Two-Level Fractional Factorial Designs

5Two-Level Fractional Factorial Designs Because the number of runs in a 2k factorial design increases rapidly as the number of factorsincreases, it is often impossible to run the full factorial design given available resources. If the experimenter can reasonably assume that certain high-order interactions (often 3-wayand higher order) are negligible, then it is often possible to estimate and test for the significanceof main effects and low-order interactions from only a fraction of the full factorial design. A design which contains a subset of factor level combinations from a full factorial design iscalled a fractional factorial design. A fractional factorial design is often used as a screening experiment involving many factorswith the goal of identifying only those factors having large effects. Once specific factors areidentified as important, they are investigated in greater detail in subsequent experiments. The successful use of two-level fractional factorial designs is based on three ideas:1. The sparsity of effects principle: When there are many variables under consideration,it is typical for the system or process to be dominated by main effects and low-orderinteractions.2. The projective property: A fractional factorial design can be projected into strongerdesigns in a subset of the significant factors.3. Sequential experimentation: If is possible to combine the runs from two or morefractional factorial designs to sequentially form a larger design that allows estimation ofall interaction effects of interest.5.12k 1 Fractional Factorial Designs Situation: There are k factors of interest each having 2 levels, but there are only enoughresources to run 1/2 of the full factorial 2k design. Thus, we say we want to run a 1/2fraction of a 2k design. This design is called a 2k 1 fractional factorial design.– Suppose there are 5 factors of interest (A, B, C, D, and E), and there are only enoughresources for 16 experimental runs. Thus, we want to run a 1/2 fraction of a 25 design.This design is called a 25 1 fractional factorial design. A 1/2 fraction can be generated from any interaction, but using the highest-order interactionis the standard. The interaction used to generate the 1/2 fraction is called the generator ofthe fractional factorial design.– When there are 3 factors, use ABC as the generator of the 23 1 design, when there are4 factors, use ABCD as the generator of the 24 1 design, . . ., when there are k factors,use ABC . . . K as the generator of the 2k 1 design. To generate a 2k 1 fractional factorial design from the highest-order interaction:(i) Select only those treatment combinations that have a plus ( ) sign in the ABC · · · Kcolumn of the 2k design, OR,(ii) Select only those treatment combinations that have a minus ( ) sign in the ABC · · · Kcolumn of the 2k design.Because the I column also contains all plus signs, we refer to I ABC · · · K orI ABC · · · Kas a defining relation for the 2k 1 design.71

Example When there are 3 factors, we have the following table of pluses and minuses.Treatmenttotalabcabcabacbc(1)I A Factorial EffectC AB AC B BC ABC AliasedeffectsI A B C I A B C Thus, for defining relation I ABC , the treatment combinations with ABC 1 formthe 23 1 design, or, the 1/2 fraction of the 23 design. The remaining 4 treatment combinationswith ABC 1 are discarded (i.e, discard the ab, ac, bc, and (1) rows). Because only 1/2 of the full factorial design is run, each of the 2k effects (including theintercept) is aliased with one other effect. That is, estimation of aliased effects are calculatedidentically and, therefore, cannot be separated from each other. To find the estimate of any model effect, use the difference in means between the and rows as we did with the 2k design. Example In the 23 1 design with defining relation I ABC, let lA , lB , and lC representestimates of the main effects. ThenlA 1[(a abc) (b c)]21[(b abc) (a c)]21 [(c abc) (a b)]2lB lCNow, let lAB , lAC , and lBC represent estimates of the two-factor interaction effects. ThenlBC 1[2]1[21 [2lAC ]lAB] Note that lA lBC , lB lAC , and lC lAB . Hence, it is impossible to separate the effectof A from BC, the effect of B from AC, and the effect of C from AB. Thus, the A and BCeffects are aliased, the B and AC effects are aliased, and the C and AB effects are aliased. In addition, note that the intercept I is aliased with ABC. If two effects E1 and E2 are aliased, then we cannot estimate each separately. That is, eachof the two columns produces an estimate of the sum of the two effects. For effect i, this isdenoted li E1 E2 for i 1, 2.72

Example: In the 23 1 design, we havelA A BClBC A BClB B ACandlAC B AClC C ABlAB C AB The alias structure for any 2k 1 design can be determined by taking the defining relationI ABC · · · K and multiplying it by any effect. The resulting effect is the aliased effect.– In the 23 1 design, to find the aliases of the main effects A, B, and C, we multiplyI ABC by A, B, and C, and reduce:A A · I A · ABC A2 BC BC A BCB B·I B ACC C ·I C AB Now consider the other 1/2 fraction having defining relation I ABC. This design containsthe treatment combinations that have minus signs for ABC · · · K. This is known as thealternate or complementary 1/2 fraction.– In the 23 1 design, the alternate 1/2 fraction are the 4 treatment combinations corresponding to rows (1), ab, ac, and bc.Treatmenttotalabacbc(1)I A B Factorial EffectC AB AC BC ABC AliasedeffectsI ABCA BCB ACC AB– The defining relation is I ABC and the alias structure using the multiplication ruleisA A · I A · ( ABC) A2 BC BC A BCB B·I B ACC C ·I C BCThus, the A and BC effects are aliased, the B and AC effects are aliased, and the C00and AB effects are aliased. The alternate fraction yields estimates lA, lBand lC0 where0lA 0lB lC0 Creating a 2k 1 fractional-factorial design is equivalent to forming 2 blocks from a 2k designand then selecting one block to run. In practice, it does not matter which fraction we use(although it is common to use the ( , ) case to avoid using the negative of an estimate. Because of aliased pairs of effects, only one effect from each pair can be included in the model.73

