Design and Analysis of MultiFactored Experiments Fractional Factorials

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

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. Lye DOE Course 2

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. Lye DOE Course 3

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 = 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. Lye DOE Course 4

John’s ¾ Four Factor Screening Design Std A B C D 1 -1 -1 2 1 1 -1 -1 3 -1 -1 4 1 -1 5 -1 1 1 -1 6 1 1 1 -1 7 -1 -1 -1 1 8 1 -1 -1 1 9 -1 1 10 1 1 -1 1 1 12 -1 1 L. M. Lye DOE Course 5

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. Lye DOE Course 6

Alias structure for factorial maineffect 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 L. M. Lye DOE Course 7

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. Lye DOE Course 8

Example: Best set up for using RGB projectors • • • Factors: Low A: Font size 10 pt B: Font Style Arial C: Background Black D: Lighting Off Response: Readability (seconds) High 18 pt Times White On • 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. Lye DOE Course 9

Effects Plot L. M. Lye DOE Course 10

ANOVA Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 1501. 58 4 375. 40 60. 64 < 0. 0001 A 1064. 08 171. 89 < 0. 0001 C 266. 67 1 266. 67 43. 08 0. 0003 D 16. 67 1 16. 67 2. 69 0. 1448 AD 168. 75 1 168. 75 27. 26 0. 0012 Residual 43. 33 7 6. 19 Cor Total 1544. 92 11 L. M. Lye DOE Course 11

Results L. M. Lye DOE Course 12

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. Lye DOE Course 13

Newer Irregular designs • There 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 nonorthogonal. • 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. L. M. Lye DOE Course 14

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. L. M. Lye DOE Course 15