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1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2006 Efficient Mixed-Level Fractional Factorial Designs: Evaluation, Augmentation and Application Yong Guo Follow this and additional works at the FSU Digital Library. For more information, please contact

2 THE FLORIDA STATE UNIVERSITY FAMU-FSU COLLEGE OF ENGINEERING EFFICIENT MIXED-LEVEL FRACTIONAL FACTORIAL DESIGNS: EVALUATION, AUGMENTATION AND APPLICATION by YONG GUO A Dissertation submitted to the Department of Industrial and Manufacturing Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2006 Copyright 2006 Yong Guo All Rights Reserved

3 The members of the Committee approve the dissertation of Yong Guo defended on April 7, James R. Simpson Professor Directing Dissertation Xufeng Niu Outside Committee Member Samuel A. Awoniyi Committee Member Joseph J. Pignatiello, Jr. Committee Member Approved: Chuck Zhang, Chair, Department of Industrial Engineering Ching-Jen Chen, Dean, FAMU-FSU College of Engineering The Office of Graduate Studies has verified and approved the above named committee members. ii

4 ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Dr. James Simpson, for his support, patience, guidance and encouragement throughout my graduate studies. I am so proud of him as my advisor, professor, boss and friend. It is not often that one finds an advisor that always finds the time for listening to the little problems and roadblocks that unavoidably crop up in the course of performing research. His technical and editorial advice was essential to the completion of this dissertation and has taught me innumerable lessons and insights on the workings of academic research in general. My thanks also go to the member of my committee member, Professor Samuel Awoniyi for supporting and helping me during my time in FAMU-FSU College of Engineering. I would like to thank Joseph Pignatiello for reading previous drafts of this dissertation and providing many valuable comments that improved the presentation and contents of this dissertation. I also thank Dr. Niu from Statistics Department for his advisement and comments for this dissertation. Micelle Zeisset is much appreciated for her truly friendship during my graduate studies. I am also grateful to my friend Wayne Wesley for staying with me in prelim exam and dissertation writing. Lisa, Francisco and Rupert in Quality Lab are appreciated and have led to many interesting and good-spirited discussions relating to this research. I would like to thank Irene, Charlie, Marcus, Noah, David, Bernie, Faqing, Fangyu for their friendship along this journey. Their encouragement was in the end what made this dissertation possible. Last, but not least, I thank other friends in the Department of Industrial Engineering and all the friends at the Rogers Hall basketball court. My parents receive my deepest gratitude and love for their supporting and understanding during my study on abroad that provided the foundation for this work. iii

5 TABLE OF CONTENTS LIST OF TABLES... vi LIST OF FIGURES...viii ABSTRACT... ix CHAPTER 1 INTRODUCTION Motivation Problem Statement Research Objective... 6 CHAPTER 2 A REVIEW OF SOME MINIMUM ABERRATION CRITERIA FOR EVALUATING FRACTIONAL FACTORIAL DESIGNS Introduction The Usual Minimum Aberration Criterion for Two-Level Designs Minimum Aberration Criterion for Designs with Two- and Four-Level Factors Other Minimum Aberration Criterion Definitions Minimum G-Aberration Criterion Generalized Minimum Aberration Criterion Minimum Moment Aberration Criterion Moment Aberration Projection Criterion Additional Minimum Aberration Definitions Conclusion CHAPTER 3 THE GENERAL BALANCE METRIC FOR FRACTIONAL FACTORIAL DESIGNS Introduction General Near-balanced Designs Features of General Balance Metric More Examples Conclusion iv

6 CHAPTER 4 OPTIMAL FOLDOVER PLANS FOR MIXED-LEVEL FRACTIONAL FACTORIAL DESIGNS Introduction Foldover Strategy for Multi-Level Factors General Balance Metric Optimal Foldovers of Mixed-Level Designs Conclusion CHAPTER 5 ANALYSIS OF MIXED-LEVEL EXPERIMENTAL DESIGNS INCLUDING QUALITATIVE FACTORS Introduction Indicator Variables for Qualitative Factors Contrast Coefficients for Qualitative Factors Classic Regression Models for Mixed-Level Designs First-Order Models First-Order Models with Interactions Second-Order Models Interactions between Qualitative Factors Regression Analysis on Fire Fighting Data Conclusion CHAPTER 6 GENERAL CONCLUSIONS AND FUTURE RESEARCH APPENDICES A. Efficient Mixed-Level Designs B. MATLAB Codes C. Glossary REFERENCES BIOGRAPHICAL SKETCH v

7 LIST OF TABLES Table 1 Full Factorial Design Table 2 Treatments Combinations for 2 Factors with 2 Levels Each... 4 Table 3 A 8-Run Mixed-Level Design... 5 Table 4 Three design options Table 5 A Mixed-Level Design with One Four-Level Factor and Four Two-Level Factors Table 6 Three Design Options Table 7 The 12-Run Plackett-Burman Design Table 8 Plus and Minus Signs for Calculating Jk ( d ) Table 9 Comparison of Two Designs Using MGA Table 10 Comparison of Two Designs Using MG 2 A Table 11 Calculation of GMA for d Table 12 Comparison of Designs in Example 2 Using GMA Table 13 Comparison of Designs in Example 1 Using MMA Table 14 Comparison of Designs in Example 2 Using MMA Table 15 All k-factor Projections and Their K-Values for d Table 16 All k-factor Projections and Their K-Values for d Table 17 Frequency Distribution of K k -Values of Factor Projections for Example Table 18 Frequency Distribution of K k -Values of Factor Projections for Example Table 19 Features of Minimum Aberration Criteria Table 20 Relationship Summery between Minimum Aberration Criteria Table 21 An Example of Coding Mixed-Level Factor Interactions Table 22 A Mixed-Level Design (6, ) Table 23 Three Design Options Table 24 Comparison of Three Design Options Table 25 Comparison of Two OA(12, 2 4 3) Designs Table 26 Comparison of Three OA(12, 2 4) designs Table 27 A Mixed-Level Design (6, ) Table 28 All Foldover Alternatives for Design (6, ) vi

