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Design and Analysis of Multi-Factored Experiments

Design and Analysis of Multi-Factored Experiments. Fractional Factorials Not Based on the Powers of 2 – Irregular Designs. Plackett-Burman Designs. The standard two-level designs provide the choice of 4, 8, 16, 32, or more runs, but only to the power of 2.

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Design and Analysis of Multi-Factored Experiments

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  1. Design and Analysis of Multi-Factored Experiments Fractional Factorials Not Based on the Powers of 2 – Irregular Designs DOE Course

  2. Plackett-Burman Designs • The standard two-level designs provide the choice of 4, 8, 16, 32, or more runs, but only to the power of 2. • In 1946, Plackett and Burman invented alternative 2 level designs that are multiples of 4. • The 12-, 20-, 24-, and 28-run PB designs are particular interest because they fill gaps in the standard designs. • Unfortunately, these designs have very messy alias structures. DOE Course

  3. PB Designs (continued) • For example, the 11 factor in the 12-run choice, which is very popular, causes the main effect to be aliased with 45 two-factor interactions. • In theory, if you are willing to accept that interactions are zero, you may get away with it. BUT, this is a very dangerous assumption. • Best to stay away from PB designs – better to use standard FFDs or those recently developed Minimum run Resolution V designs. • PB designs are available in Design-Expert but avoid it!. DOE Course

  4. More Irregular Fraction Designs • It is possible to do other “irregular” fractions and still maintain a relatively high resolution. However, these designs are not orthogonal. • An example of this design is the ¾ replication for 4 factors. It can be created by identifying the standard quarter-fraction, and then selecting two more quarter-fractions. i.e. 4 + 4 + 4 = 12 runs. • This is a 12-run resolution V design. See next few pages on the design and alias structure. • These designs were developed by Peter John (1961, 1962, 1971). DOE Course

  5. John’s ¾ Four Factor Screening Design DOE Course

  6. Alias Structure for Factorial Model Intercept = intercept – ABD [A] = A – ACD [B] = B – BCD [C] = C – ABCD [D] = D – ABCD [AB] = AB – ABCD [AC] = AC –BCD [AD] = AD –BCD [BC] = BC – ACD [BD] = BD – ACD [CD] = CD – ABD [ABC] = ABC -ABD DOE Course

  7. Alias structure for factorial main-effect model [Intercept] = intercept – 0.333 CD – 0.333 ABC + 0.333 ABD [A] = A – 0.333 BC – 0.333 BD – 0.333 ACD [B] = B – 0.333 AC – 0.333 AD - 0.333 ACD [C] = C – 0.5 AB [D] = D – 0.5 AB DOE Course

  8. Warning: Irregular fractions may produce irregularities in effect estimates • Irregular fractions have somewhat peculiar alias structures. E.g. when evaluated for fitting a two-factor interaction model, they exhibit good properties: main effect aliased with three-factor interaction, etc. • But, if you fit only the main effects, they become partially aliased with one or more two-factor interactions. Main effects can get inflated by any large 2 factor interactions. Insignificant main effects may be selected as a result. • Check the p-values in ANOVA for the selected model terms. If there are no interactions, or they are relatively small, then no anomaly. • Normally not a problem because you would never restrict yourself to main effects only. DOE Course

  9. Factors: LowHigh A: Font size 10 pt 18 pt B: Font Style Arial Times C: Background Black White D: Lighting Off On Response: Readability (seconds) Readability – time to transcribe a series of random numbers displayed on the screen by a group of students. We will use a irregular fraction design with 12 runs. Example: Best set up for using RGB projectors DOE Course

  10. Effects Plot DOE Course

  11. ANOVA Analysis of variance table [Partial sum of squares]Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 1501.58 4 375.40 60.64 < 0.0001 A1064.0811064.08171.89< 0.0001C266.671266.6743.080.0003D16.67116.672.690.1448AD168.751168.7527.260.0012Residual 43.33 7 6.19Cor Total 1544.92 11 DOE Course

  12. Results DOE Course

  13. Conclusion • Bigger font – better readability in general • Lights on is better with 18 pt but lights off is better if Font is size 10. • Saved 4 runs by using irregular fraction design. • Design-Expert can construct ¾ fraction when the number of factors is 4, 5, or 6. For 7 factors the fraction is 3/8; for 8 factors the fraction is 3/16; and for 9, 10, and 11 factors the fraction is 1/8, 1/16, and 3/64, respectively. DOE Course

  14. Newer Irregular designs • There are also newer minimum run Resolution IV and V designs available in Design-Expert 7. E.g. 6 Factors in 22 runs, 10 factors in 56 runs, etc. These are generated by computer. Alias structure is complicated and the designs are slightly non-orthogonal. • Another approach to obtain irregular fractions is by use of a semi-foldover where only half the number of runs are necessary compared to a full foldover. • E.g. 24-1 = 8 runs + semi-foldover = 12 runs. • See case study of Hawkins and Lye (2006) • Semi-foldovers can be done using DX-7. DOE Course

  15. Recommendations • Avoid the use of low resolution (Res III) minimum run designs such as Plackett-Burman designs. Unless you can assume all interactions are zero and that time and budget is really tight. • Irregular fraction design can be used with some caution. This is usually not too serious a problem. But check alias structure. • New min run Res V designs can be used to save on runs without compromising too much. DOE Course

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