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1 STA 536 – Nonregular Designs: Construction and Properties Chapter 8: Non-regular Designs: Construction and Properties regular designs: 2 kp and 3 kp 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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2 STA 536 – Nonregular Designs: Construction and Properties 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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3 STA 536 – Nonregular Designs: Construction and Properties OA(12, 2 11 )

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4 STA 536 – Nonregular Designs: Construction and Properties 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)

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5 STA 536 – Nonregular Designs: Construction and Properties Design Matrix and Response Data, Blood Glucose Experiment OA(18, 2x3 7 )

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6 STA 536 – Nonregular Designs: Construction and Properties orthogonal arrays In Tables 7.2 and 7.4, the design used does not belong to the 2 kp series (Chapter 5) or the 3 kp 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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7 STA 536 – Nonregular Designs: Construction and Properties Examples: OA(12, 2 11 ) in Table 7.2. OA(18, ) in Table kp, 3 kp and Latin squares are (regular) OAs. A 2 kp R design is an OA(N = 2 kp, 2 k, t = R 1).

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8 STA 536 – Nonregular Designs: Construction and Properties 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, s 1 m1 · · · s γ mγ ) has strength t = 2.

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9 STA 536 – Nonregular Designs: Construction and Properties 7.2 Some Advantages of Nonregular Designs Over the 2 kp and 3 kp Series of Designs Run size economy. Suppose 8-11 factors at two levels are to be studied. Using an OA(12,2 11 ) will save 4 runs over a 16-run 2 kp design. Similarly, suppose 5-7 factors at three levels are to be studied. Using an OA(18,3 7 ) will save 9 runs over a 27-run 3 kp design. Flexibility. Many OAs exist for flexible combinations of factor levels.

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10 STA 536 – Nonregular Designs: Construction and Properties Facts on regular designs The run size of a 2 k or 2 kp design must be 4, 8, 16, 32,… Max number of factors to be studied are 3, 7, 15, 31,... The run size of a 3 k or 3 kp 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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11 STA 536 – Nonregular Designs: Construction and Properties To study 7 two-level factors, can use 2 73 IV (16 runs) 2 74 III (8 runs, saturated, no df for error estimation) 12-run OA in Table 7.2. To study 8-11 two-level factors A regular design needs at least 16 runs (2 84, ). A nonregular design in Table 7.2 has 12 runs.

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12 STA 536 – Nonregular Designs: Construction and Properties To study 7 three-level factors A regular design needs at least 27 runs (3 74 ). 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 OAs Any 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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13 STA 536 – Nonregular Designs: Construction and Properties 7.3 A Lemma on Orthogonal Arrays

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14 STA 536 – Nonregular Designs: Construction and Properties 7.4 Plackett-Burman Designs and Halls Designs Statistical Properties of OAs For an OA(N, 2 N1 ) A = (a ij ), consider the main effects model: with xi = ±1. The model matrix is an N × N matrix X = (1 A). Because A is an OA, X is an orthogonal matrix: X T X = XX T = N I N The least squares estimate of The covariance matrix of the estimates is

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15 STA 536 – Nonregular Designs: Construction and Properties Therefore, for an OA(N, 2 N1 ), assuming interactions are negligible, All main effects are estimable. The estimates of main effects are independent.

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16 STA 536 – Nonregular Designs: Construction and Properties Hadamard matrix Definition: A Hadamard matrix of order N, denoted by H N, is an N ×N orthogonal matrix with entries 1 or 1, that is H N T H N = H N H N T = NI N We can always normalize (or standardize) a Hadamard matrix so that its first column consists of 1s. Then the remaining N 1 columns is an OA(N, 2 N1 ). An OA(N, 2 N1 ) 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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17 STA 536 – Nonregular Designs: Construction and Properties Hadamard conjecture: If N is a multiple of 4, a Hadamard matrix of order N exists. For N = 2 k, it is true. If H N is a Hadamard matrix of order N, then is a Hadamard matrix of order 2N. It is true for N 256

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18 STA 536 – Nonregular Designs: Construction and Properties Plackett-Burman designs are special OA(N, 2 N1 ) or Hadamard matrices Table 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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19 STA 536 – Nonregular Designs: Construction and Properties Plackett-Burman designs

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20 STA 536 – Nonregular Designs: Construction and Properties Halls 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 design. Type II–V are nonregular OA(16, 2 15 ) (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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21 STA 536 – Nonregular Designs: Construction and Properties 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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22 STA 536 – Nonregular Designs: Construction and Properties 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, ) and OA(12, ) For OA(N, ), N is a multiple of l.c.m.( , 2 2 ) = 12. There is no OA(12, k ) with k > 4. For OA(12, ), there are 11 (3 1) 4(2 1) = 5 df left for error estimations. OA(12, ) is a nearly orthogonal array. pairs of columns (4, 6) and (5, 7) are not orthogonal.

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23 STA 536 – Nonregular Designs: Construction and Properties 7.5 A Collection of Useful Mixed-Level OAs 2. Table 7C.2 OA(18, ) and OA(18, ). For OA(N, ), N is a multiple of l.c.m.( , 3 2 ) = 18. OA(18, ) can estimate all the MEs plus the interaction between columns 1 and 2. OA(18, ) is saturated (for the main effects model). 3. Table 7C.3 OA(20, ) and OA(20, ) 4. Tables 7C.4 and 7C.5 OA(24, ) and OA(24, ).

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24 STA 536 – Nonregular Designs: Construction and Properties 7.5 A Collection of Useful Mixed-Level OAs 5. Tables 7C.6 and 7C.7 OA(36, ) and OA(36, ). 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, ) and OA(36, ). 7. Table 7C.10 OA(48, ). saturated. 8. Table 7C.11 OA(50, ) and OA(50, ) 9. Table 7C.12 OA(54, ) and OA(54, ).

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25 STA 536 – Nonregular Designs: Construction and Properties 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 Ma, C.X, Fang, K.T., Liski E. A new approach in constructing orthogonal and nearly orthogonal arrays, METRIKA 50 (3): Xu, H. (2002). An algorithm for constructing orthogonal and nearly orthogonal arrays with mixed levels and small runs.

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26 STA 536 – Nonregular Designs: Construction and Properties 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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27 STA 536 – Nonregular Designs: Construction and Properties Optimal Choice of Nonregular Designs Generalized minimum aberration criterion Ma CX, Fang KT, A note on generalized aberration in factorial designs, METRIKA 53 (1): Xu, H. and Wu, C. F. J. (2001). Generalized minimum aberration for asymmetrical fractional factorial designs. Annals of Statistics, 29,

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