The successful use of fractional factorial designs is based on three key ideas: 1)The sparsity of effects principle. When there are several variables, the system or process is likely to be driven primarily by some of the main effects an low order interactions. 2)The projection property. Fractional factorial designs can be projected into stronger designs in the subset of significant factors. 3)Sequential experimentation. Fractional Factorial

For a 2 4 design (factors A, B, C and D) a one-half fraction, 2 4-1, can be constructed as follows: Choose an interaction term to completely confound, say ABCD. Using the defining contrast L = x 1 + x 2 + x 3 + x 4 like we did before we get:

Fractional Factorial Lx1x1 x2x2 x3x3 x4x4 mod 2 Lx1x1 x2x2 x3x3 x4x

Fractional Factorial Hence, our design with ABCD completely confounded is as follows: a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD 0 Y Y Y Y ABCD 1 Y Y Y Y a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d1 ABCD 0 Y Y Y Y ABCD 1 Y Y Y Y The fractional factorial design a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD 1 Y Y Y Y a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d1 ABCD 1 Y Y Y Y 01110

Fractional Factorial Each calculated sum of squares will be associated with two sources of variation. Sourc e Prin. Frac. AliasSourcePrin. Frac. Alias AABCDA 2 BCDBCDBCABCDAB 2 C 2 DAD BABCDAB 2 CDACDBDABCDAB 2 CD 2 AC CABCDABC 2 DABDCDABCDABC 2 D 2 AB DABCDABCD 2 ABC ABCDA2B2C2DA2B2C2DD ABABCDA 2 B 2 CDCDABDABCDA 2 B 2 CD 2 C ACABCDA 2 BC 2 DBDACDABCDA 2 BC 2 D 2 B ADABCDA 2 BCD 2 BCBCDABCDAB 2 C 2 D 2 A

Fractional Factorial Lets clean a bit: SourceAliasSourceAlias ABCDBCAD BACDBDAC CABDCDAB DABC D ABCDABDC ACBDACDB ADBCBCDA

Fractional Factorial Lets reorganize: SourceAlias ABCD BACD CABD ABCD ACBD BCAD ABCD Complete 2 3 Design

Fractional Factorial So to analyze a fractional factorial design we need to run a complete 2 3 factorial design (ignoring one of the factors) and analyze the data based on that design and re-interpret it in terms of the design.

Fractional Factorial Resolution: Many resolutions the three listed in the book are: 1.Resolution III designs: No main effect is aliased with any other main effect, they are aliased with two factor interactions and two factor interactions are aliased with each other. Example 2 III 3-1 with ABC as the principle fractions. 2.Resolution IV designs: No main effect is aliased with any other main effect or any two factor interaction, but two factor interactions are aliased with each other. Example, 2 IV 4-1 with ABCD as the principle fraction. 3.Resolution V designs. No main effect or two-factor interactions is aliased with any other main effect or two- factor interaction, but two-factor interactions are aliased with three factor interactions. Example, 2 V 5-1 with ABCDE as the principle fraction.

Fractional Factorial Example: a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d

Assuming all factors are fixed, the linear model is as follows: Fractional Factorial

If we still cant run this design all at once, we can block; that is we can implement a group-interaction confounding step. We can confound the highest level interaction of the 2 3 design, as we did before. Fractional Factorial

Partial confounding: Group-Interaction Confounded designs Confounding ABC replicate 1: Lx1x1 x2x2 x3x3 mod

a0b0c0a0b0c0 a0b0c1a0b0c1 a0b1c0a0b1c0 a1b0c0a1b0c0 a0b1c1a0b1c1 a1b0c1a1b0c1 a1b1c0a1b1c0 a1b1c1a1b1c1 Block 0 Y 0000 Y 0011 Y 0101 Y 0110 Block 1 Y 0001 Y 0010 Y 0100 Y 0111 Partial confounding: Confounding ABC replicate 1: Group-Interaction Confounded designs

Fractional Factorial For a 2 4 design (factors A, B, C and D) a one-quarter fraction, 2 4-2, can be constructed as follows: Choose two interaction terms to confound, say ABD and ACD, these will serve as our principle fractions. The third interaction, called the generalized interaction, that we confounded in the way is: A 2 BCD 2 = BC. Need two defining contrasts L 1 = x 1 + x x 4 and L 2 = x x 3 + x 4

