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: e w (WP error) –Within plot error: (SP error) Model: The WP and SP error terms have mutually independent normal distributions with standard deviations σ w and σ s

7. The Design Situation: –Have k factors: k 1 WP factors and k 2 SP factors –Wish to explore impact in 2 k-p trials –Have a 2 k1-p1 fractional factorial for the WP factors –Require p=p 1 +p 2 generators –Called a 2 (k 1 + p 2 )-(k 1 + p 2 ) 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 (k 1 + p 2 )-(k 1 + p 2 ) 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: σ w (WP error). –Within plot error: σ s (SP error). WP Effects compared to: aσ s 2 + bσ s 2 SP effects compared to : bσ s 2 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. This is a 16-run design…have 15 effects to estimate Which effects are test as WP effects? SP effects? I=ABCpq Have a 2 3 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