STAT 5200 Handout #28: Fractional Factorial Design (Ch. 18)

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1 STAT 5200 Handout #28: Fractional Factorial Design (Ch. 18) For factorial designs where all factors have 2 levels, it is possible to systematically exclude certain factor level combinations and still make meaningful conclusions. However, this must be done in a carefully structured way at the design stage. A 2 k factorial design can be fractioned by introducing confounding (or aliasing) of higherorder interactions. A 2 k q fractional factorial design has k factors (each at two levels) that uses 2 k q experimental units (and factor level combinations). In the example below, k=9 and q=5. Example: (from p. 479 of text; also Taguchi and Wu 1980) In an experiment studying how various factors affect weld strength, nine factors (at two levels each) were identified. The full design would be a design (with possible replicates at each combination); in shorthand, this would be a 2 9 design. There are 512 possible combinations of factor levels, but the researchers couldn t realistically afford so many replicates. Instead, they carefully created a design requiring only 16 experimental units, summarized in the following table (here factors are coded A-J [skipping I]): Factors and levels (+/-) Generators (I) Y A B C D E F G H J ABCDE BCDF ACDG CDH ABDJ base factors (note full Defining relation: I = ABCDE = BCDF = ACDG = CDH = ABDJ = AEF = BEG = ABEH = CEJ = ABFG = factorial BFH = ACFJ = AGH = BCGJ = ABCHJ = CDEFG = structure ACDEFH = BDEFJ = BCDEGH = ADEGJ = DEHJ = embedded ABCDFGH = DFGJ = ADFHJ = BDGHJ = EFGH = ABCEFGJ = BCEFHJ = ACEGHJ = CFGHJ here) 1

2 How this design was constructed (following p. 478 of text): 1. Choose q generators and get the aliases of identity I Generators are the effects that we confound in the design; they are also sometimes called defining contrasts or simply words. Appendix C.5 (p ) in the text gives suggested generators for various values of k and q (in C.5, p=q). These can be used to create a defining relation, or a set of terms such that the product of any two terms in the set is also in the set (reducing exponents mod 2; so treat A 2 = I; think of this as identity or 1). Using two terms from our example s defining relation, BEG BDEFJ = B 2 DE 2 FGJ = DFGJ. Only higher-order interactions (degree three or more) should be in the defining relation because these terms cannot be estimated, and this same set is used to find the aliases for all lower-order terms. For example, the aliases of A can be found by multiplying A by the defining relation (again reducing exponents mod 2): I = ABCDE = BCDF = = AEF = = AGH = A I = A ABCDE = A BCDF = = A AEF = = A AGH = A = BCDE = ABCDF = = EF = = GH = 2. Find a set of k q base factors that has an embedded complete factorial Choose any set of k q factors that does not contain an alias of I as a subset. There are possibly many such sets. In our example (where k=9 and q=5, so we want a set of 4 base factors), there are 126 (9 choose 4) total sets of four factors, and only ten of those sets appear in the defining relation. ABCD includes four factors and does not appear in the defining relation, so we could (and did) let A, B, C, D be our base factors. 3. Write all factor-level combinations (+/-) of the base factors in standard order. 4. Find the aliases of the remaining q factors in terms of interactions of the k q base factors. For example, since I = AEF, then E I = E AEF, or E = AF. 5. Determine the plus/minus pattern for the q remaining factors from their aliased interactions. For example, the E column in our design is the A column times the F column. 2

3 A few notes on 2 k q fractional factorial designs: o Since we usually can t interpret the highest order interactions (and we often can t afford a full factorial design), we might be willing to assume the higherorder interactions are all zero in a fractional factorial design. o The resolution of the fractional design is the number of letters in the shortest alias of I. o The 2 k q fractional factorial design is one block of a confounded 2 k factorial. There are 2 q blocks, defined by the sign combinations (+/-) of the q generators. In our example here, we used the block where all generators have positive signs. Other blocks (and sets of generators) are possible. This confounding (and associated aliasing) means certain terms in the model are completely confounded with each other. See this confounding (with any terms, particularly those involving only the base factors) using the defining relation. For example, to see what is confounded with ABCD, multiply it by the defining relation: I = = ABCDFGH = = ABCDE = ABCD I = = ABCD ABCDFGH = = ABCD ABCDE = ABCD = = FGH = = E = When we analyze a fractional factorial as an unreplicated 2 k q (using only the base factors and the interactions by hand ), there are 0 degrees of freedom for error. Instead of pooling interactions (which we could do, essentially losing any information on even two-way interactions), we can use a graphical approach to identify large effects. data welding; input Y A B C D E F G H J; cards; ; 3

