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Chapter 8: Non-regular Designs: Construction and Properties

regular designs: 2k−p and 3k−p

constructed through defining relations among factors.

any two factorial effects can either be estimated independently of each other or are fully aliased.

nonregular designs: orthogonal arrays do not have defining contrast subgroups. some factorial effects are partially aliased (0

< |correlation| < 1).

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8.1 Two Experiments: Weld Repaired Castings and Blood Glucose Testing

Weld Repaired Castings Experiment used a 12-run design to study the effects of seven factors on

the fatigue life of weld repaired castings. The response is the logged lifetime of the casting The goal of the

experiment was to identify the factors that affect the casting lifetime.

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OA(12, 211)

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Blood glucose testing experiment to study the effect of 1 two-level factor and 7 three-level

factors on blood glucose readings made by a clinical laboratory testing device.

used an 18-run mixed-level orthogonal array. factor F combines two variables, sensitivity and absorption

(because the 18-run design cannot accommodate eight three-level factors)

5STA 536 – Nonregular Designs: Construction and Properties

Design Matrix and Response Data, Blood Glucose Experiment

OA(18, 2x37)

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orthogonal arrays

In Tables 7.2 and 7.4, the design used does not belong to the 2k−p series (Chapter 5) or the 3k−p series (Chapter 6), because the latter would require run size as a power of 2 or 3. These designs belong to the class of orthogonal arrays.

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Examples:

OA(12, 211) in Table 7.2.OA(18, 2137) in Table 7.4.2k−p, 3k−p and Latin squares are

(regular) OAs.A 2k−p

R design is an OA(N = 2k−p, 2k, t = R − 1).

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Symmetrical and Asymmetrical OAs

Symmetrical OAs: all factors have the same number of levels (i.e., γ=1).

Asymmetrical (or mixed-level) OAs: γ> 1.

Convention: An OA(N, s1

m1 · · · sγmγ )

has strength t = 2.

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7.2 Some Advantages of Nonregular Designs Over the 2k−p and 3k−p Series of Designs Run size economy. Suppose 8-11 factors at two levels

are to be studied. Using an OA(12,211) will save 4 runs over a 16-run 2k−p design. Similarly, suppose 5-7 factors at three levels are to be studied. Using an OA(18,37) will save 9 runs over a 27-run 3k−p design.

Flexibility. Many OA’s exist for flexible combinations of factor levels.

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Facts on regular designs

The run size of a 2k or 2k−p design must be 4, 8, 16, 32,… Max number of factors to be studied are 3, 7, 15,

31, ...The run size of a 3k or 3k−p design must be 9, 27, 81, …

Max number of factors to be studied are 4, 13, 40, …The gaps in the run sizes becomes larger and larger.

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To study 7 two-level factors, can use27−3

IV (16 runs)27−4

III (8 runs, saturated, no df for error estimation)

12-run OA in Table 7.2.

To study 8-11 two-level factorsA regular design needs at least 16 runs (28−4,

211−7).A nonregular design in Table 7.2 has 12 runs.

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To study 7 three-level factorsA regular design needs at least 27 runs (37−4).A nonregular design in Table 7.4 has 18 runs.The 18-run OA in Table 7.4 can accommodate 1

two-level factor.Mixed-level OAs are flexible in accommodating

various combinations of factors with different numbers of levels.

An important property of OAsAny two factorial effects represented by the

columns of an OA can be estimated and interpreted independently of each other (assuming interaction effects are negligible).

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7.3 A Lemma on Orthogonal Arrays

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7.4 Plackett-Burman Designs and Hall’s Designs

Statistical Properties of OAs For an OA(N, 2N−1) A = (aij), consider the main effects

model:

with xi = ±1. The model matrix is an N × N matrixX = (1 A).

Because A is an OA, X is an orthogonal matrix:

XTX = XXT = N IN The least squares estimate of The covariance matrix of the estimates is

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Therefore, for an OA(N, 2N−1), assuming interactions are negligible,

All main effects are estimable.The estimates of main effects are

independent.

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Hadamard matrix

Definition: A Hadamard matrix of order N, denoted by HN, is an N ×N orthogonal matrix with entries 1 or −1, that is

HNTHN = HNHN

T= NIN We can always normalize (or standardize) a

Hadamard matrix so that its first column consists of 1’s. Then the remaining N −1 columns is an OA(N, 2N−1).

