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The General 2k-p Fractional Factorial Design

• 2k-1 = one-half fraction, 2k-2 = one-quarter fraction, 2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p fraction

• Design matrix for a 2k-p: Add p columns to the basic design; select p independent generators

• Defining relation: generating relations + generalized interactions => aliases

• Important to select generators so as to maximize resolution, see Table 8-14 page 305

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The General 2k-p Design: Resolution may not be sufficient to distinguish between designs • Minimum aberration designs: to achieve fewest number of

words of min. length

• Word length patterns: {4, 4, 4} {4, 4, 6} {4, 5, 5}

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The General 2k-p Fractional Factorial Design

• Usually try to achieve highest resolution and minimum aberration designs

• Table 8-14: Selection of 2k-p for k 15 factors• Appendix Table X: alias structures

• Example 8-5• Seven factors• Estimate seven main effects and some 2-factor

interactions• 3-factor and higher order interactions are

negligible

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Example 8-5• Why not choose R = 7?

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The General 2k-p Fractional Factorial Design• Techniques for resolving the ambiguities in

aliased effects:o Use prior knowledge

One effect is not likely to be importantInteractions between factors in different

subprocesses may not be importanto Conduct follow-up experiments at different

factor settings than the original experiment (Pages 167-175, Wu and Hamada) Method of adding orthogonal runsOptimal design approachFold-over technique

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The General 2k-p Fractional Factorial Design

• Projection • Any subset of rk-p factors which doesn’t

appear as a word in the complete defining relation forms a full factorial design

• Any subset of rk-p factors which appears as a word in the complete defining relation forms a replicated fractional factorial design

• A design of resolution R contains full factorials in any R – 1 of the factors

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The General 2k-p Fractional Factorial Design

• Reconsider Example 8-5 ( )• It projects into a full factorial in any four (4) of

the original seven (7) factors that is not a word in the defining relation (28 such subsets or designs)

372 IV

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The General 2k-p Fractional Factorial Design

• Blocking • Consider the fractional design

262 IV

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The General 2k-p Fractional Factorial Design

• Run the design in two blocks of eight treatment combinations each

• Select an interaction to confound• Choose L = x1 + x2 + x4 = 0

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Example 8-6• Machining an impeller – determine effects on

profile deviation (standard deviation of the difference is the response)

• Eight factors were selected (many of them may have little effects – but not sure)

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Example 8-6• The machine has four spindles – four blocks• Three-factor and higher interactions are not

important, but two-factor interactions may be important

• Table 8-14 => two designs appear appropriate:with 16 runs, or with 32 runs482

IV382

IV

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13

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• Choose • Use a log

transformation• Large effects: A,

B, and AD + BG• Difficult to

separate AD and BG Engineering

knowledge, or Adding more runs

• Assume AD is important

382 IV

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16

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Summary

• The main practical motivation for choosing a fractional factorial design is run size economy. Aliasing of effects is the price paid for economy.

• A 2k-p design is determined by its defining contrast subgroup, which consists of 2p-1 defining words and the identity I. The length of its shortest word is the resolution of the design.

• A main effect or two-factor interaction is said to be clear if none of its aliases are main effects or two-factor interactions and strongly clear if none of its aliases are main effects or two-factor or three-factor interactions.

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• Useful rules for resolution IV or V designs: In any resolution IV design, the main effects are clear In any resolution V design, the main effects are

strongly clear and the two-factor interactions are clear.Among the resolution IV designs with given k and p,

those with the largest number of clear two-factor interactions are the best.

• Projective property of resolution: the projection of a resolution R design onto any R-1 factors is a full factorial in the R-1 factors

• Unless there is specific information on the relative importance among the factorial effects, the minimum aberration criterion should be used for selecting good 2k-p designs. Minimum aberration automatically implies maximum resolution.

• A blocked 2k-p design with more clear effects is considered to be better.

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• The analysis of fractional factorial experiments is the same as in the full factorial experiments except that the observed significance of an effect should be attributed to the combination of the effect and all its aliased effects.

• Optimal factor settings can be determined by the signs of significant effects and the corresponding interaction plots.

• Two approaches to resolve ambiguities in aliased effects are:Use a priori knowledge to dismiss some of the aliased

effects. If a lower order effect is aliased with a higher order effect, the latter may be dismissed.

Run a follow-up experiment