1. DEMA, August 12, 20081 A design problem 18 runs Five factors

2. DEMA, August 12, 20082 A design problem Block 1Block 2 Block 3

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

4. DEMA, August 12, 20084 Contents Optimality criteria for strength-2 designs and blocking Searching an ordered design catalog Conclusions

5. DEMA, August 12, 20085 Generalized Word Length Pattern n factors (A 1, A 2, …, A n ) A p : sum of squared and standardized inner products of q and (p-q)-factor interactions Generalizes WLP for regular designs. Generalizes G 2 -aberration for two- level designs Xu and Wu (2001), Annals

6. DEMA, August 12, 20086 Application to introductory design Including the blocking factor: OA(18; 3 6 ; 2) Excluding the blocking factor: OA(18; 3 5 ; 2) subtraction (A 3, A 4 ) = (13, 13.5) (A 3, A 4 ) = ( 5, 7.5) ________________ (A 21, A 31 )= (8, 6) Confounding 2fi/3fi with blocks

7. DEMA, August 12, 20087 Three blocking criteria If we can recover inter-block information: W 1 : ttt << tttt << ttb << tttb If there is no hope to recover inter-block information: W 2 : ttt << ttb << tttt << tttb To improve error estimation: W 3 : ttt << -ttb << tttt << tttb

8. DEMA, August 12, 20088 Searching an ordered design catalog Schoen (2007): all combinatorially non-isomorphic 18-run arrays Ordered according to GWLP 2, 3 or 6 blocks

9. DEMA, August 12, 20089 Simple selection Minimization of ttt words (all criteria): 5.0.1 is the unique array with minimum ttt W 1 (ttt << tttt) is satisfied if 3 6 designs project into minimum aberration 3 5 6.0.1, 6.0.5, 6.0.8 project into 5.0.1 Minimization of ttb (W 2 ): Choosing 6.0.1 minimizes A 3 (6 factors) – A 3 (5 factors) Maximization of ttb (W 3 ): Choosing 6.0.8 maximizes A 3 (6 factors) – A 3 (5 factors)

10. DEMA, August 12, 200810 Some blocked 3 5 arrays

11. DEMA, August 12, 200811 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 2 a ). This permits blocking in five blocks of size 4.

12. DEMA, August 12, 200812 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.