A Complete Catalog of Geometrically non-isomorphic OA18

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A Complete Catalog of Geometrically
non-isomorphic OA18

• Kenny Ye
• Albert Einstein College of Medicine

June 10, 2006, ????
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Outline

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

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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)

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Two geometric non-isomorphic designs
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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

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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

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Example

• The Indicator Function
• Variable Exchange Exchange A B
• Level Reversing on factor B

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Grouping of the coefficientsExample
Coefficient Index t Group
111 1
222 2
111 121 211 3 3 3
122 212 221 4 4 4
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Total Number of Geometrically Non-Isomorphic OA18s
OA(18, 3m) OA(18, 3m) OA(18, 3m) OA(18, 3m) OA(18, 3m) 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) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m)
Non-Iso. Designs 119 1836 1332 1617 726
Maximum Designs 0 852 0 0 0
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Comparison to incomplete classification
OA(18, 3m) OA(18, 3m) OA(18, 3m) OA(18, 3m) OA(18, 3m) 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) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m) OA(18, 21 3m)
Complete 119 1836 1332 1617 726
Beta-WLP(CY2004) 118 1293 1274 1406 556
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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

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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)

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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)

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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.

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EDP another measure of (linear) model
discrimination

• Expected Distance of Predictions
• D(H1 H2)(H1 H2)
• Maximize trace(D)

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MMPD and AEPD

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

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Model Discrimination Properties

• Three-factor 2nd order models
• MMPD gt 0.75 in all the design
• Complete Aliasing of 4-factor 2nd order models

Without 2 level With 2-level
4 factors 1/1836
5 factors 2/332 13/1332
6 factors 13/478 56/1617
7 factors 5/284 10/726
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Estimation Capacity, OA(18,3m)

• Number of full capacity designs

3 factor model 4-factor model
3 factors 11/13
4 factors 122/133 98/133
5 factors 276/332 182/332
6 factors 19/478 67/478
7 factors 0/284 0/284
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Estimation Capacity, OA(18,213m)

• Number of full capacity designs

3 factor model 4-factor model
4 factors 116/119 109/119
5 factors 1253/1836 979/1836
6 factors 1008/1332 369/1332
7 factors 649/1617 67/1617
8 factors 0/726 0/726
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Acknowledgement

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