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## Chapter 8 Two-Level Fractional Factorial Designs

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**8.1 Introduction**• The number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly • After assuming some high-order interactions are negligible, we only need to run a fraction of the complete factorial design to obtain the information for the main effects and low-order interactions • Fractional factorial designs • Screening experiments: many factors are considered and the objective is to identify those factors that have large effects.**Three key ideas:**• The sparsity of effects principle • There may be lots of factors, but few are important • System is dominated by main effects, low-order interactions • The projection property • Every fractional factorial contains full factorials in fewer factors • Sequential experimentation • Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation**8.2 The One-half Fraction of the 2k Design**• Consider three factor and each factor has two levels. • A one-half fraction of 23 design is called a 23-1 design**In this example, ABC is called the generator of this**fraction (only + in ABC column). Sometimes we refer a generator (e.g. ABC) as a word. • The defining relation: I = ABC • Estimate the effects: • A = BC, B = AC, C = AB**Aliases:**• Aliases can be found from the definingrelationI = ABC by multiplication: AI = A(ABC) = A2BC = BC BI =B(ABC) = AC CI = C(ABC) = AB • Principal fraction: I = ABC**The Alternate Fraction of the 23-1 design:**I = - ABC • When we estimate A, B and C using this design, we are really estimating A – BC, B – AC, and C – AB, i.e. • Both designs belong to the same family, defined by I = ± ABC • Suppose that after running the principal fraction, the alternate fraction was also run • The two groups of runs can be combined to form a full factorial – an example of sequential experimentation**The de-aliased estimates of all effects by analyzing the**eight runs as a full 23 design in two blocks. Hence • Design resolution: A design is of resolution R if no p-factor effect is aliased with another effect containing less than R – p factors. • The one-half fraction of the 23 design with I = ABC is a design**Resolution III Designs:**• me = 2fi • Example: A 23-1 design with I = ABC • Resolution IV Designs: • 2fi = 2fi • Example: A 24-1 design with I = ABCD • Resolution V Designs: • 2fi = 3fi • Example: A 25-1 design with I = ABCDE • In general, the resolution of a two-level fractional factorial design is the smallest number of letters in any word in the defining relation.**The higher the resolution, the less restrictive the**assumptions that are required regarding which interactions are negligible to obtain a unique interpretation of the data. • Constructing one-half fraction: • Write down a full 2k-1 factorial design • Add the kth factor by identifying its plus and minus levels with the signs of ABC…(K – 1) • K = ABC…(K – 1) => I = ABC…K • Another way is to partition the runs into two blocks with the highest-order interaction ABC…K confounded.**Any fractional factorial design of resolution R contains**complete factorial designs in any subset of R – 1 factors. • A one-half fraction will project into a full factorial in any k – 1 of the original factors**Example 8.1:**• Example 6.2: A, C, D, AC and AD are important. • Use 24-1 design with I = ABCD**This design is the principal fraction, I = ABCD**• Using the defining relation, • A = BCD, B=ACD, C=ABD, D=ABC • AB=CD, AC=BD, BC=AD**A, C and D are large.**• Since A, C and D are important factors, the significant interactions are most likely AC and AD. • Project this one-half design into a single replicate of the 23 design in factors, A, C and D. (see Figure 8.4 and Page 296)**Example 8.2:**• 5 factors • Use 25-1 design with I = ABCDE (Table 8.5) • Every main effect is aliased with four-factor interaction, and two-factor interaction is aliased with three-factor interaction. • Table 8.6 (Page 298) • Figure 8.6: the normal probability plot of the effect estimates • A, B, C and AB are important • Table 8.7: ANOVA table • Residual Analysis • Collapse into two replicates of a 23 design**Sequences of fractional factorial: Both one-half fractions**represent blocks of the complete design with the highest-order interaction confounded with blocks.**Example 8.3:**• Reconsider Example 8.1 • Run the alternate fraction with I = – ABCD • Estimates of effects • Confirmation experiment**8.3 The One-Quarter Fraction of the 2k Design**• A one-quarter fraction of the 2k design is called a 2k-2 fractional factorial design • Construction: • Write down a full factorial in k – 2 factors • Add two columns with appropriately chosen interactions involving the first k – 2 factors • Two generators, P and Q • I = P and I = Q are called the generating relations for the design • All four fractions are the family.**The complete defining relation: I = P = Q = PQ**• P, Q and PQ are called words. • Each effect has three aliases • Aone-quarter fraction of the 26-2 with I = ABCE and I = BCDF. The complete defining relation is I = ABCE = BCDF = ADEF**Another way to construct such design is to derive the four**blocks of the 26 design with ABCE and BCDF confounded , and then choose the block with treatment combination that are + on ABCE and BCDF • The 26-2 design with I = ABCE and I = BCDF is the principal fraction. • Three alternate fractions: • I = ABCE and I = - BCDF • I = -ABCE and I = BCDF • I = - ABCE and I = -BCDF**This fractional factorial will project into**• A single replicate of a 24 design in any subset of four factors that is not a word in the defining relation. • A replicate one-half fraction of a 24 in any subset of four factors that is a word in the defining relation. • In general, any 2k-2 fractional factorial design can be collapsed into either a full factorial or a fractional factorial in some subset of r ≦ k –2 of the original factors.**Example 8.4:**• Injection molding process with six factors • Design table (see Table 8.10) • The effect estimates, sum of squares, and regression coefficients are in Table 8.11 • Normal probability plot of the effects • A, B, and AB are important effects. • Residual Analysis (Pages 307-309)**8.4 The General 2k-p Fractional Factorial Design**• A 1/ 2p fraction of the 2k design • Need p independent generators, and there are 2p –p – 1 generalized interactions • Each effect has 2p – 1 aliases. • A reasonable criterion: the highest possible resolution, and less aliasing • Minimum aberration design: minimize the number of words in the defining relation that are of minimum length.