FACTORIAL AND FRACTIONAL FACTORIAL DESIGNS WITH RANDOMIZATION RESTRICTIONS - A PROJECTIVE GEOMETRIC APPROACH Pritam Rarljan B.Stat., Indian Stat,istical Instit,ut,e,2001 hl.Stat.. Indian Statistical Iiistitut,e. 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOROF PHILOSOPHY in the Department of St8atjist'icsand Actuarial Science @ Prit.am Ranjan 2007 SIMON FRASER UNIVERSITY Summer 2007 All rights reserved. This work may not be reproduced in whole or in part,, by phot,ocopy or other mea,ns, without the permission of the a,utlior. APPROVAL Name: Pritam R.anjan Degree: Doct,or of Philosophy Title of thesis: Factsorial and Fractional Fact~orialDesigns with Ran- domization Restrictions - A Project,ive Geometric Ap- proach Examining Committee: Dr. Richard Lockhart Chair Dr. Derek Bingham. Senior Supervisor Dr. Randy Sit,t,er,Silpervisor Dr. Boxin Tang. Suptrvisor Dr. Tom Lougllin, SFU Examincr Dr. Kenny Ye! Ext,ernal Exami~~er, Albert Einst,ein College of Medicine Date Approved : SIMON FRASER UNR~ERSIW~~brary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the "Institutional Repository" link of the SFU Library website <www.lib.sfu.ca> at: <http:llir.lib.sfu.calhandle/l892/112>) and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author's written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence. The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. Simon Fraser University Library Burnaby, BC, Canada Revised: Spring 2007 Abstract Two-level factorial and fract,ional factsot-id designs have playcd a prominent role in the theory and pract,ice of experimental design. Though commonly used in indust.ria1 experiments to identify the significant effects, it is often undesimble to perform t,he trials of a. factorial design (or, fractional factorial design) in a complet,ely random order. Instmead,restrictions are imposed on tJhe randomization of experirne~it~alruns. In recent years, considerable attentlion has been devot,ed to fact(oria1and fractional fa~t~orialplans with different randomization restrict,ions (e.g., nested designs, split,-plot designs, split-split-plot designs, strip-plot designs, split-lot designs, and combinatiorls thereof). Bingham et al. (2006) proposed an approach to represent. t,he randomization structlure of factorial designs with randomization restri~t~ions.This thesis introduces a related, but more general, rcpresent,ation referred to as randomization defining con- trast subspaces (RDCSS). The RDCSS is a projective geometric f~rmulat~ionof mn- domization defining contrast subg~oups(RDCSG) defined in Bingham et al. (2006) and allows for t,heoretical st,udy. For factorial designs with different randomization struckures, the mere existence of a design is not straightforward. Here, the t'heoretical results are developed for the existence of fact,orial designs wit,h randomization restrictions within this unified framework. Our theory brings t,ogether results from finite projective geomet,ry to establish the existence and construction of such designs. Specifically, for the existence of a set of disjoint, RDCSSs, several results are proposed using (t - 1)-spreads and partial (t- 1)-spreads of PG(p- I,?). Furthermore, t'he t'heory developed here offers a sy~t~emat~icapproach for the const,ructtion of t,wo-level full factorial designs and regular fractional factsorial designs with randomization restrictions. Finally, when t,he ~ondit~ionsfor the existmemeof a set of disjoint RDCSSs are vio- lated, the data analysis is highly influenced fro111 the overlapping pat,tern among the RDCSSs. Under t,hese circumstances, a geometric structure called star is proposed for a set of (t - 1)-dimensional subspaces of PG(p - 1,q), wherc 1 < t < p. This c~periment~alplan permits the assessment of a relatively larger nnmber of fact,orial effects. The necessary and sufficient conditions for the exist,ence of stars and a collec- tion of stars are dso developed here. In particular, stars ~onstit~uteuseful designs for practitioners because of their flexith structure and easy construction. Dedication To my teachers, parents and sisters. Acknowledgments There are ma,ny people who deserve thanks for helping me in many different ways to pursue my career in academics. The last four years in this department has been