8 1 1 Table 29 Statistics of EA(15, ) Table 30 Comparison of the Combined Design with EA(30, ) Table 31 Optimal Foldover Plans for EAs in terms of GBM and B Table 32 Decomposition of Factor A Using Contrast Coefficients Table 33 Two Orthogonal Decomposition Options Table 34 Four Fire Fighting System Methods Table 35 Descriptive Statistics of Factors and Responses Table 36 Contrast Coefficients for Surface and Method Table 37 Analysis of Variance of the Fire Data (All Terms) Table 38 Analysis of Variance of the Fire Data (Significant Terms) Table 39 Model Terms Estimation vii

9 LIST OF FIGURES Figure 1 The components of a process Figure 2 Application of different minimum aberration criteria Figure 3 Two (12, 2 ) designs Figure 4 Two OA(12, 2 3) designs Figure 5 Coincidence matrices for Example Figure 6 Balance relationship among types of columns Figure 7 Two OA(12, 2 3) designs Figure 8 Three OA(12, 2 4) designs Figure 9 Rotate a five-level factor Figure 10 EA(15, ) Figure 11 EA(30, ) Figure 12 Interaction of qualitative factors A and B Figure 13 Plot of extinguishment time verse method Figure 14 Plot of extinguishment time verse surface Figure 15 Probability plot of extinguishment time Figure 16 Probability plot of transformed extinguishment time viii

10 ABSTRACT In general, a minimum aberration criterion is used to evaluate fractional factorial designs. This dissertation begins with a comprehensive review and comparison of minimum aberration criteria definitions regarding their applications, relationships, advantages, limitations and drawbacks. A new criterion called the general balance metric, is proposed to evaluate and compare mixed-level fractional factorial designs. The general balance metric measures the degree of balance for both main effects and interaction effects. This criterion is related to, and dominates orthogonality criteria as well as traditional minimum aberration criteria. Besides, the proposed criterion provides immediate feedback and comprehensively assesses designs and has practical interpretations. The metric can also be used for the purpose of design augmentation to improve model fit. Based upon the proposed criterion, a method is proposed to identify the optimal foldover strategies for efficient mixed-level designs. The analysis of mixedlevel designs involving qualitative factors can be achieved through indicator variables or contrast coefficients. A regression model is developed to include qualitative factor interactions which have been previously ignored. ix

11 CHAPTER 1 INTRODUCTION An experiment is a test or series of tests conducted under controlled conditions made to demonstrate a known truth, examine the validity of a hypothesis, or determine the efficacy of action previously untried. In an experiment, one or more input process variables are changed deliberately in order to observe the effect that changes have on one or more response variables. Experiments are performed a number of times in order to evaluate the output response variables under the different input process variable conditions. The design of experiments is an efficient method for planning experiments so that the data obtained can be analyzed to yield valid and objective conclusions. The method for conducting designed experiments begins with determining the objectives of an experiment and selecting the process factors for the study. A designed experiment requires establishing a detailed experimental plan in advance of conducting the experiment, which results in a streamlined approach in the data collection stage. Appropriately choosing experimental designs maximizes the amount of information that can be obtained for a given amount of experimental effort. Experimental designs are used to investigate industrial systems or processes. A typical process model is given in Figure 1. Purposeful changes are made to the controllable input factors of a process so as to observe and identify the reasons for changes that may be observed in the output responses. The noise factors are considered as random effects that cannot be controlled. Experimental data are used to derive a statistical empirical model linking the outputs and inputs. These empirical models generally contain first and second-order terms. For more information regarding the statistical empirical model, see Montgomery (2005). 1

12 Noise Factors Controllable Process Output Responses Input Factors Figure 1 The components of a process. Many experiments involve the study of the effects of two or more factors on one or more output responses. Full factorial designs are test matrices that contain all possible combinations of the levels of the factors. For example, if factor A has a levels and factor B has b levels, then the two-factor full factorial design contains ab combinations. Table. 1 shows another example, a full factorial design with three factors: one with two levels, one with three and one with five. The shorthand notation for this design is ( ), which displays the factor levels as the base numbers and the number of factors with that many levels as the exponent. One purpose of factorial designs is to study the effects of these factors on the response. The main effect of a factor is defined to be the change in response produced by a change in the level of the factor. The term main effect is used because it refers to the primary factors of interest in the experiment. A main effect reflects the individual impact of each factor. One-factor-at-a-time testing is an extensively used experimentation strategy. This method consists of selecting a starting point setting for each factor, then successively varying the settings of each factor over its range, with the other factors held constant (Montgomery, 2005). Compared with one-factor-at-a-time experiments, factorial designs are more efficient. Factorial designs allow the effects of a factor to be estimated at several levels of the other factors because the difference in response between the levels of one factor may not be the same at all levels of the other factor. Therefore, factorial designs are necessary when interactions may be present. 2

