HTC................................ ................................................. 61 9 The maximum prediction variance for the 25 factorial experiment with X i being HTC................................................................................. 62 10 The maximum prediction variance for the 26 factorial experiment with X i being HTC.................................................................................. 63 11 The maximum prediction variance for the 27 factorial experiment with X i being HTC.................................................................................. 64 12 The coefficients a, bn, b^,..., biC for factorial experiments with a randomized run order and c HTC factors............................................
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