Fractional Factorial Designs2k-p Fractional Factorial DesignsIntroductory Example of a 2k-p DesignAnalysis of 27-4 DesignEffects and Confidence Intervals for 2k-p DesignsPreparing the Sign Table for a 2k-p DesignSign Table for k-p FactorsAssigning Remaining FactorsConfoundingThe Confounding ProblemAn Example of ConfoundingAnalyzing the Confounding ExampleConfounding NotationChoices in Fractional Factorial DesignAlgebra of ConfoundingRules of Confounding AlgebraExample: 23-1 ConfoundingsGenerator PolynomialsExample of Finding Generator PolynomialTurning Basic Terms into Generator PolynomialFinishing Generator PolynomialDesign ResolutionDefinitions Leading to ResolutionDefinition of ResolutionFinding ResolutionChoosing a ResolutionWhite SlideFractional Factorial DesignsAndy WangCIS 5930-03Computer SystemsPerformance Analysis22k-p FractionalFactorial Designs•Introductory example of a 2k-p design•Preparing the sign table for a 2k-p design•Confounding•Algebra of confounding•Design resolution3Introductory Exampleof a 2k-p Design•Exploring 7 factors in only 8 experiments:RunA B C D E F G1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 14Analysis of 27-4 Design•Column sums are zero:•Sum of 2-column product is zero:•Sum of column squares is 27-4 = 8•Orthogonality allows easy calculation of effects:jxiij0ljxxiilij0887654321yyyyyyyyqA5Effects and Confidence Intervals for 2k-p Designs•Effects are as in 2k designs:•% variation proportional to squared effects•For standard deviations, confidence intervals:–Use formulas from full factorial designs–Replace 2k with 2k-piiipkxyq216Preparing the Sign Table for a 2k-p Design•Prepare sign table for k-p factors•Assign remaining factors7Sign Table for k-p Factors•Same as table for experiment with k-p factors–I.e., 2(k-p) table–2k-p rows and 2k-p columns–First column is I, contains all 1’s–Next k-p columns get k-p chosen factors–Rest (if any) are products of factors8Assigning Remaining Factors•2k-p-(k-p)-1 product columns remain•Choose any p columns–Assign remaining p factors to them–Any others stay as-is, measuring interactions9Confounding•The confounding problem•An example of confounding•Confounding notation•Choices in fractional factorial design10The Confounding Problem•Fundamental to fractional factorial designs•Some effects produce combined influences–Limited experiments means only combination can be counted•Problem of combined influence is confounding–Inseparable effects called confounded11An Example of Confounding•Consider this 23-1 table:•Extend it with an AB column:I A B C1 -1 -1 11 1 -1 -11 -1 1 -11 1 1 1I A B C AB1 -1 -1 1 11 1 -1 -1 -11 -1 1 -1 -11 1 1 1 112Analyzing theConfounding Example•Effect of C is same as that of AB:qC = (y1-y2-y3+y4)/4qAB = (y1-y2-y3+y4)/4•Formula for qC really gives combined effect:qC+qAB = (y1-y2-y3+y4)/4•No way to separate qC from qAB–Not problem if qAB is known to be small13Confounding Notation•Previous confounding is denoted by equating confounded effects:C = AB•Other effects are also confounded in this design:A = BC, B = AC, C = AB, I = ABC–Last entry indicates ABC is confounded with overall mean, or q014Choices in Fractional Factorial Design•Many fractional factorial designs possible–Chosen when assigning remaining p signs–2p different designs exist for 2k-p experiments•Some designs better than others–Desirable to confound significant effects with insignificant ones–Usually means low-order with high-order15Algebra of Confounding•Rules of the algebra•Generator polynomials16Rules ofConfounding Algebra•Particular design can be characterized by single confounding–Traditionally, use I = wxyz... confounding•Others can be found by multiplying by various terms–I acts as unity (e.g., I times A is A)–Squared terms disappear (AB2C becomes AC)17Example:23-1 Confoundings•Design is characterized by I = ABC•Multiplying by A gives A = A2BC = BC•Multiplying by B, C, AB, AC, BC, and ABC:B = AB2C = AC, C = ABC2 = AB,AB = A2B2C = C, AC = A2BC2 = B,BC = AB2C2 = A, ABC = A2B2C2 = I•Note that only first line is unique in this case18Generator Polynomials•Polynomial I = wxyz... is called generator polynomial for the confounding•A 2k-p design confounds 2p effects together–So generator polynomial has 2p terms–Can be found by considering interactions replaced in sign table19Example of FindingGenerator Polynomial•Consider 27-4 design•Sign table has 23 = 8 rows and columns•First 3 columns represent A, B, and C•Columns for D, E, F, and G replace AB, AC, BC, and ABC columns respectively–So confoundings are necessarily:D = AB, E = AC, F = BC, and G = ABC20Turning Basic Terms into Generator Polynomial•Basic confoundings are D = AB, E = AC, F = BC, and G = ABC•Multiply each equation by left side:I = ABD, I = ACE, I = BCF, and I = ABCGorI = ABD = ACE = BCF = ABCG21Finishing Generator Polynomial•Any subset of above terms also multiplies out to I–E.g., ABD times ACE = A2BCDE = BCDE•Expanding all possible combinations gives 16-term generator (book is wrong):I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = BDFG = CEFG= ABCDEFG22Design Resolution•Definitions leading to resolution•Definition of resolution•Finding resolution•Choosing a resolution23Definitions Leadingto Resolution•Design is characterized by its resolution•Resolution measured by order of confounded effects•Order of effect is number of factors in it–E.g., I is order 0, ABCD is order 4•Order of confounding is sum of effect orders–E.g., AB = CDE would be of order 524Definition of Resolution•Resolution is minimum order of any confounding in design•Denoted by uppercase Roman numerals–E.g, 25-1 with resolution of 3 is called RIII–Or more compactly,1-5III225Finding Resolution•Find minimum order of effects confounded with mean–I.e., search generator polynomial•Consider earlier example: I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = BDFG = ABDG= CEFG = ABCDEFG•So it’s an RIII design26Choosing a Resolution•Generally, higher resolution is better•Because usually higher-order interactions are smaller•Exception: when low-order interactions are known to be small–Then choose design