Fractional Factorial L1L1 x1x1 x2x2 0x4x4 mod 2 L2L2 x1x1 0x3x3 x4x

Fractional Factorial L1L1 x1x1 x2x2 0x4x4 mod 2 L2L2 x1x1 0x3x3 x4x

Fractional Factorial L1L1 L2L2 abcdL1L1 L2L2 abcd

Hence, our design with ABCD completely confounded is as follows: a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD 0(00) Y Y ABCD 1(11) Y Y ABCD 2(01) Y Y ABCD 3(10) Y Y a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d1 ABCD 0(00) Y Y ABCD 1(11) Y Y ABCD 2(01) Y Y ABCD 3(10) Y Y 31011

Fractional Factorial One of the possible one-quarter designs is: a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD 2(01) Y Y a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d1 ABCD 2(01) Y Y 21101

Fractional Factorial Each calculated sum of squares will be associated with four sources of variation. SourcePrin. Frac.Alias AABD,ACD,BCA 2 BD, A 2 CD, ABCBD,CD,ABC BABD,ACD,BCAB 2 D, ABCD, B 2 CAD,ABCD,C CABD,ACD,BCABCD, AC 2 D, BC 2 ABCD,AD,B DABD,ACD,BCABD 2, ACD 2, BCDAB,AC,BCD ABABD,ACD,BCA 2 B 2 D, A 2 BCD, AB 2 CD,BCD,AC ACABD,ACD,BCA 2 BCD, A 2 C 2 D, ABC 2 BCD,D,AB

Fractional Factorial Each calculated sum of squares will be associated with four sources of variation. SourcePrin. Frac.Alias ADABD,ACD,BCA 2 BD 2, A 2 CD 2, ABCB,C,ABC BDABD,ACD,BCAB 2 D 2, ACD 2, B 2 CDA,AC,CD CDABD,ACD,BCABCD 2, AC 2 D 2, BC 2 DABC,A,BD ABCABD,ACD,BCA 2 B 2 CD, A 2 BC 2 D, AB 2 C 2 CD,BD,A BCDABD,ACD,BCAB 2 CD 2, ABC 2 D 2, B 2 C 2 DAC,B,D ABCDABD,ACD,BCA 2 B 2 CD 2, A 2 BC 2 D 2, AB 2 C 2 DC,B,AD

Fractional Factorial The above is not quite satisfactory because we are aliasing some of the main effects with other main effects; i.e. the resolution is not good enough!!!

Fractional Factorial What happens after analyzing the data: Can do a confirmatory experiment, complete the block!!

Fractional Factorial L1L1 x1x1 x2x2 0x4x4 mod 2 L2L2 x1x1 0x3x3 x4x

Fractional Factorial L1L1 x1x1 x2x2 0x4x4 mod 2 L2L2 x1x1 0x3x3 x4x

Fractional Factorial L1L1 L2L2 abcdL1L1 L2L2 abcd

Hence, our design with ABCD completely confounded is as follows: a0b0c0d0a0b0c0d0 a0b0c0d1a0b0c0d1 a0b0c1d0a0b0c1d0 a0b1c0d0a0b1c0d0 a1b0c0d0a1b0c0d0 a0b0c1d1a0b0c1d1 a0b1c0d1a0b1c0d1 a1b0c0d1a1b0c0d1 ABCD 0(00) ABCD 1(11) ABCD 2(01) ABCD 3(10) a0b1c1d0a0b1c1d0 a1b0c1d0a1b0c1d0 a1b1c0d0a1b1c0d0 a0b1c1d1a0b1c1d1 a1b0c1d1a1b0c1d1 a1b1c0d1a1b1c0d1 a1b1c1d0a1b1c1d0 a1b1c1d1a1b1c1d1 ABCD 0(00) ABCD 1(11) ABCD 2(01) ABCD 3(10)

Fractional Factorial Each calculated sum of squares will be associated with four sources of variation. SourcePrin. Frac.Alias AABD,ACD,BC B C D ABABD,ACD,BC ACABD,ACD,BC

Fractional Factorial Each calculated sum of squares will be associated with four sources of variation. SourcePrin. Frac.Alias ADABD,ACD,BC BDABD,ACD,BC CDABD,ACD,BC ABCABD,ACD,BC BCDABD,ACD,BC ABCDABD,ACD,BC