13 Table. 1Full Factorial Design Run Factor A Factor B Factor C A specific case of general factorial designs is the 2 k factorial design. That is, these designs have k factors, each at only two levels. These levels may be either quantitative or qualitative. Normally + is used to represent the high level and is used to represent the low level in the 2-level factorial designs. A complete replicate of such a design 3

14 requires 2 k observations and is called a 2 k factorial design. Table 2 gives an example for k=2 in three replicates. Table 2. Treatments Combinations for 2 Factors with 2 Levels Each Factor Treatment Replicate A B Combination I II III A low, B low A high, B low A low, B high A high, B high The interaction effect AB is defined as the average change in response between the effect of A at the high level of B and the effect of A at the low level of B. The methods used for generating 2 k factorial designs are straightforward. Each column represents a factor. The levels for the first column follow The levels for the second column follow the pattern of For the n th column, the pattern will be + + and the number of minus signs and plus signs is n for each. Many experimental design textbooks and software packages emphasize the use of factorial and fractional factorial designs in which all factors in the experiment have two levels, often called 2 k-p designs, where k is the number of factors, p is the degree of fractionation, and 2 k-p is the number of runs. It is true that technological experiments often have only quantitative factors; however, it is not uncommon for technological experiments to also include factors that are qualitative in nature. There are often more than two levels of such factors. In order to include factors that have more than two levels, mixed-level designs are used, which have become more practically used in the field of design of experiments. For example, an experimental design (Table 3) considers five factors: four factors with two levels and one factor with four levels. 4

15 Table 3. A 8-Run Mixed-Level Design Two-level factors Four-level factor Run A B C D E Motivation The full factorial mixed-level design could be very large in terms of run number, depending on the number of factors and the factor levels. For example, a mixed-level design considers three factors: a three-level factor, a five-level factor, and a seven-level factor. The full factorial design contains a total of 105 runs. Therefore, it may be desirable to use fractional factorial mixed-level designs instead of the full factorial. Some mixed-level designs are available in the literature. Orthogonal and near orthogonal mixed-level designs are discussed by Sloane (2006) and Xu (2002). In the case that balanced designs are not available, a good solution then is to use near-balanced efficient mixed-level designs (Guo, Simpson, and Pignatiello 2005). However, different fractions from a full factorial may have the same balance and orthogonality property. An important consideration is how to further select the best fractional factorial mixed-level designs. In situations where we have little or no knowledge about the effects that are potentially important, it is appropriate to use the minimum aberration criterion. 1.2 Problem Statement The usual definition of minimum aberration (MA) criterion for regular two-level designs was introduced by Fries and Hunter (1980). A type of mixed-level design, 4 m 2 n-p, can be developed by using the usual MA criterion (Wu and Hamada 2000, Ankenman 1999, and Montgomery 2005). Even though the usual definition of minimum aberration 5

16 works well for designs with two-level factors and four-level factors, it is not possible to extend this usual definition to other applications, such as two-level non-regular designs, multi-level designs and mixed-level designs, since the principle of the usual MA is based upon design generators. Therefore, new MA definitions have been developed to meet these requirements. Some definitions include the minimum G-aberration criterion (Deng and Tang 1999), the minimum G 2 -aberration criterion (Tang and Deng 1999, Ingram and Tang 2005), the generalized minimum aberration criterion (Xu and Wu 2001), the minimum moment aberration criterion (Xu 2003), the moment aberration projection (Xu and Deng 2005), the minimum generalized aberration criterion (Ma and Fang 2001), and a general criterion of minimum aberration (Cheng and Tang 2005). In general, the currently existing minimum aberration criteria are complicated and not easy to apply in industrial situations. 1.3 Research Objectives The first objective of this dissertation is to review the existing minimum aberration criteria. Examples including non-regular two-level designs and mixed-level designs are used for comparing these criteria. The goal is to introduce these minimum aberration criteria to practitioners with practical examples so that the practitioners can know the relationships, advantages and drawbacks of these minimum aberration criteria. The second objective is to develop a new minimum aberration criterion, the general balance metric, for mixed-level fractional factorial designs. The performance of this criterion will be compared with other criteria. The third objective is to fold over efficient mixed-level designs using the general balance metric. The purpose is to decompose aliased model terms. With this method, find the optimal foldover plans for given mixed-level designs via algorithms. Finally, provide optimal foldover plans for existing efficient mixed-level designs. The fourth objective is to analyze mixed-level designs involving qualitative factor interactions via contrast coefficients. The goal is to analyze qualitative factor interactions from the point of view of model building and to propose a regression model that includes qualitative factor interactions. 6

17 This dissertation is structured as follows. Chapter 2 reviews the existing minimum aberration criteria. Chapter 3 develops a new criterion, called general balance metric. Chapter 4 proposes a method to fold over mixed-level fractional factorial designs. Chapter 4 identifies qualitative factor interactions and develops a regression model to incorporate qualitative factor interactions. Finally, general conclusions from these research topics will be discusses in Chapter 6. 7

