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Lecture 13

Lecture 13. Today: 4.3-4.6 Next day: Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question on D-optimality. Optimal Design Approach (4.4.2). Algorithm:. Assignment Question. Suppose in the cable shrinkage example, effects A, E and AC=BE are identified as signifincat

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Lecture 13

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  1. Lecture 13 • Today: 4.3-4.6 • Next day: • Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question on D-optimality

  2. Optimal Design Approach (4.4.2) • Algorithm:

  3. Assignment Question • Suppose in the cable shrinkage example, effects A, E and AC=BE are identified as signifincat • To resolve the aliasing of the interaction effects, a follow-up experiment with 4 trials is to be performed • What 4 trails should be performed? • Use the D-optimality criterion and report the value of Dmax

  4. Additional Features of a Fractional Factorial • Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects or 2fi’s • Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s or 3fi’s

  5. Blocking Fractional Factorial Designs • Can perform a 2k-p fractional factorial design in 2q blocks • That is, k factors are investigated in 2k-p runs with 2q blocks • The design is constructed by assigning p treatment factors and q blocking factors to interactions between (k-p) of the factors

  6. Example • An experimenter wishes to explore the impact of 6 factors (A-F) on the response of a system • There exists enough resources to run 16 experiment trials in 4 blocks • A 26-2 fraction factorial design in 22 blocks is required

  7. Example • Design: • Fractional factorial: E=ABC; F=ABD • Blocking: b1=ACD; b2=BCD • Defining Contrast sub-group:

  8. Example

  9. Comment • Must be careful when choosing the interactions to assign the factors • Fractional factorial: E=AB; F=ABD • Blocking: b1=ACD; b2=BCD • Defining Contrast sub-group:

  10. Additional Features • Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects, 2fi’s or block effects • Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s, 3fi’s or block effects • As before, block by factor interactions are negligible • Analysis is same as before • Appendix 4 has blocked fractional factoria designs ranked by number of clear effects

  11. Fractional Factorial Split-Plot Designs • It is frequently impractical to perform the fractional factorial design in a completely randomized manner • Can run groups of treatments in blocks • Sometimes the restrictions on randomization take place because some factors are hard to change or the process takes place in multiple stages • Fractional factorial split-plot (FFSP) design may be a practical option

  12. Performing FFSP Designs • Randomization of FFSP designs different from fractional factorial designs • Have hard to change factors (whole-plot or WP factors) and easy to change factors (sub-plot or SP factors) • Experiment performed by: • selecting WP level setting, at random. • performing experimental trials by varying SP factors, while keeping the WP factors fixed.

  13. Example • Would like to explore the impact of 6 factors in 16 trials • The experiment cannot be run in a completely random order because 3 of the factors (A,B,C) are very expensive to change • Instead, several experiment trials are performed with A, B, and C fixed…varying the levels of the other factors

  14. Design Matrix

  15. Impact of the Randomization Restrictions • Two Sources of randomization  Two sources of error • Between plot error: ew (WP error) • Within plot error: (SP error) • Model: • The WP and SP error terms have mutually independent normal distributions with standard deviations σw and σs

  16. The Design • Situation: • Have k factors: k1 WP factors and k2 SP factors • Wish to explore impact in 2k-p trials • Have a 2k1-p1fractional factorial for the WP factors • Require p=p1+p2 generators • Called a 2(k1+ p2)-(k1+ p2) FFSP design

  17. Constructing the Design • For a 2(k1+ p2)-(k1+ p2) FFSP design, have generators for WP and SP designs • Rules: • WP generators (e.g., I=ABC ) contain ONLY WP factors • SP generators (e.g., I=Apqr ) must contain AT LEAST 2 SP factors • Previous design: I=ABC=Apqr=BCpqr

  18. Analysis of FFSP Designs • Two Sources of randomization  Two sources of error • Between plot error: σw (WP error). • Within plot error: σs (SP error). • WP Effects compared to: aσs2+ bσs2 • SP effects compared to : bσs2 • df for SP df for WP. • Get more power for SP effects!!!

  19. WP Effect or SP Effect? • Effects aliased with WP main effects or interactions involving only WP factors tested as a WP effect. • E.g., pq=ABCD tested as a WP effect. • Effects aliased only with SP main effects or interactions involving at least one SP factors tested as a SP effect . • E.g., pq=ABr tested as a SP effect.

  20. Ranking the Designs • Use minimum aberration (MA) criterion

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