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
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The successful use of fractional factorial designs is based on three key ideas: 1)The sparsity of effects principle. When there are several variables,
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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
Fractional Factorial
For a 24 design (factors A, B, C and D) a one-half fraction, 24-1, can be constructed as follows:
Choose an interaction term to completely confound, say ABCD.
Using the defining contrast L = x1 + x2 + x3 + x4 like we did before we get:
Fractional Factorial
L x1 x2 x3 x4 mod 2
L x1 x2 x3 x4 mod 2
0000 0 0 0 0 0 0110 0 1 1 0 0
0001 0 0 0 1 1 1010 0 1 0 1 0
0010 0 0 1 0 1 1100 1 1 0 0 0
0100 0 1 0 0 1 0111 0 1 1 1 1
1000 1 0 0 0 1 1011 1 0 1 1 1
0011 0 0 1 1 0 1101 1 1 0 1 1
0101 0 1 0 1 0 1110 1 1 1 0 1
1001 1 0 0 1 0 1111 1 1 1 1 0
Fractional FactorialHence, our design with ABCD completely confounded is as follows:
Each calculated sum of squares will be associated with two sources of variation.
Source
Prin. Frac.
Alias Source Prin. Frac.
Alias
A ABCD A2BCD BCD BC ABCD AB2C2D AD
B ABCD AB2CD ACD BD ABCD AB2CD2 AC
C ABCD ABC2D ABD CD ABCD ABC2D2 AB
D ABCD ABCD2 ABC ABC ABCD A2B2C2D D
AB ABCD A2B2CD CD ABD ABCD A2B2CD2 C
AC ABCD A2BC2D BD ACD ABCD A2BC2D2 B
AD ABCD A2BCD2 BC BCD ABCD AB2C2D2 A
Fractional Factorial
Lets clean a bit:
Source Alias Source Alias
A BCD BC AD
B ACD BD AC
C ABD CD AB
D ABC ABC D
AB CD ABD C
AC BD ACD B
AD BC BCD A
Fractional Factorial
Lets reorganize:
Source Alias
A BCD
B ACD
C ABD
AB CD
AC BD
BC AD
ABC D
Complete 23 Design
Fractional Factorial
So to analyze a 24-1 fractional factorial design we need to run a complete 23 factorial design (ignoring one of the factors) and analyze the data based on that design and re-interpret it in terms of the 24-1 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 2III
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, 2IV
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, 2V
Assuming all factors are fixed, the linear model is as follows:
Fractional Factorial
1,..., ; 1,..., ; 1,..., ; 1,..., ; 1,...,
ijklm i j k l m ijklij kl ik jl jk ilY
i a j b k c l d m n
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 23 design, as we did before.
For a 24 design (factors A, B, C and D) a one-quarter fraction, 24-2, 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: A2BCD2 = BC.
Need two defining contrasts
L1 = x1 + x2 + 0 + x4
and
L2 = x1 + 0 + x3 + x4
Fractional Factorial
L1 x1 x2 0 x4 mod 2
L2 x1 0 x3 x4 mod 2
0000 0 0 0 0 0 0000 0 0 0 0 0
0001 0 0 0 1 1 0001 0 0 0 1 1
0010 0 0 0 0 0 0010 0 0 1 0 1
0100 0 1 0 0 1 0100 0 0 0 0 0
1000 1 0 0 0 1 1000 1 0 0 0 1
0011 0 0 0 1 1 0011 0 0 1 1 0
0101 0 1 0 1 0 0101 0 0 0 1 1
1001 1 0 0 1 0 1001 1 0 0 1 0
Fractional Factorial
L1 x1 x2 0 x4 mod 2
L2 x1 0 x3 x4 mod 2
0110 0 1 0 0 1 0110 0 0 1 0 1
1010 1 0 0 0 1 1010 1 0 1 0 0
1100 1 1 0 0 0 1100 1 0 0 0 1
0111 0 1 0 1 0 0111 0 0 1 1 0
1011 1 0 0 1 0 1011 1 0 1 1 1
1101 1 1 0 1 1 1101 1 0 0 1 0
1110 1 1 0 0 0 1110 1 0 1 0 0
1111 1 1 0 1 1 1111 1 0 1 1 1
Fractional Factorial
L1 L2 a b c d L1 L2 a b c d
0 0 0 0 0 0 1 0 0 1 0 0
1 0 0 1 0 0 1 1
0 1 1 1 1 0 1 0
1 1 1 0 1 1 0 1
1 1 0 0 0 1 0 1 0 0 1 0
1 0 0 0 0 1 0 1
0 1 1 0 1 1 0 0
1 1 1 1 1 0 1 1
Fractional FactorialHence, our design with ABCD completely confounded is as follows: