A Complete Catalog of Geometrically non-isomorphic OA18

Kenny Ye

Albert Einstein College of Medicine

June 10, 2006, 南開大學

Outline

Construction of the Complete Catalog of OA18

Design Properties of OA18 for Response Surface Studies Model-Discrimination Model-Estimation

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)

Two geometric non-isomorphic designs

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

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

Example The Indicator Function

Variable Exchange: Exchange A & B

Level Reversing on factor B

Grouping of the coefficientsExample:

Coefficient Index t Group

111 1

222 2

111

121

211

3

3

3

122

212

221

4

4

4

Total Number of Geometrically Non-Isomorphic OA18s

OA(18, 3m)

# 3-level factors 3 4 5 6 7

# Non-Iso. Designs 13 133 332 478 284

# Maximum Designs 0 44 0 0 0

OA(18, 21 3m)

# Non-Iso. Designs 119 1836 1332 1617 726

# Maximum Designs 0 852 0 0 0

Comparison to incomplete classification

OA(18, 3m)

# 3-level factors 3 4 5 6 7

Complete 13 133 332 478 284

Q-Crit (TMG2000) 13 129 320 440 223

Beta-WLP(CY2004) 13 129 320 440 253

OA(18, 21 3m)

Complete 119 1836 1332 1617 726

Beta-WLP(CY2004) 118 1293 1274 1406 556

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

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)

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)

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.

EDP: another measure of (linear) model discrimination

Expected Distance of Predictions

D=(H1 – H2)(H1 – H2) Maximize trace(D)

MMPD and AEPD

Min-Max Prediction Difference (MMPD)

Average Expected Prediction Difference (AEPD)

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 level With 2-level

4 factors 1/18365 factors 2/332 13/13326 factors 13/478 56/16177 factors 5/284 10/726

Estimation Capacity, OA(18,3m)

Number of full capacity designs3 factor model 4-factor model

3 factors 11/134 factors 122/133 98/1335 factors 276/332 182/3326 factors 19/478 67/4787 factors 0/284 0/284

Estimation Capacity, OA(18,213m)

Number of full capacity designs

3 factor model 4-factor model

4 factors 116/119 109/1195 factors 1253/1836 979/18366 factors 1008/1332 369/13327 factors 649/1617 67/16178 factors 0/726 0/726

Acknowledgement

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

dissertation of Ko-Jen Tsai