L. M. LyeDOE Course1 Design and Analysis of Multi-Factored Experiments Fractional Factorials Not Based on the Powers of 2 – Irregular Designs.

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L. M. LyeDOE Course1 Design and Analysis of Multi-Factored Experiments Fractional Factorials Not Based on the Powers of 2 – Irregular Designs

L. M. LyeDOE Course2 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.

L. M. LyeDOE Course3 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!.

L. M. LyeDOE Course4 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 = 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).

L. M. LyeDOE Course5 John’s ¾ Four Factor Screening Design StdABCD

L. M. LyeDOE Course6 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

L. M. LyeDOE Course7 Alias structure for factorial main- effect model [Intercept] = intercept – CD – ABC ABD [A] = A – BC – BD – ACD [B] = B – AC – AD ACD [C] = C – 0.5 AB [D] = D – 0.5 AB

L. M. LyeDOE Course8 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.

L. M. LyeDOE Course9 Example: Best set up for using RGB projectors Factors: LowHigh A: Font size10 pt18 pt B: Font StyleArialTimes C: BackgroundBlackWhite D: LightingOffOn 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.

L. M. LyeDOE Course10 Effects Plot

L. M. LyeDOE Course11 ANOVA Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquares DFSquareValueProb > F Model < A < C D AD Residual Cor Total

L. M. LyeDOE Course12 Results

L. M. LyeDOE Course13 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.

L. M. LyeDOE Course14 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 = 8 runs + semi-foldover = 12 runs. See case study of Hawkins and Lye (2006) Semi-foldovers can be done using DX-7.

L. M. LyeDOE Course15 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.