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Figure 7 Fractional Factorials

Figure 7 Fractional Factorials. This first example (C=AB) is in the book. c1 c2 a1 b1  b2  a 2 b1  b2 . Condition GM A B C AB AC BC ABC a 1 b 1 c 1 1 -1 -1 -1 1 1 1 -1 a 1 b 1 c 2 1 -1 -1 1 1 -1 -1 1

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Figure 7 Fractional Factorials

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  1. Figure 7Fractional Factorials This first example (C=AB) is in the book c1 c2 a1 b1  b2  a2 b1  b2  Condition GM A B C AB AC BC ABC a1b1c1 1 -1 -1 -1 1 1 1 -1 a1b1c2 1 -1 -1 1 1 -1 -1 1 a1b2c1 1 -1 1 -1 -1 1 -1 1 a1b2c2 1 -1 1 1 -1 -1 1 -1 a2b1c1 1 1 -1 -1 -1 -1 1 1 a2b1c2 1 1 -1 1 -1 1 -1 -1 a2b2c1 1 1 1 -1 1 -1 -1 -1 a2b2c2 1 1 1 1 1 1 1 1 Condition GM A B C AB AC BC ABC a1b1c2 1 -1 -1 1 1 -1 -1 1 a1b2c1 1 -1 1 -1 -1 1 -1 1 a2b1c1 1 1 -1 -1 -1 -1 1 1 a2b2c2 1 1 1 1 1 1 1 1

  2. This first example (C=AB) is in the book Figure 7, continued "confounding" or "aliases": start with confounding grand mean and highest order interaction effect in full 23 in 1/2 23 where GM = ABC GM GM = ABC A A(ABC) = A2BC = BC A & BC are aliases B B(ABC) = AB2C = AC B & AC are aliases C C(ABC) = ABC2 = AB C & AB are aliases AB AB(ABC) = A2B2C = C AC AC(ABC) = A2BC2 = B BC BC(ABC) = AB2C2 = A ABC ABC(ABC) = A2B2C2 = GM

  3. Figure 8Fractional Factorial: Planning GMAC GM & AC are aliases A A(AC) = A2C= C A & C are aliases B B(AC) = ABC = ABC B & ABC are aliases C C(AC) = AC2= A AB AB(AC) = A2BC = BC AB & BC are aliases AC AC(AC) = A2C2= GM BC BC(AC) = ABC2= AB ABC ABC(AC) = A2BC2 = B c1 c2 a1 b1  b2  a2 b1  b2  Condition GM A B C AB AC BC ABC a1b1c1 1 -1 -1 -1 1 1 1 -1 a1b2c1 1 -1 1 -1 -1 1 -1 1 a2b1c2 1 1 -1 1 -1 1 -1 -1 a2b2c2 1 1 1 1 1 1 1 1

  4. Fractional Factorials • so ANOVA table: source df • A (or BC) 1 • B (or AC) 1 • C (or AB) 1 • need error term • need to assume higher order interactions are not significant or interesting

  5. Fractional Factorials 2nd ex/ 24 factorial, fraction confound GM = ABCD Source in 24in fractional factorial A A(ABCD) = A2BCD = BCD B B(ABCD) = AB2CD = ACD C ABC2D = ABD D ABCD2 = ABC AB A2B2CD = CD AC A2BC2D = BD AD A2BCD2 = BC BC AB2C2D = AD BD AB2CD2 = AC CD ABC2D2 = AB ABC A2B2C2D = D ABD A2B2CD2 = C ACD A2BC2D2 = B BCD AB2C2D2 = A ABCD GM So, assuming you expect main effects A,B,C, & D, couldn't untangle them from BCD, ACD, ABD, ABC. Furthermore, aliasing/confounding of AB = CD, AC = BD, AD = BC. Clearly if you expect some higher-order interactions, better think hard before using this design. (used in conjoint…)

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