Northwestern University Department of Industrial Engineering and Management Sciences Assembled Designs for Dispersion Ef

1. Northwestern UniversityDepartment of Industrial Engineering and Management SciencesAssembled Designs for Dispersion Effects Bruce E. Ankenman, Northwestern University Ana Ivelisse Avilés, NIST José C. Pinheiro, Novartis This material is based upon work supported under a National Science Foundation Graduate Fellowship Grant and Grant # 738578.

2. A Robustness Experiment using an Assembled Design Objective: Find settings of the control factors that minimize the permeability and minimize the the batch-to-batch variability and the sample-to-sample variability

3. Consider Structure 1 Consider Structure 2 Sample 1 Batch 1 Sample 1 Sample 2 Batch 1 Sample 2 Batch 2 Batch 3 Sample 1 Sample 2 Batch 2 Sample 1 Sample 1 Level 2 Max Size Level 2 Level 1 W/C Ratio Level 1 Level 1 Grade Level 2 An Assembled Design A hybrid design that places a hierarchical nested design (HND) at each design point in a crossed factor design. Parameters: r = # of design points in the crossed factor design n = # of observations per design point (assume constant) s = # of different HND’s (structures) used in the design q = # of variance components (equals # of levels of nesting) BT = Total number of Batches Splitting generators are used to place the HNDs r=8 n=4 (2,1,1) s=2 q=2 BT=2 (2,2) Split gen.=ABC

4. Must have design points ordered in some way. Example of Notation for Assembled Designs Example: 23 design, Splitting Generator = ABC, r=8, n=4, s=2, q=2.

5. Level 2 Max Size Level 2 Level 1 W/C Ratio Level 1 Level 1 Grade Level 2 An Experiment for Robust Design We want to estimate effects of Grade, W/C ratio, and Max Size on the permeability of concrete. (Location Effects) We want to estimate effects of Grade, W/C ratio, and Max Size on the variability of the permeability from batch-to-batch and from sample-to-sample. (Dispersion Effects)

6. Design Issues For Assembled Designs How do we split the degrees of freedom between the fixed effects, the batch-to-batch variance and the sample-to-sample variance? How do we choose which structures to use? How do we decide where to put the structures?

7. ui is a vector of normally distributed independent random effect parameters associated with the ith variance component The Model Dispersion effects Location effects y is a vector of nr observations. X is an nrr matrix of all estimable location effects and the constant column. b is a vector of r unknown coefficient parameters including the constant term. Zi is an indicator matrix associated with the ith variance component.

8. The dispersion effect parameter on the ith variance component by the jth regressor. Structure of Dispersion Effect Model(Wolfinger and Tobias, 1998) Dispersion effect parameters Design Matrix for Dispersion Effects (Depends on the factor settings)

9. Example of Dispersion Effect Model Factor 1 has a dispersion effect on the 2nd variance component (sample-to-sample).

10. Example of Dispersion Effect Model

11. Example of Dispersion Effect Model If factor 1 is at the low level. If factor 1 is at the high level. If g21 is large and positive, then the sample-to-sample variance increases as factor 1 is changed from low to high.

12. Parameters of Interest The standard deviation ratio • If the batch-to-batch standard deviation is about twice the sample-to-sample standard deviation then The scale ratio on the batch-to-batch variation for the jth regressor • If the batch-to-batch standard deviation at the high level of Factor A is about three times larger than the batch-to-batch standard deviation at the low level of Factor A then The scale ratio on the batch-to-batch variation for the jth regressor • If the sample-to-sample standard deviation at the high level of Factor A is about three times larger than the sample-to-sample standard deviation at the low level of Factor A then III

13. Estimating the Location Effects

14. Using Maximum Likelihood Estimates of the Variance Components and the Dispersion Effects Where aij depends on partial derivatives of the likelihood function with respect to each gi.

15. Information Matrix(ML estimates) III

16. For q variance components, we have provided expressions for: The model The information matrix of the ML estimates of the location effects estimates The information matrix of the of the ML estimates of the variance components and dispersion effect estimates

17. What Criteria should be used? What criterion makes the most sense? D, A, . . . Should there be some weighting? Location and dispersion effects not on same scale or equally important. What about REML? Closed form expressions for information matrix? What designs are comparable designs? Cost.

18. The practitioner is unlikely to have accurate knowledge about which dispersion effects are important. The practitioner might only be required to provide the number of design points, r, and information about the cost of running the experiment. Level 2 Max Size Level 2 Level 1 W/C Ratio Level 1 Level 1 Grade Level 2 Cost Basis for Comparable Designs

19. Cost Structure III