18 CHAPTER 2 A REVIEW OF MINIMUM ABERRATION CRITERIA FOR EVALUATING FRACTIONAL FACTORIAL DESIGNS 2.1 Introduction In the last 20 years, significant attention has been paid on developing new minimum aberration criteria and constructing minimum aberration designs using those criteria. The concept of minimum aberration was first introduced by Fries and Hunter (1980) as a way of selecting the best two-level fractional factorial designs from those with equal maximum resolution. The resolution for two-level fractional factorial designs was proposed by Box and Hunter (1961). Minimum aberration designs have the best alias structure and possess robust properties (Cheng, Steinberg and Sun 1999 and Tang and Deng 1999). Regular two-level designs indicate those designs who are constructed by design generators (Motgomery 2005). Regular two-level designs are denoted by 2 m-q, and have simple alias structures. Non-regular designs have more flexible design sizes than regular designs, but their alias structures are more complicated. Examples of non-regular designs are Plackett-Burman designs (Deng and Tang, 1999) and supersaturated designs (Xu 2003, Xu and Wu 2005). Since the usual minimum aberration criterion definition can not be applied directly to non-regular designs, it was necessary to develop new minimum aberration definitions for evaluating non-regular designs. Extensive work on non-regular two-level fractional factorial designs was developed by Chen and Hedayat (1996), Chen (1992), Bingham and Sitter (1999), Sitter, Chen and Feder (1997), Huang, Chen and Voelkel (1998), Tang and Wu (1996), Ma and Fang (2001), Wu and Zhu (2003), Cheng and Tang (2005) and Xu and Deng (2005). 8

19 As an extension of two-level fractional factorial designs, Franklin (1984) and Suen, Chen and Wu (1997) discuss the construction of multi-level minimum aberration designs. Xu and Wu (2001) proposed a generalized minimum aberration for mixed-level (asymmetrical) fractional factorial designs. Wu and Zhang (1993) and Ankenman (1999) used minimum aberration designs in two-level and four-level mixed-level designs. Mukerjee and Wu (2001) developed minimum aberration designs for mixed-level fractional factorial designs involving factors with two or three distinct levels. In the following section, the usual minimum aberration criterion for two-level designs is briefly reviewed. Then the application of this criterion to special cases of mixed-level designs with two- and four-level factors is discussed. In the subsequent section, other proposed minimum aberration criteria are reviewed and compared via examples. The last section discusses the conclusions and suggestions. For the reader s convenience, we use consistent notation throughout the paper. A complete glossary can be found in Appendix C The Usual Minimum Aberration Criterion for Two-Level Designs A two-level 2 m-q design is defined to be a fractional factorial design with m factors, each at two levels, consisting of 2 m-q runs. Therefore, it is a 2 -q fraction of the 2 m full factorial design in which the fraction is determined by q generators, where a generator consists of letters which are the names of the factors denoted by A, B,. The number of letters in a word is its word length and the word formed by the q defining words is called the defining relation. For a 2 m-q design, let A ( ) contrast subgroup. The vector k d be the number of words of length k in the defining ( 1, 2,, m ) ( ) = ( ) ( ) ( ) W d A d A d A d is called the word length pattern of the design d (Fries and Hunter, 1980). The resolution of a 2 m-q design, R, is defined to be the smallest r such that A ( ) r d 1, that is, the length of the shortest word in the defining contrast subgroup. For any two 2 m-q designs d 1 and d 2, let r be the smallest integer such that A ( d ) A ( d ) r 1 r 2. Then d 1 is said to have less 9

20 aberration than d 2 if A ( d ) A ( d ) <. If no design exists with less aberration than d 1, r 1 r 2 then d 1 has minimum aberration. Consider a experiment, with three design options. Table 4 provides the design generators for three designs along with their defining relations. In this example, d 3 has less aberration than d 1 or d 2 because the first unequal number in word length pattern is in the fourth position and d 3 has the smallest number in that position. Design d 3 is the minimum aberration design. Other 2 m-q minimum aberration designs and their design generators are presented in Montgomery (2005). Montgomery (2005) gives a slightly different formatted word length pattern from Wu and Zhang (1993) s. Instead of using numbers of words of length k in the defining contrast subgroup, Montgomery (2005) directly shows the length of each word in the defining contrast group (Table 4). Table 4. Three design options Design Options d 1 d 2 d 3 Generators F=ABC, G=BCD F=ABC,G=ADE F=ABCD, G=ABDE Defining Relations I=ABCF=BCDG=ADFG I=ABCF=ADEG=BCDEFG I=ABCDF=ABDEG=CEFG WLP Wu and Zhang (0, 0, 0, 3, 0, 0) (0, 0, 0, 2, 0, 1) (0, 0, 0, 1, 2, 0) (1993) WLP Montgomery (2005) {4, 4, 4} {4, 4, 6} {4, 5, 5} 2.3 Minimum Aberration Criterion for Designs with Two- and Four- Level Factors The application of the usual minimum aberration (MA) criterion can be expanded for designs other than the regular two-level designs using certain schemes. The literature (Addelman (1962), Wu and Hamada (2000), Ankenman (1999) and Montgomery (2005)) 10

21 shows that multi-level factors can be replaced by two-level factors, thereby taking advantage of two-level fractional factorial design alternatives. This method was successfully used for four-level factors. Wu and Hamada (2000) proposed a formal procedure for replacing four-level factors with two-level factors. The idea is to replace one four-level factor with three two-level factors by the following rule. X A B AB Now consider an experiment with five factors, one with four levels, and four with two levels. The full factorial contains 64 runs, but an 8-run fractional is of interest. This design (Table 5) illustrates that although only two, two-level factors are used to replace a four-level factor, the interaction of these two two-level factors is also used. Thus, this four-level factor is replaced by three single degree-of-freedom two-level factors. Table 5. A Mixed-Level Design with One Four-Level Factor and Four Two-Level Factors Run X 1 X 2 X 1 X 2 B C D E Run X B C D E Wu and Zhang (1993) present three designs (one four-level factor A mixed with four two-level factors, B, C, D, and E). Let 1, 2, 3, 4 be four columns of the 2 4 full factorial design. Let (1, 2, 1 2) be the four-level factor A. Factor B uses column 3 and C is column 4. Then factors D and E are formed from combinations of columns 1, through 4 11