An OA(N, 2N−1) is equivalent to a Hadamard matrix of order N.

A necessary condition for the existence of a Hadamard matrix of order N is N = 1, 2, or a multiple of 4.

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Hadamard conjecture:

If N is a multiple of 4, a Hadamard matrix of order N exists.

For N = 2k, it is true. If HN is a Hadamard matrix of order N, then

is a Hadamard matrix of order 2N. It is true for N ≤ 256

http://www.research.att.com/njas/hadamard/

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Plackett-Burman designs are special OA(N, 2N−1) or Hadamard matrices

Table 7.2. 12-run P-B designs cyclically shift the first row (generator) to the left 10

times. add a row of −’s.

Appendix 7A (p. 330). cyclically shift the first row (generator) to the right 10

times. add a row of −’s.

For N=12, 20, 24, 36, 44, P-B designs are cyclic (see Table 7.5 and Appendix 7A).

For N = 28, see Appendix 7A (p. 332).

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Plackett-Burman designs

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Hall’s designs are Hadamard matrices of order 16 and 20.

Hall (1961): 5 Hadamard matrices of order 16, called Types I-V. Type I is a regular 215−11 design. Type II–V are nonregular OA(16, 215) (see Appendix

7B). Hall (1965): 3 Hadamard matrices of order 20, called

Types Q, N, P. Type Q is equivalent to the 20-run P-B design.

The number of inequivalent Hadamard matrices are

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Remarks: Nonregular designs such as P-B designs

have complex aliasing among factorial effects.are traditionally used for screening main effects

(assuming interactions are negligible).have some interesting hidden projection

properties.enable to estimate a few interactions (with

effect sparsity) (see Chap. 8).

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7.5 A Collection of Useful Mixed-Level OAs

Appendix 7C gives a collection of mixed-level OAs with 12–54 runs and 2–6 levels.

1. Table 7C.1 OA(12, 3124) and OA’(12, 3126) For OA(N, 3124), N is a multiple of l.c.m.(3121,

22) = 12. There is no OA(12, 312k) with k > 4. For OA(12, 3124), there are 11 − (3 − 1) −

4(2 − 1) = 5 df left for error estimations. OA’(12, 3126) is a nearly orthogonal array.

pairs of columns (4, 6’) and (5, 7’) are not orthogonal.

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7.5 A Collection of Useful Mixed-Level OAs

2. Table 7C.2 OA(18, 2137) and OA(18, 6136). For OA(N, 2137), N is a multiple of l.c.m.(2131,

32) = 18. OA(18, 2137) can estimate all the ME’s plus

the interaction between columns 1 and 2. OA(18, 6136) is saturated (for the main

effects model).3. Table 7C.3 OA(20, 5128) and OA’(20, 51210)4. Tables 7C.4 and 7C.5 OA(24, 31216) and OA(24,

61214).

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7.5 A Collection of Useful Mixed-Level OAs

5. Tables 7C.6 and 7C.7 OA(36, 211312) and OA(36, 312211). They are arranged to minimize the number of level

changes for two level (Table 7C.6) and three-level (Table 7C.7) factors, respectively.

They are saturated.6. Tables 7C.8 and 7C.9 OA(36, 3763) and OA(36, 2863).7. Table 7C.10 OA(48, 211412).

saturated.8. Table 7C.11 OA(50, 21511) and OA’(50, 23511)9. Table 7C.12 OA(54, 21325) and OA(54, 61324).

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7.5 A Collection of Useful Mixed-Level OAs

More OAs and nearly OAs A library of OAs with run size 100 is available online at

http://neilsloane.com/doc/cent4.html Ma, C.X, Fang, K.T., Liski E. A new approach in

constructing orthogonal and nearly orthogonal arrays,METRIKA 50 (3): 255-268 2000

Xu, H. (2002). An algorithm for constructing orthogonal and nearly orthogonal arrays with mixed levels and small runs.

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7.5 A Collection of Useful Mixed-Level OAs

* especially useful

Learn to choose and use the design tables in the collection.

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Optimal Choice of Nonregular Designs

Generalized minimum aberration criterion

Ma CX, Fang KT, A note on generalized aberration in factorial designs, METRIKA 53 (1): 85-93 2001

Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs.  Annals of Statistics, 29, 1066-1077.