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A design problem

A design problem. 18 runs. Five factors. A design problem. Block 3. Block 1. Block 2. A blocking strategy for Orthogonal Arrays of strength 2. Contents. Optimality criteria for strength-2 designs and blocking Searching an ordered design catalog Conclusions. n factors

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A design problem

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  1. A design problem 18 runs Five factors

  2. A design problem Block 3 Block 1 Block 2

  3. A blocking strategy for Orthogonal Arrays of strength 2 Eric Schoen, TNO Science & Industry (Delft, Holland) / U. of Antwerp (Belgium)

  4. Contents • Optimality criteria for strength-2 designs and blocking • Searching an ordered design catalog • Conclusions

  5. n factors (A1, A2, …, An) Ap: sum of squared and standardized inner products of q and (p-q)-factor interactions Generalizes WLP for regular designs. Generalizes G2-aberration for two-level designs Xu and Wu (2001), Annals Generalized Word Length Pattern

  6. Including the blocking factor: OA(18; 36; 2) Excluding the blocking factor: OA(18; 35; 2) subtraction (A3, A4) = (13, 13.5) (A3, A4) = ( 5, 7.5) ________________ (A21, A31)= (8, 6) Confounding 2fi/3fi with blocks Application to introductory design

  7. Three blocking criteria If we can recover inter-block information: W1: ttt << tttt << ttb << tttb If there is no hope to recover inter-block information: W2: ttt << ttb << tttt << tttb To improve error estimation: W3: ttt << -ttb << tttt << tttb

  8. Schoen (2007): all combinatorially non-isomorphic 18-run arrays Ordered according to GWLP 2, 3 or 6 blocks Searching an ordered design catalog

  9. Minimization of ttt words (all criteria): 5.0.1 is the unique array with minimum ttt W1 (ttt << tttt) is satisfied if 36 designs project into minimum aberration 35 6.0.1, 6.0.5, 6.0.8 project into 5.0.1 Minimization of ttb (W2): Choosing 6.0.1 minimizes A3(6 factors) – A3 (5 factors) Maximization of ttb (W3): Choosing 6.0.8 maximizes A3(6 factors) – A3 (5 factors) Simple selection

  10. Some blocked 35 arrays

  11. Application to two-level arrays • Existing method: combine two-level columns to a four-level column. • Does not work for N=20. • However, we can generate OA(20; 5 x 2a). • This permits blocking in five blocks of size 4.

  12. Conclusions • Blocking of orthogonal arrays. • Classification with GWLP. • GWLP catalog including blocking factor. • Projections into arrays with one factor less. • Three blocking criteria, including maximization of ttb words.

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