22 according to three different schemes (Table 6). Those schemes result in different aberration scores for each design. Among all three designs, design d 1 has minimum aberration because design A3( d 1) =1 but A3( d2) A3( d3) aberration than d 3 since A ( d ) A ( d ) = 0< = = =2. Furthermore, d 2 has less Table 6. Three Design Options A A 1 A 2 A 3 B C D E Defining Relations WLP d I = A 1 BCD = A 2 BE = A 3 CDE (0, 0, 1, 2, 0) d I = A 1 CD = A 2 BE = A 3 BCDE (0, 0, 2, 0, 1) d I = A 3 BDE = BCE = A 3 CD (0, 0, 2, 1, 0) 2.4 Other Minimum Aberration Criterion Definitions Even though the usual definition of minimum aberration (MA) works well for designs with two-level factors and four-level factors, it is difficult to extend this usual definition to other applications, such as two-level non-regular designs, multi-level designs and mixed-level designs, since the principle of the usual MA is based upon design generators. Therefore, new MA definitions have been developed to meet these requirements. Some definitions include the minimum G-aberration criterion (Deng and Tang (1999)), the minimum G 2 -aberration criterion (Tang and Deng (1999), Ingram and Tang (2005)), the generalized minimum aberration criterion (Xu and Wu (2001)), the minimum moment aberration criterion (Xu (2003)), the moment aberration projection (Xu and Deng (2005)), the minimum generalized aberration criterion (Ma and Fang (2001)), and a general criterion of minimum aberration (Cheng and Tang (2005)). The application of these MA criteria is shown in Figure 2. 12

23 Mixed-level designs Multi-level designs Non-regular two-level designs Regular two-level designs Usual minimum aberration Minimum G- aberration Minimum G 2 - aberration Generalized minimum aberration Minimum moment aberration Moment aberration projection Figure 2 Application of different minimum aberration criteria. Minimum generalized aberration criterion General criterion of minimum aberration To facilitate explanation, it is useful to work with example designs that cannot be evaluated nor compared using the usual MA. Two examples are considered for the reminder of this discussion. Example 1 Consider the 12-run Plackett-Burman design given in Table 7. Two sub-designs can be formed from this base design. The first design d 1 contains columns 1-4 and 10, second design d 2 uses only columns 1-5 (Xu and Deng (2005)). 13

24 Table 7. The 12-Run Plackett-Burman Design Run Figure 3 gives these two sub-designs (12, 2 5 ), representing 5 two-level factors in 12 runs. In general, k1 k2 k ( nl, 1 l T 2 l T ) denotes an n-run fractional factorial design, involving k 1 factors with l 1 levels, k 2 factors with l 2 levels so and so on. d 1 d 2 A B C D E A B C D E Figure 3. Two (12, 2 5 ) designs. 14

25 Example 2 A second example considers two mixed-level designs (Figure 4). Each design has 5 factors in a total of 12 runs. Factor A has three levels and factors B, C, D, and E has two levels each. The factor levels are coded as 1, 2, 3,, representing the first-, second-, third- level of that factor. A B C D E A B C D E d 3 d 4 Figure 4 Two OA(12, 2 4 3) designs Minimum G-Aberration Criterion The first MA criterion is the minimum G-aberration (MGA) criterion for two-level designs, proposed by Deng and Tang (1999). Ingram and Tang (2005) used this criterion to construct two-level designs with 24 runs. For an experimental design matrix, d, let n represent the number of rows (runs) and m be the number of columns (factors). Let [ ] s= c1, c2,, ck represent a k-column subset matrix from d and c ij is the i th element of column c j. Define n ( ) 1 2 J d = c c c for k = k 1,, m. k i i i i= 1 15

26 Jk ( d) corresponds to all k-factor interactions, 1 through m. Let F ( d ) be the vector that contains the frequencies of the different J ( d ) k values for design d. Define the confounding frequency vector of d as F( d) F ( d) F ( d) F ( d) r = 1, 2,, m ( ) ( ). If ( ) < F ( d ) F d F d 1 r 2 r 1 r 2. Let r be the smallest number for which F d, d 1 is said to has less G-aberration than d 2. This criterion was developed strictly for two-level designs. MGA can be used to compare the designs in Example 1. First, calculate plus and minus signs for interactions by multiplying the appropriate preceding columns, row by J ( d ) ( ) row. The calculation of is shown in Table 8. It is found that J d has three k possible values, (8, 4, 0), listed in a decreasing order. So 1 Fk ( d ) of J ( d ) =, followed by frequencies of J ( d ) =, and ( ) 0 k 1 8 k k 1 contains the frequencies J d. For the main k 1 = effects and two-factor interaction effects, J ( d ) J ( d ) = =. Therefore, F1( d 1) = ( 0,0,5) and F2( d 1) = ( 0, 0,10). That is all five J1( d 1) are equal to 0 for main ( ) effects and all ten J d are equal to 0 for two-factor interactions. For three-factor and 2 1 four-factor interactions, all F4( d 1) = ( 0,5,0) J ( d ) 4 1 ( ) ( ) so F5 d 1 = 1, 0, 0.. That is all ten J ( d ) k ( d ) J3( d 1) and J4( d 1) equal 4, so 3( 1) = ( 0,10, 0) F d and in three-factor interactions equal to 4 and all five in four-factor interactions is 4. For five-factor interactions, the only J d is 8, In a similar way, J is calculated and there are the same possible numbers (8, J d are 4, 0). The frequencies of the different ( ) k 2 ( ) ( 0,0,5 ),( 0,0,10 ),( 0,10,0 ),( 0,5,0), ( 0,0,0 ) F d 2 = The comparison of two designs is given in Table 9. It can be seen that design d 2 is better ( ) ( d ) since J d =0, but J = ( ) 5 1 k 16