**Minimizing aberration of resolution R ensures that a design**has the minimum # of main effects aliased with interactions of order R – 1, the minimum # of two-factor interactions aliased with interactions of order R – 2, …. • Table 8.14**Example 8.5**• Estimate all main effects and get some insight regarding the two-factor interactions. • Three-factor and higher interactions are negligible. • designs in Table 8-14 • 16-run design: main effects are aliased with three-factor interactions and two-factor interactions are aliased with two-factor interactions • 32-run design: all main effects and 15 of 21 two-factor interactions**Analysis of 2k-p Fractional Factorials:**• For the ith effect: • Projection of the 2k-p Fractional Factorials • Project into any subset of r ≦ k – p of the original factors: a full factorial or a fractional factorial (if the subsets of factors are appearing as words in the complete defining relation.) • Very useful in screening experiments • For example 16-run design: Choose any four of seven factors. Then 7 of 35 subsets are appearing in complete defining relations.**Blocking Fractional Factorial:**• Appendix Table XII • Consider the fractional factorial design with I = ABCE = BCDF = ADEF. Select ABD (and its aliases) to be confounded with blocks. (see Figure 8.18) • Example 8.6 • There are 8 factors • Four blocks • Effect estimates and sum of squares (Table 8.17) • Normal probability plot of the effect estimates (see Figure 8.19)**A, B and AD + BG are important effects**• ANOVA table for the model with A, B, D and AD (see Table 8.18) • Residual Analysis (Figure 8.20) • The best combination of operating conditions: A –, B + and D –**8.5 Resolution III Designs**• Designs with main effects aliased with two-factor interactions • A saturated design has k = N – 1 factors, where N is the number of runs. • For example: 4 runs for up to 3 factors, 8 runs for up to 7 factors, 16 runs for up to 15 factors • In Section 8.2, there is an example, design. • Another example is shown in Table 8.19: design I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG**This design is a one-sixteenth fraction, and a principal**fraction. I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG= AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG • Each effect has 15 aliases.**Assume that three-factor and higher interactions are**negligible. • The saturated design in Table 8.19 can be used to obtain resolution III designs for studying fewer than 7 factors in 8 runs. For example, for 6 factors in 8 runs, drop any one column in Table 8.19 (see Table 8.20)**When d factors are dropped , the new defining relation is**obtained as those words in the original defining relation that do not contain any dropped letters. • If we drop B, D, F and G, then the treatment combinations of columns A, C, and E correspond to two replicates of a 23 design.**Sequential assembly of fractions to separate aliased**effects: • Fold over of the original design • Switching the signs in one column provides estimates of that factor and all of its two-factor interactions • Switching the signs in all columns dealiases all main effects from their two-factor interaction alias chains – called a full fold-over**Example 8.7**• Seven factors to study eye focus time • Run design (see Table 8.21) • Three large effects • Projection? • The second fraction is run with all the signs reversed • B, D and BD are important effects**The defining relation for a fold-over design**• Each separate fraction has L + U words used as generators. • L: like sign • U: unlike sign • The defining relation of the combining designs is the L words of like sign and the U – 1 words consisting of independent even products of the words of unlike sign. • Be careful – these rules only work for Resolution III designs**Plackett-Burman Designs**• These are a different class of resolution III design • Two-level fractional factorial designs for studying k = N – 1 factors in N runs, where N = 4 n • N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, … • The designs where N = 12, 20, 24, etc. are called nongeometric PB designs • Construction: • N = 12, 20, 24 and 36 (Table 8.24) • N = 28 (Table 8.23)**The alias structure is complex in the PB designs**• For example, with N = 12 and k = 11, every main effect is aliased with every 2FI not involving itself • Every 2FI alias chain has 45 terms • Partial aliasing can greatly complicate interpretation • Interactions can be particularly disruptive • Use very, very carefully (maybe never)**Projection: Consider the 12-run PB design**• 3 replicates of a full 22 design • A full 23 design + a design • Projection into 4 factors is not a balanced design • Projectivity 3: collapse into a full fractional in any subset of three factors.**Example 8.8:**• Use a set of simulated data and the 11 factors, 12-run design • Assume A, B, D, AB, and AD are important factors • Table 8.25 is a 12-run PB design • Effect estimates are shown in Table 8.26 • From this table, A, B, C, D, E, J, and K are important factors. • Interaction? (due to the complex alias structure) • Folding over the design • Resolve main effects but still leave the uncertain about interaction effects.**8.6 Resolution IV and V Designs**• Resolution IV: if three-factor and higher interactions are negligible, the main effects may be estimated directly • Minimal design: Resolution IV design with 2k runs • Construction: The process of fold over a design (see Table 8.27)**Fold over resolution IV designs: (Montgomery and Runger,**1996) • Break as many two-factor interactions alias chains as possible • Break the two-factor interactions on a specific alias chain • Break the two-factor interactions involving a specific factor • For the second fraction, the sign is reversed on every design generators that has an even number of letters**Resolution V designs: main effects and the two-factor**interactions do not alias with the other main effects and two-factor interactions.