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Lecture 14

Lecture 14. Today: Next day: Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question on D-optimality. Fractional Factorial Split-Plot Designs. It is frequently impractical to perform the fractional factorial design in a completely randomized manner

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Lecture 14

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

  2. 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

  3. 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.

  4. 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

  5. Design Matrix

  6. 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

  7. 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

  8. Running the Design • Randomly select a level setting of the WP factors and fix their levels • With the WP levels fixed, run experimental trials varying the level settings of the SP factors in random order • Can view WP as a completely randomized design • Can view SP as a randomized block design with the blocks defined by the WP factors

  9. 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

  10. 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!!!

  11. 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.

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

  13. Example • Experiment is performed to study the geometric distortion of gear drives • The response is “dishing” of the gear • 5 factors thought to impact response: • A: Furnace track • B: Tooth size • C: Part position • p: Carbon potential • q: Operating Mode

  14. Example • Because of the time taken to change the levels of some of the factors, it is more efficient to run experiment trials keeping factors A-C fixed and varying the levels of p and q • A 2(3+2)-(0+1) FFSP design was run ( I=ABCpq )

  15. Example

  16. Example • This is a 16-run design…have 15 effects to estimate • Which effects are test as WP effects? SP effects? • I=ABCpq • Have a 23 design for the WP effects: A,B,C,AB,AC,BC,ABC=pq are tested as WP effects • SP effects: p,q,Ap,Aq,Bp,Bq,Cp,Cq • Need separate qq-plots for each set of effects

  17. QQ-Plots

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