27 17 Run A B C D E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE ABCDE cc i1 i2 cik i= 1 ( ) 12 J d = c c c k i1 i2 ik i= 1 Table 8. Plus and Minus Signs for Calculating J ( d ) k

28 Table 9. Comparison of Two Designs Using MGA MGA Frequencies of J-values (8, 4, 0) d 1 [(0, 0, 5) 1, (0, 0, 10) 2, (0, 10, 0) 3, (0, 5, 0) 4, (1, 0, 0) 5 ] d 2 [(0, 0, 5) 1, (0, 0, 10) 2, (0, 10, 0) 3, (0, 5, 0) 4, (0, 0, 0) 5 ] The use of a frequency vector may not be convenient when large computations are involved. The minimum G 2 -aberration (MG 2 A) criterion is proposed by Tang and Deng (1999) as a relaxed version of MGA. Define where s B k ( d) ( d) 2 J, for k=1,, m, k = s= k n = k indicates that the summation covers all subset matrices with k columns. The criterion of MG 2 A sums the square of the normalized Jk ( d ) for k =1,, m. For two designs d 1 and d 2, suppose that r is the smallest integer such that r ( ) ( ). Then d B d B d2 1 is said to have less G 2 -aberration than d 2 if B d < B d. 1 r Now, apply MG 2 A for the designs in Example 1. For both d 1 and d 2, B 3 ( d) B 1 ( d) 4 1 = 10 = = 5 = , B ( d) 2 0 = 10 = =1.111 and ( ) 2 B 4 d 4 5 = 5 = = r ( ) ( ) 1 r 2 However, for 5-factor interactions, the two designs diverge with different B 5 -values, which are, B ( d ) = = =0.444 and ( ) B d 5 2 = = A summary of the results (Table 10) shows that d 2 is better than d 1 by MG 2 A. 18

29 Table 10. Comparison of Two Designs Using MG 2 A B d, B d, B d, B d, B d MG 2 A ( 1( ) 2( ) 3( ) 4( ) 5( )) d 1 (0, 0, 1.111, 0.556, 0.444) d 2 (0, 0, 1.111, 0.556, 0) The MG 2 A criterion is equivalent to the usual MA for evaluating regular designs. The MG 2 A is computationally much easier than the MGA because it uses a single number for each interaction, instead of a frequency vector. As a result, MG 2 A can be helpful for comparing or evaluating large designs. The MG 2 A criterion can be further generalized into the minimum G e -aberration criterion (Ingram and Tang (2005)). That is B k ( d) ( s) e J. k = s= k n With values of e larger than 2, more emphasis is put on lower order interactions Generalized Minimum Aberration Criterion Xu and Wu (2001) proposed a generalized minimum aberration (GMA) criterion for multi-level and mixed-level designs. For a design d, the ANOVA model has the following form Y = X α + X α + + X α + ε, m m ( ) where Y is the response, α k is the vector of all k-factor interactions and X k = x k ij is the matrix of contrast coefficients forα. Let The A ( ) k ( 1 2 k n 2 2 ( k ij. j i= 1 A ( d) n x k = ) d are invariant with respect to the choice of orthogonal contrasts. The vector A ( d), A ( d),, Am ( d) ) is called the generalized word length pattern. Then the generalized minimum aberration criterion is to sequentially minimize A ( ) m. k d for k=1,, 19

30 For two-level fractional factorial designs, the criterion of GMA is equivalent to MG 2 A in mathematical form. Since the two designs in Example 1 are two-level designs, GMA will give the exact same results as MG 2 A. For the mixed-level designs from Example 2. Normalized orthogonal polynomials are used as the contrast coefficients Factor Level Contrast Coefficient Factor Level Contrast Coefficient Table 11 shows the model matrix of X 1 and X 2, and the calculation of the corresponding A 1 and A 2 for design d 3. A comparison of d 3 versus d 4 using GMA is given in Table 12. Since A1( d 3) = A1( d 4) = 0 but A2( d3) A2( d4) Therefore, d 4 is better than d 3 by the GMA. >, design d 4 has less aberration than d 3. 20

31 Table 11. Calculation of GMA for d 3 X 1 A 1 A 2 B C D E 21 Sum A 1 (d 3 ) ( ) / 12 2 = 0 X 2 A 1 B A 1 C A 1 D A 1 E A 2 B A 2 C A 2 D A 2 E BC BD BE CD CE DE Sum A 2 (d 3 ) ( ) / 12 2 =

