1 The General 2 k-p Fractional Factorial Design 2 k-1 = one-half fraction, 2 k-2 = one-quarter fraction, 2 k-3 = one-eighth fraction, …, 2 k-p = 1/ 2 p fraction Design matrix for a 2 k-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

2 The General 2 k-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}

3 The General 2 k-p Fractional Factorial Design Usually try to achieve highest resolution and minimum aberration designs Table 8-14: Selection of 2 k-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

4 Example 8-5 Why not choose R = 7?

5 The General 2 k-p Fractional Factorial Design Techniques for resolving the ambiguities in aliased effects: oUse prior knowledge One effect is not likely to be important Interactions between factors in different subprocesses may not be important oConduct follow-up experiments at different factor settings than the original experiment (Pages , Wu and Hamada) Method of adding orthogonal runs Optimal design approach Fold-over technique

6 The General 2 k-p Fractional Factorial Design Projection Any subset of r  k-p factors which doesn’t appear as a word in the complete defining relation forms a full factorial design Any subset of r  k-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

7 The General 2 k-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)

8 The General 2 k-p Fractional Factorial Design Blocking Consider the fractional design

9 The General 2 k-p Fractional Factorial Design Run the design in two blocks of eight treatment combinations each Select an interaction to confound Choose L = x 1 + x 2 + x 4 = 0

10 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)

11 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 runs

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14 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

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17 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 2 k-p design is determined by its defining contrast subgroup, which consists of 2 p -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.

18 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 2 k-p designs. Minimum aberration automatically implies maximum resolution. A blocked 2 k-p design with more clear effects is considered to be better.

19 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