an enjoyable and unforgettable experience for me. First and foremost, I cannot thank enough t,o my senior supervisor, Dr. Derek Bingham, for his support, guidance and encouragcment in every possible way. In particular, I will always be grateful t,o him for his friendship. Many thanks to my commit,tce members, Dr. Kenny Ye, Dr. Tom Loughin, Dr. Boxin Tang and Dr. Randy Sit,t,er for their useful comments and suggestions that led to significant im- provement in the thesis. I would thaak Dr. Petr Lisonck for his help on get.ting me statled in the area of Projective Geometry. Of course, the graduat,e st,udents of this depa'rt,ment play a very important role in making my stsayin this departrncnt really wonderful. Special thanks tto Chunfanp Crystal and Matk for t'heir friendship. They were there for me whenever I neede, them. I would also like t,o thank Soumik Pal and Abhyuday Mandal, my friends fror Indian Statistical Institute, for their support and encouragement which m~tivat~edm t,o do Ph.D. I would not be here without their help and support. Finally, and most importantly, I would like to thank my parents and sisters fc their support and the sa,crifics they made throughout t,he course of my st,udies. Contents .. Approval 11 Abstract iii Dedication v Acknowledgments vi Contents vii List of Tables ix List of Figures xi 1 Introduction 1 2 Preliminaries and Notations 5 2.1 Fact'orial and fractional factorial designs . 6 2.1.1 Fhct,ional factorial designs . 7 2.2 Fact(oria1and fractional fact,orial designs with randomization rest,rictions 12 2.2.1 Block designs . 12 2.2.2 Split)-plot designs . 14 2.2.3 St,rip-plot designs . 17 vii 2.2.4 Split.-lot designs .......................... 2.3 Finite projective geo~net~ricrepresentatmion ............... 2.4 Randomizat.ion re~t~rict~ionsand subspaces ............... 3 Linear Regression Model and RDCSSs 3.1 Unified Model ............................... 3.2 M~t~ivationfor disjoint RDCSSs ..................... 4 Factorial designs and Disjoint Subspaces 4.1 Existence of RDCSSs ........................... 4.1.1 RDCSSs and (t - 1)-spreads ................... 4.1.2 RDCSSs and disjoint subspaces ................. 4.2 Construction of Disjoint Subspaces ................... 4.2.1 RDCSSs and (t - 1)-spreads ................... 4.2.2 Partial (t - 1)-spreads ...................... 4.2.3 Disjoint subspaces of different sizes ............... 4.3 Fractional factmial designs ........................ 4.4 Further applications ........................... 5 Factorial Designs and Stars 5.1 Minimum overlap ............................. 5.2 Overlapping stlrategy ........................... 5.2.1 Stjars ................................ 5.2.2 Balanced stars and minimal (t - 1)-covers ........... 5.2.3 Finite galaxies ........................... 5.3 Discussion ................................. 6 Summary and Future Work Bibliography viii List of Tables 2.1 Factorial effect estimates for the chemical experiment .......... 10 2.2 The arrangement of 64 experimental units in 4 blocks .......... 13 2.3 The analysis of variance table for a split.-plot. design .......... 16 2.4 The analysis of variance table for a stripplot design .......... 19 2.5 A design matrix for a 24 full fadorial experiment............. 22 2.6 The analysis of variance table for the 2"plit.- lot example ....... 23 3.1 The ANOVA t.able for t.he 25 split.-lot design in a two-st.age process . 39 4.1 The elements of P using cyclic const.ruction............... 52 4.2 The 2-spread obtained using t.he cyclic construct.ion........... 55 4.3 The 3-spread Sf obtained aft.er applying Mnon S" ........... 59 4.4 The ANOVA t.able for the 27 full fact.orial design ............ 60 4.5 The 2-spread of PG(5, 2) aft.er transformation .............. 62 4.6 The 2-spread of PG(5, 2) after applying the ~ollineat~ionmatrix M . 64 4.7 The ANOVA table for the bat.t.ery cell experiment ........... 66 4.8 The grouping of factorial effects for t.he bat.t.ery cell experiment .... 67 4.9 The ANOVA t,able for the chemical experiment.............. 68 4.10 The grouping of effects for the chemical experiment............ 69 5.1 The ANOVA table for t.he 2"-13 split.-lot. design in a 18-stage process . 76 5.2 The dishibution of factorial effects for the batt.ery cell experiment ..