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

FullEC OA with strength 3 2Nankai University, July Outline Motivating example Motivating example Strength or efficiency? Strength or efficiency? Selection of arrays Selection of arrays Details of best arrays Details of best arrays

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

FullEC OA with strength 3 4Nankai University, July Full Estimation Capacity Most of the factors will be active. Most of the factors will be active. Many active interactions. Many active interactions. Design should permit estimation of all main effects (me) and all 2-factor interactions (2fi). Design should permit estimation of all main effects (me) and all 2-factor interactions (2fi). Parameters in wood technology example: Parameters in wood technology example:  1 intercept  9 me  32 2fi

FullEC OA with strength 3 5Nankai University, July Design options Choose orthogonal array (OA) of maximum strength compatible with run-size & factor specifications. 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. Choose array that maximizes D-efficiency for the p parameters of the full model.

FullEC OA with strength 3 6Nankai University, July Orthogonal arrays Example: OA(4, 2 3, 2). Example: OA(4, 2 3, 2). Symbols arranged in N rows and n columns. Symbols arranged in N rows and n columns. Rows: runs of a design. Rows: runs of a design. Columns: factors of a design. Columns: factors of a design. Strength t : each t-tuple of symbols occurs equally often. Strength t : each t-tuple of symbols occurs equally often

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

FullEC OA with strength 3 8Nankai University, July Optimization of D-efficiency X 1 : matrix with intercept + me ( length  N ) X 1 : matrix with intercept + me ( length  N ) X 2 : matrix with 2fi ( length  N ) X 2 : matrix with 2fi ( length  N ) X = [X 1 X 2 ] X = [X 1 X 2 ] D = |X’X| 1/p /N D = |X’X| 1/p /N 0  D  1 0  D  1 Maximum D minimizes volume of confidence region. Maximum D minimizes volume of confidence region.

FullEC OA with strength 3 9Nankai University, July Efficiency of me estimation Primary goal: efficient estimation of main effects to separate active from inactive me. Primary goal: efficient estimation of main effects to separate active from inactive me. D 1 = |X 1 ’(I - X 2 (X 2 ’X 2 ) -1 X 2 ’)X 1 | 1/p1 /N D 1 = |X 1 ’(I - X 2 (X 2 ’X 2 ) -1 X 2 ’)X 1 | 1/p1 /N residual sum of squares and products after projecting X 1 on X 2 residual sum of squares and products after projecting X 1 on X 2 Orthogonal arrays of strength 3 have D 1 = 1. Orthogonal arrays of strength 3 have D 1 = 1.

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

FullEC OA with strength 3 11Nankai University, July Selection of arrays Strength is best Generate all non- isomorphic (48; 4 x 3 x 2 4 ; 3): 19 arrays. Generate all non- isomorphic OA(48; 4 x 3 x 2 4 ; 3): 19 arrays. Select FullEC arrays: 14 arrays. Select FullEC arrays: 14 arrays. Select array with maximum D 2 : 1 array. Select array with maximum D 2 : 1 array. D-efficiency is best Generate 50 random arrays. Improve D-efficiency with modified Fedorov algorithm. Keep the best array.

FullEC OA with strength 3 12Nankai University, July Details of best arrays strength D1D1D1D1 D2D2D2D

FullEC OA with strength 3 13Nankai University, July Some other 48-run cases 6 x 2 5 t=3: 30 non-isomorphic arrays. t=3: 30 non-isomorphic arrays. No FulEC array. No FulEC array. Best D 2 -efficiency Best D 2 -efficiency Strength 0. Strength 0. D 1 -efficiency D 1 -efficiency x 2 7 t=3: 3056 ni arrays. 209 FulEC array. Best D 2 given t=3: Best D 2 -efficiency Strength 0. D 1 -efficiency 0.71.

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

FullEC OA with strength 3 15Nankai University, July Contact information (see also abstract) (see also abstract)