32 Table 12. Comparison of Designs in Example 2 Using GMA A d, A d, A d, A d, A d GMA ( 1( ) 2( ) 3( ) 4( ) 5( )) d 3 (0, , , ) d 4 (0, , , ) Minimum Moment Aberration Criterion The MGA, MG 2 A, and GMA criteria all require contrast coefficients of factors. Xu (2003) developed a minimum moment aberration criterion (MMA), which does not need contrast coefficients. For a design matrix d, let d ij be the elements of i th row and j th column. The coincidence between two elements d ij and d lj is defined by δ ( d, d ), where δ ( d, d ) = 1 if d ij = d lj and 0 otherwise. The value of δ d, d measures the ij lj m ( ij lj ) coincidence between the i th and l th rows of d. The k th power moment is defined by Xu (2003) as j= 1 m 1 Kk( d) = [ n( n 1)/2 ] δ ( di j, dlj) 1 i l n j= 1. For two designs d 1 and d 2, d 1 is said to have less moment aberration than d 2 if there exists an r such that Kr( d1) < Kr( d2) and Kt( d1) = Kt( d 2 ) for all t=1,, r-1. Therefore, d 1 is said to have minimum moment aberration if there is no other design with less moment aberration than d 1. Figure 5 is the coincidence matrix for the designs in Example 1, where the (ith, jth) element indicating the coincidence between the ith row and jth row. Since the lower triangular matrix is symmetric to the upper triangular matrix, only the elements in upper triangular matrix are used for calculation. Table 13 gives the calculation results, which shows that design d 2 is better than d 1. k ij lj 22

33 Run (a) Coincidence Matrix for d 1 Run (b) Coincidence Matrix for d 2 Figure 5 Coincidence matrices for Example 1. Table 13. Comparison of Designs in Example 1 Using MMA K d, K d, K d, K d, K d MMA ( 1( ) 2( ) 3( ) 4( ) 5( )) d 1 (2.3, 5.9, 16.8, 51.4, 167.7) d 2 (2.3, 5.9, 16.8, 51.4, 165.9) 23

34 The application of MMA can also cover multi-level or mixed-level designs. For designs in Example 2, design d 4 is better than d 3 with criterion MMA (Table 14). MMA investigates the relationship between runs (rows) instead of factors (columns). Therefore, MMA can be used for any design and it is computationally quick. Although MMA has the capability to discriminate among designs, the k-th power moment in the MMA definition is not related to k-factor interactions. Table 14. Comparison of Designs in Example 2 Using MMA K d, K d, K d, K d, K d MMA ( 1( ) 2( ) 3( ) 4( ) 5( )) d 3 (2.1, 5.3, 14.8, 43.9, 134.8, 427.1) d 4 (2.1, 5.0, 13.0, 35.9, 103.9, 312.3) Moment Aberration Projection Criterion In order to address the drawback that k-th power moment is not corresponding to k- factor interactions, Xu and Deng (2005) proposed a criterion, called the moment aberration projection (MAP). MAP uses the coincidence matrix for all factor projections. For a given k m (1 k m), there are k-factor projections. The frequency distribution of Kk-values of k these projections is called the k-dimensional K-value distribution and is denoted by F k (d). For two designs d 1 and d 2, suppose that r is the smallest integer such that the r-dimensional K-value distributions are different, that is, F r (d 1 ) F r (d 2 ). Hence, d 1 is said to have less MAP than d 2 if F r (d 1 ) < F r (d 2 ). Moreover, the criterion of MAP was developed for twolevel non-regular designs and it also can be used in multi-level and mixed-level designs. For the designs in Example 2, the one-, two-, three-, four-, five-factor projections and their K-values are shown in Table 15 and Table 16. The summarized K-value frequency distributions are given in Table 17. According to F 2, design d 4 is better than d 3. MAP values are also generated for Example 1 (Table 18) to compare designs d 1 and d 2. 24

35 Table 15. All k-factor Projections and Their K-Values for d 3 k-factor Projections K-Value k=1 A B C D E K k=2 AB AC AD AE BC BD BE CD CE DE K k=3 ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE K k=4 ABCD ABCE ABDE ACDE BCDE K k=5 ABCDE K Table 16. All k-factor Projections and Their K-Values for d 4 k-factor Projections K-Value k=1 A B C D E K k=2 AB AC AD AE BC BD BE CD CE DE K k=3 ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE K k=4 ABCD ABCE ABDE ACDE BCDE K k=5 ABCDE K Table 17. Frequency Distribution of K k -Values of Factor Projections for Example 2 k-factor Projections: (K-Values) Frequency Distribution d 3 d 4 F 1 : (30, 18) (4, 1) (4, 1) F 2 : (100, 88, 84, 60) (1, 1, 4, 4) (0, 0, 6, 4) F 3 : (402, 342, 330, 294, 270, 246, 222) (2, 2, 0, 1, 1, 0, 4) (0, 0, 4, 0, 0, 2, 4) F 4 : (2180, 1728, 1408, 1252, 1152) (1, 0, 2, 2, 0) (0, 1, 0, 0, 4) F 5 : (8898, 6858) (1, 0) (0, 1) 25

36 Table 18 Frequency Distribution of K k -Values of Factor Projections for Example 1 k-factor Projections: (K-Values) Frequency Distribution d 1 d 2 F 1 : F 2 : F 3 : F 4 : F 5 : (11070, 10950) (1, 0) (0, 1) Unlike MMA which uses all moments for the whole design, MAP uses k moments for k-factor projections. As a result, k-factor projections correspond to k-factor interactions. Therefore, MAP reflects the interaction alias structure better than MMA. However, MAP uses a distribution vector instead of a single value, making itself more cumbersome in terms of application Additional Minimum Aberration Definitions In addition to the definitions already discussed, there are two other minimum aberration criteria proposed in literature, which will be briefly introduced here. The first criterion is the minimum generalized aberration (MGA) of Ma and Fang (2001), and Fang, Ge, Liu and Qin (2003), which is based upon code theory. This criterion can be used for multi-level designs. For a p-level design d, let {( ) H } E d = n c d d c d = k c d D, for k =0,, m, 1 k( ), : (, ),, where d H (c, d) is the hamming distance between two runs c and d, which is the number of places where they differ. The vector (E 0 (d),, Em(d)) is called the distance distribution of d. In algebraic coding theory, hamming distance can be calculated by n δ ij, where δ ij is the coincidence between two rows in the criterion of MMA. where g g g The vector ( A1 ( d), A2 ( d),, Am ( d) ) is called the generalized word length pattern, 26

