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(18,3 m ) Check geometric isomorphism Adding one factor at a time Add the two-level column to the OA(18,3 m ). 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. Example The Indicator Function Variable Exchange: Exchange A & B Level Reversing on factor B

8. Grouping of the coefficients Example: Coefficient Index tGroup 1111 2222 111 121 211 333333 122 212 221 444444

9. Total Number of Geometrically Non-Isomorphic OA18s OA(18, 3 m ) # 3-level factors34567 # Non-Iso. Designs13133332478284 # Maximum Designs044000 OA(18, 2 1 3 m ) # Non-Iso. Designs119183613321617726 # Maximum Designs0852000

10. Comparison to incomplete classification OA(18, 3 m ) # 3-level factors34567 Complete13133332478284 Q-Crit (TMG2000)13129320440223 Beta-WLP(CY2004)13129320440253 OA(18, 2 1 3 m ) Complete119183613321617726 Beta-WLP(CY2004)118129312741406556

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 H 1 – H 2 MDP is no greater than 1.

15. EDP: another measure of (linear) model discrimination Expected Distance of Predictions D=(H 1 – H 2 )(H 1 – H 2 ) 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 Without 2 levelWith 2-level 4 factors 1/1836 5 factors 2/33213/1332 6 factors 13/47856/1617 7 factors 5/28410/726

18. Estimation Capacity, OA(18,3 m ) Number of full capacity designs 3 factor model4-factor model 3 factors 11/13 4 factors 122/13398/133 5 factors 276/332182/332 6 factors 19/47867/478 7 factors 0/284

19. Estimation Capacity, OA(18,2 1 3 m ) Number of full capacity designs 3 factor model4-factor model 4 factors 116/119109/119 5 factors 1253/1836979/1836 6 factors 1008/1332369/1332 7 factors 649/161767/1617 8 factors 0/726

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