Orthogonal arrays of strength 3 with full estimation capacities

1. Orthogonal arrays of strength 3 withfull estimation capacities Eric D. Schoen; Man V.M. Nguyen

2. Outline • Motivating example • Strengthor efficiency? • Selection of arrays • Details of best arrays Nankai University, July 10 2006

3. Example from wood technology • Response: strength of wood-glue-wood bond • Factors: • glue (4 levels) • wood (3 levels) • moisture content, pretreatment, pressure, storage temperature (2 levels) • About 50 runs Nankai University, July 10 2006

4. Full Estimation Capacity • Most of the factors will be active. • Many active interactions. • Design should permit estimation of all main effects (me) and all 2-factor interactions (2fi). • Parameters in wood technology example: • 1 intercept • 9 me • 32 2fi Nankai University, July 10 2006

5. Design options • Choose orthogonal array (OA) of maximum strength compatible with run-size & factor specifications. • Choose array that maximizes D-efficiency for the p parameters of the full model. Nankai University, July 10 2006

6. Orthogonal arrays • Example: OA(4, 23, 2). • Symbols arranged in N rows and n columns. • Rows: runs of a design. • Columns: factors of a design. • Strength t : each t-tuple of symbols occurs equally often. Nankai University, July 10 2006

7. Properties for strength t = 2, 3, 4 • t = 2: orthogonality among main effects. • t = 3: additional orthogonality me  2fi. • t = 4: additional orthogonality among 2fi. • In example: run sizes multiple of 24, 48, 96 for strength t = 2, 3, 4. • Find 48-run array of strength 3 and FullEC. Nankai University, July 10 2006

8. Optimization of D-efficiency • X1: matrix with intercept + me ( length N ) • X2: matrix with 2fi ( length N ) • X = [X1 X2 ] • D = |X’X|1/p/N • 0  D 1 • Maximum D minimizes volume of confidence region. Nankai University, July 10 2006

9. Efficiency of me estimation • Primary goal: efficient estimation of main effects to separate active from inactive me. • D1 = |X1’(I - X2(X2’X2)-1X2’)X1|1/p1/N • residual sum of squares and products after projecting X1 on X2 • Orthogonal arrays of strength 3 have D1 = 1. Nankai University, July 10 2006

10. Efficiency of 2fi estimation • Primary goal: distinguishing among 2fi. • D2 = |X2’(I - X1(X1’X1)-1X1’)X2|1/p2/N • |X’X|= (ND2)p2.|X1’X1| • If an array has higher |X’X| than has a strength-3 array, it must have higher D2-efficiency too. Nankai University, July 10 2006

11. Strength is best Generate all non-isomorphic OA(48; 4 x 3 x 24; 3): 19 arrays. Select FullEC arrays: 14 arrays. Select array with maximum D2: 1 array. D-efficiency is best Generate 50 random arrays. Improve D-efficiency with modified Fedorov algorithm. Keep the best array. Selection of arrays Nankai University, July 10 2006

12. Details of best arrays Nankai University, July 10 2006

13. 6 x 25 t=3: 30 non-isomorphic arrays. No FulEC array. Best D2-efficiency 0.77. Strength 0. D1-efficiency 0.72. 3 x 27 t=3: 3056 ni arrays. 209 FulEC array. Best D2 given t=3: 0.77. Best D2-efficiency 0.80. Strength 0. D1-efficiency 0.71. Some other 48-run cases Nankai University, July 10 2006

14. Conclusion Separation of main effects & 2fi? • Use FullEC array of strength 3. • Choose array with highest D2-efficiency. Activity of 2fi? • Use FullEC array with optimum D2-efficiency. Nankai University, July 10 2006

15. Contact information • eric.schoen@tno.nl • e.d.schoen@tue.nl • (see also abstract) Nankai University, July 10 2006