A Complete Catalog of Geometrically non-isomorphic OA18

1. A Complete Catalog of Geometrically non-isomorphic OA18 Kenny Ye Albert Einstein College of Medicine June 10, 2006, 南開大學

2. Outline • Construction of the Complete Catalog of OA18 • Design Properties of OA18 for Response Surface Studies • Model-Discrimination • Model-Estimation

3. Geometric Isomorphism, Cheng and Ye (AOS 2004) • For experiments with quantitative factors, properties of factorial designs depends on their geometric structure • Two designs are geometrically isomorphic if one can be obtained by a series of two kinds of operations: • Variable Exchange • Level Reversing • Tsai, Gilmore, Mead (Biometrika 2000) • Clark and Dean (Statistica Sinica 2001)

4. Two geometric non-isomorphic designs

5. Construction of the complete catalog of OA18 • Construct all geometrically non-isomorphic cases of OA(18,3m) • Check geometric isomorphism • Adding one factor at a time • Add the two-level column to the OA(18,3m). • Main difficulty: isomorphism checking

6. Determine Geometric Isomorphism using Indicator Function • Indicator Function, Cheng and Ye(2004) • A factorial design is uniquely represented by a linear combination of orthonormal contrasts defined on a full factorial design • Variable exchange rearranges the position of the coefficients within sub-groups • level reversal changes the sign of the coefficients

7. The Indicator Function Variable Exchange: Exchange A & B Level Reversing on factor B Example

8. Grouping of the coefficientsExample:

9. Total Number of Geometrically Non-Isomorphic OA18s

10. Comparison to incomplete classification

11. Combinatorial Non-isomorphic OA18s • Indicator function approach is not efficient for isomorphism checking • Subset of the geometrically non-isomorphic OA18s • In practice, the larger catalog is enough • Currently working with AM Dean to further classify into combinatorial isomorphism

12. Response Surface Method • Original two-step approach • Factor screening • Response surface exploration • 3-level factorial designs for selecting response surface models - Cheng and Wu(2001 Statistica Sinica)

13. Design properties for response surface studies • Three-level factorial designs can be used by response surface studies (Cheng and Wu, SS 2001) • Fitting second order polynomial model on projections • Estimation efficiency (Xu, Cheng, Wu Technometrics 2004) • Estimation Capacity • Information Capacity (Average Efficiency) • Model Discrimination Criteria (Jones, Li, Nachtsheim, Ye, JSPI, 2005)

14. MDP: a measure of (linear) model discrimination • Maximum difference of predictions • Computation: Find the largest absolute eigenvalues of H1 – H2 • MDP is no greater than 1.

15. EDP: another measure of (linear) model discrimination • Expected Distance of Predictions • D=(H1 – H2)(H1 – H2) • Maximize trace(D)

16. MMPD and AEPD • Min-Max Prediction Difference (MMPD) • Average Expected Prediction Difference (AEPD)

17. Model Discrimination Properties • Three-factor 2nd order models • MMPD > 0.75 in all the design • Complete Aliasing of 4-factor 2nd order models

18. Estimation Capacity, OA(18,3m) • Number of full capacity designs

19. Estimation Capacity, OA(18,213m) • Number of full capacity designs

20. Acknowledgement • Joint work with Ko-Jen Tsai and William Li • Much of the work is in the Ph.D. dissertation of Ko-Jen Tsai