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A blocking strategy for Orthogonal Arrays of strength 2

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

Contents

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

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 PatternIncluding 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 designThree 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

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

Ordered according to GWLP

2, 3 or 6 blocks

Searching an ordered design catalogMinimization of arraysttt 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 selectionSome blocked 3 arrays5 arrays

Application to two-level arrays 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.

Conclusions arrays

- 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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