4 /* Define interactions among base factors 'by hand' */ data welding; set welding; AB = A*B; AC = A*C; AD = A*D; BC = B*C; BD = B*D; CD = C*D; ABC = A*B*C; ABD = A*B*D; ACD = A*C*D; BCD = B*C*D; ABCD = A*B*C*D; /* Run 'by hand' full factorial model on base factors */ proc glm data=welding; class A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD; model Y = A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD; estimate 'A' A 1-1; estimate 'B' B 1-1; estimate 'C' C 1-1; estimate 'D' D 1-1; estimate 'AB' AB 1-1; estimate 'AC' AC 1-1; estimate 'AD' AD 1-1; estimate 'BC' BC 1-1; estimate 'BD' BD 1-1; estimate 'CD' CD 1-1; estimate 'ABC' ABC 1-1; estimate 'ABD' ABD 1-1; estimate 'ACD' ACD 1-1; estimate 'BCD' BCD 1-1; estimate 'ABCD' ABCD 1-1; ods output Estimates=out1; title1 'Full Factorial on Base Factors (by hand)'; Full Factorial on Base Factors (by hand) Source DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total

5 Parameter Estimate Standard Error t Value Pr > t A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD /* Construct normal probability plot of effect estimates */ proc rank data=out1 normal=blom out=out2; var Estimate; ranks nest; proc sgplot data=out2; scatter x=estimate y=nest / markerchar=parameter markercharattrs=(size=12); xaxis label='effect'; yaxis label='normal Score'; title1 'Normal Probability Plot of the Effects'; Note that only ABCD (confounded with E, among others) and BCD (confounded with F, among others) appear to have large effects. Most likely, we are seeing that main effects E and F are significant. 5

6 /* Compare with additive model approach where all interactions are pooled into error */ proc glm data=welding plots=diagnostics; class A B C D E F G H J; model Y = A B C D E F G H J; title1 'Fractional Factorial Design (with all interactions pooled)'; Fractional Factorial Design (with all interactions pooled) Source DF Type III SS Mean Square F Value Pr > F A B C D E <.0001 F G H J

7 /* Automate this fractional design */ proc factex; factors A B C D E F G H J; size design=16; /* # of experimental units 2**(k-q) */ model resolution=3; /* # letters in shortest alias of I */ examine aliasing confounding design; title1 'Fractional Factorial Design'; Fractional Factorial Design (alternative aliasing) Design Points ExpNumber A B C D E F G H J Factor Confounding Rules E = A*B*C*D F = B*C*D G = A*C*D H = C*D J = A*B*D Aliasing Structure A = E*F = G*H B = E*G = F*H C = D*H = E*J D = C*H E = A*F = B*G = C*J F = A*E = B*H G = A*H = B*E H = A*G = B*F = C*D J = C*E A*B = D*J = E*H = F*G A*C = D*G = F*J A*D = B*J = C*G A*J = B*D = C*F B*C = D*F = G*J D*E = H*J 7

8 /* Check original generators (based on 'Factor Confounding Rules' in FACTEX output) */ data temp; set welding; ABCDE = A*B*C*D * E ; BCDF = B*C*D * F ; ACDG = A*C*D * G ; CDH = C*D * H ; ABDJ = A*B*D * J ; proc print data=temp; var ABCDE BCDF ACDG CDH ABDJ; title1 'Checking Generators'; title2 '(This design uses block where generators all positive)'; Checking Generators (This design uses block where generators all positive) Obs ABCDE BCDF ACDG CDH ABDJ

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