dema, august 12, 20081 a design problem 18 runs five factors

DEMA, August 12, 20081

A design problem

0 0 1 1 2 2 0 0 1 1 2 2 0 0 1 1 2 20 0 1 1 2 2 1 1 2 2 0 0 2 2 0 0 1 10 1 0 2 1 2 1 2 0 1 0 2 0 2 1 2 0 10 1 0 2 1 2 2 1 1 0 2 0 2 0 2 1 1 00 1 2 1 2 0 0 2 1 0 1 2 2 1 2 0 0 1

18 runs

Fiv

e fa

cto

rs


A design problem

0 0 1 1 2 20 0 1 1 2 20 1 0 2 1 20 1 0 2 1 20 1 2 1 2 0

0 0 1 1 2 21 1 2 2 0 01 2 0 1 0 22 1 1 0 2 00 2 1 0 1 2

0 0 1 1 2 22 2 0 0 1 10 2 1 2 0 12 0 2 1 1 02 1 2 0 0 1

Block 1 Block 2 Block 3

Eric Schoen, TNO Science & Industry (Delft, Holland) / U. of Antwerp (Belgium)

A blocking strategy for Orthogonal Arrays of strength 2


Contents

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


Generalized Word Length Pattern

• 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


Application to introductory design

• 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


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


Searching an ordered design catalog

• Schoen (2007): all combinatorially non-isomorphic 18-run arrays

• Ordered according to GWLP• 2, 3 or 6 blocks

a 3a 3a21 613a

2 1 3 23 4 15 34 12 48 15 10 19 16 8 12 17 3 3 -


Simple selection

• 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)


Some blocked 35 arrays

Array ttt, tttt ttb, tttb

5.0.1/6.0.1 5, 7.5 5, 15

5.0.1/6.0.8 5, 7.5 8, 6

5.0.10/6.0.5 8, 1.5 5, 12


Application to two-level arrays

a 512a

2 33 104 155 386 307 48 19 -

N = 20 • 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.


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.

dema, august 12, 20081 a design problem 18 runs five factors

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