37 and m [ ] 1 j= 0 g A ( d) = n( q 1) P( j; m) E ( d), i=1,, m i i j r j r k m k Pj ( k; m) = ( 1) ( q 1) r= 0 r j r is the Krawtchouk polynomial. The MGA g criterion is to sequentially minimize A ( ) i j d for i=1,, m. The MGA was proposed for multi-level designs. The mathematical principles behind MGA are theoretical and its application can be covered by the other criteria. However, for readers who are interested in combining criteria of minimum aberration and uniformity, Ma and Fang (2001) is a good paper to review. One final measure to mention is the general criterion of minimum aberration (GCMA) of Cheng and Tang (2005). For two-level regular fractional factorial designs, γ 0 is the overall mean, and γ 1 is a set of effects to be estimated. The fitted model is then Y = γ I + Wγ + ε, where Y is the vector of responses, W1 is the model matrix corresponding to γ 1 and ε is the vector of uncorrelated random errors. Besides γ 1, the remaining effects may not be negligible. Suppose that these remaining effects can be divided into J-1 groups, γ, 2 γ, via previous experience. However, these effects groups have to be ordered in such a way that the effects in γ j are more important than those in γ j + 1 for j = 2,, J 1. The true, J model is Y = γ I + Wγ + W γ + + W γ + ε, J J where W j is the model matrix corresponding to γ j for j = 1,, J. The general criterion of minimum aberration is defined as sequentially minimizing a vector of ( N N ),, 2 J, where N j is the number of effects inγ j that are aliased with those in γ 1, for j = 2,, J. If γ j are used for the j-factor interactions, the relationship of general criterion of minimum aberration and the usual minimum aberration can be established as ( 1) ( 1) N j = j+ Aj + 1+ m j+ A j 1 for j = 2,, m 1, where m is the number of factors and Nm = Am 1. 27

38 This criterion is established aiming at unifying different versions of MA criteria. It is true that GCMA can be reduced to a more practical criterion such as GMA. However, it is not convenient to make use of a criterion, which needs to satisfy model assumptions. For preliminary readers whose goal is to find appropriate criterion to evaluate two-level nonregular designs, GCMA is not recommended. However, for advanced readers, we suggest them referring the original paper for more details. 2.5 Conclusions Minimum aberration criteria have been used to compare two-level regular fractional factorial designs. However, it is theoretically difficult to extend the usual minimum aberration definition to two-level non-regular, multi-level or mixed-level design situations. Therefore, many statisticians have placed their efforts in developing new minimum aberration criteria for handling these situations. This chapter reviews and compares existing definitions with the intent to introduce these definitions to engineers and industrial scientists. Table 19 provides a summary of some features for applying these minimum aberration criteria. MA Notation Table 19. Features of Minimum Aberration Criteria Contrast Features Coefficient J d for each k MGA F Yes Uses frequency vector of k ( ) MG 2 A B Yes Uses sum of squares of the normalized J ( d) GMA A Yes MMA K No MAP F No MGAC g A No GCMA N Yes Needs orthonormal contrast coefficients for factors Equals MG 2 A when evaluating two-level designs Uses coincidence relations between rows K k d does not reflect factor interactions But ( ) Uses coincidence matrix for each k-factor subset Corresponds to k-factor interactions Uses frequency vector of K-value for each k Complicated mathematical principles Connected to uniformity A general MA that can derive some other MA criteria Assume prior knowledge of effects to rank γ1, γ2, γ J k 28

39 In terms of evaluating two-level designs, the MGA and related the MG 2 A criteria are appropriate for both regular and non-regular designs. The MG 2 A is simpler and easier to compute than MGA. The GMA is equivalent to MG 2 A for two-level designs. However, the GMA criterion also applies to multi-level or mixed-level designs, if contrast coefficients are used. The MMA is another criterion that can be used to evaluate multi- or mixed-level designs. The MMA criterion does not require contrast coefficients. The MAP criterion provides more detailed information than the MMA criterion but the MAP criterion is more complicated. The relationships between MA criteria are shown in Table 20. Table 20 Relationship Summery between Minimum Aberration Criteria MA Criteria Relationships MGA MG 2 A Both use J characteristics of k factor interaction. MGA provides frequency but MG 2 A is sum of squares of normalized J MG 2 A GMA Exact same for two-level designs MMA MAP Both depend on coincidence matrix. MMP counts frequency of k- factor interactions but MMA sums k-power MAP MGA Both are represent as frequency vectors MGAC MMA MGAC uses Hamming distance dh and MMA uses coincidenceδ, where dh = m δ In conclusion, minimum aberration discriminates many types of fractional factorial designs based upon their alias relations. The proposition of different MA criteria satisfies different design situations. Once better understood, these MA criteria can be widely used to help practitioner for the real problem solving. 29