Example: A 24 1 design (a 1/2 fraction of a 24 design) Using generator I ABCD, the following table contains the 8 combinations ABCD 1.This is one possible 24 1 design.I A B C D ABCD Notation Using generator I ABCD, the following table contains the 8 combinations ABCD 1.This is a second possible 24 1 design.I A B C D ABCD Notation In the 24 1 design, to find the aliases of the A, B, C, D, AB, AC, . . . we multiply I ABCDby A, B, C, D, AB, AC, . . . and then reduce:A A · I A · ABCD A2 BCD BCDlA lBCD A BCDB B·I lB lACD B ACDC C ·I lC lABD C ABDD D·I lD lABC D ABCAB AB · I AB · ABCD A2 B 2 CD CDlAB lCD AB CDAC AC · I AC · ABCD A2 BC 2 D BDlAC lBD AC BDAD AD · I lAD lBC 74

5.2Design ResolutionA design is resolution R if no p-factor effect is aliased with another effect containing less thanR p factors. Resolution III Designs: No main effect is aliased with any other main effect, but at least onemain effect is aliased with a two-factor interaction. Resolution IV Designs: No main effect is aliased with any other main effect or two-factor interaction, but at least one two-factor interaction is aliased with another two-factor interaction. Resolution V Designs: No main effect or two-factor interaction is aliased with any other maineffect or two-factor interaction, but at least one two-factor interaction is aliased with a threefactor interaction. For example:– The 23 1 design with defining relation I ABC is resolution III and is denoted 23 1III .– The 24 1 design with defining relation I ABCD is resolution IV and is denoted 24 1IV .– The 25 1 design with defining relation I ABCDE is resolution V and is denoted 25 1V . When choosing a design, we want the highest resolution possible to minimize the number ofinteractions that must be considered negligible in order to obtain a unique interpretation ofthe data.5.32k 2 Fractional Factorial Designs There are k factors of interest each having 2 levels. There are only enough resources to run1/4 of the full factorial 2k design. Thus, we say we want to run a 1/4 fraction of a 2kdesign. This design is called a 2k 2 fractional factorial design.– Example: There are 6 factors of interest (A, B, C, D, E, F ). There are only enoughresources for 16 experimental runs which is 1/4 of a 26 design. Thus, we want to run a1/4 fraction of a 26 design. This design is called a 26 2 fractional factorial design. When selecting a 1/4 fraction, we want to be sure that we select design points that will enableus to estimate effects of interest. Generation of such a design (if it exists) requires carefully choosing two interactions to generatethe design of maximum resolution and then decide on the sign of each generator. These two signed interactions are called the generators of the 2k 2 fractional factorial design,and with their generalized interactions form the complete defining relation for the design.– Suppose there are 6 factors and we choose ABCE and BCDF to be the generators ofthe 26 2 design. Then there are 4 ways to assign signs to these:( , ) I ABCE and I BCDF( , ) I ABCE and I BCDF( , ) I ABCE and I BCDF( , ) I ABCE and I BCDFEach of these four assignments will generate a unique 1/4 fraction of a 26 design.– Suppose we choose the ( , ) case: I ABCE and I BCDF . The generalizedinteraction ABCE · BCDF . Thus, the complete definingrelation is given by I ABCE BCDF .75

The resolution of any design equals the length of the shortest ‘word’ (excluding I) in thecomplete defining relation.– In the 26 2 design with complete defining relation I ABCE BCDF ADEF , thelength of the shortest ‘word’ is four. Therefore, this is a 26 2IV fractional factorial design. A 2k 2 fractional factorial design is formed by selecting one of the four sign pairs ( ( , ),( , ), ( , ) or ( , )) and the selecting only those treatment combinations that matchthe pair of signs in the columns corresponding to the two generating effects. This can beaccomplished in two ways:(i) List all 2k combinations and selecting the rows with plus signs in the two columns corresponding to the two generators.(ii) Or, more simply, list all 2k 2 combinations for a factorial design having k 2 factors.Then create columns for the remaining two factors based on the effects in the completedefining relation. Example: Generate the 26 2 design with defining relation I ABCE BCDF ADEFusing method (ii).– Generate a 24 design in the first four factors (A, B, C, and D). Then, choose aliases forE and F that can be expressed only in terms of A, B, C, and D. Use these to determinethe columns for E and F .E ABC BCDEF ADFF – Now, multiply the appropriate columns to form columns for E and F .A 24 DesignB C D E F 01653752 Because only 1/4 of the full factorial design is run, each of the 2k effects (including the intercept) is aliased with three other effects. That is, estimation of aliased effects are calculatedidentically and, therefore, cannot be separated from each other. The alias structure for any 2k 2 design can be determined by taking the defining relation andmultiplying it by any effect. The resulting four effects are all aliased.76