Nonregular Designs: Construction and Properties - Orthogonal Arrays and Their Advantages, Lab Reports of Statistics

Download Nonregular Designs: Construction and Properties - Orthogonal Arrays and Their Advantages and more Lab Reports Statistics in PDF only on Docsity! Chapter 7 Nonregular Designs: Construction and Prop- erties • 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). 7.1 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. Table 7.1 Factors and Levels, Cast Fatigue Experiment Level Factor − + A. initial structure as received β treat B. bead size small large C. pressure treat none HIP D. heat treat anneal solution treat/age E. cooling rate slow rapid F. polish chemical mechanical G. final treat none peen 1 Table 7.2 Design Matrix and Lifetime Data, Cast Fatigue Experiment Factor Logged Run A B C D E F G 8 9 10 11 Lifetime 1 + + − + + + − − − + − 6.058 2 + − + + + − − − + − + 4.733 3 − + + + − − − + − + + 4.625 4 + + + − − − + − + + − 5.899 5 + + − − − + − + + − + 7.000 6 + − − − + − + + − + + 5.752 7 − − − + − + + − + + + 5.682 8 − − + − + + − + + + − 6.607 9 − + − + + − + + + − − 5.818 10 + − + + − + + + − − − 5.917 11 − + + − + + + − − − + 5.863 12 − − − − − − − − − − − 4.809 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) Table 7.3 Factors and Levels, Blood Glucose Experiment Level Factor 0 1 2 A. wash no yes B. microvial volume (ml) 2.0 2.5 3.0 C. caras H2O level (ml) 20 28 35 D. centrifuge RPM 2100 2300 2500 E. centrifuge time (min) 1.75 3 4.5 F. (sensitivity, absorption) (0.10,2.5) (0.25,2) (0.50,1.5) G. temperature (0C) 25 30 37 H. dilution ratio 1:51 1:101 1:151 2 • 12-run OA in Table 7.2. To study 8-11 two-level factors • A regular design needs at least 16 runs (28−4, 211−7). • A nonregular design in Table 7.2 has 12 runs. To study 7 three-level factors • A 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 OAs • Any two factorial effects represented by the columns of an OA can be estimated and interpreted independently of each other (assuming inter- action effects are negligible). 7.3 A Lemma on Orthogonal Arrays Lemma 7.1. For an OA(N, s1 m1 · · · sγmγ , t), its run size N must be divisible by the least common multiple (l.c.m.) of s1 k1s2 k2 · · · sγkγ , for all possible combinations of ki with ki ≤ mi and k1 + k2 + · · ·+ kγ = t. Examples • OA(N, 2137, 2): N is a multiple of l.c.m.(2131, 32) = 18. • OA(N, 2237, 2): N is a multiple of l.c.m.(22, 2131, 32) = 36. • OA(N, 2137, 3): N is a multiple of l.c.m.(2132, 33) = 54. • OA(N, 4131210, 2): N is a multiple of l.c.m.(4131, 4121, 3121, 22) = 24. 5 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: y = β0 + x1β1 + x2β2 + · · ·+ xN−1βN−1 + , with xi = ±1. The model matrix is an N ×N matrix X = (1A). Because A is an OA, X is an orthogonal matrix: XTX = XXT = NIN The least squares estimate of β = (β0, β1, . . . , βN−1) T = (XTX)−1XTY = N−1XTY. The covariance matrix of the estimates is cov(β) = σ2(XTX)−1 = σ2IN/N. Therefore, for an OA(N, 2N−1), assuming interactions are negligible, • All main effects are estimable. • The estimates of main effects are independent. Definition: A Hadamard matrix of order N , denoted by HN , is an N×N orthogonal matrix with entries 1 or −1, that is HTNHN = HNH T N = 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. 6 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 H2N = ( HN HN HN −HN ) is a Hadamard matrix of order 2N . • It is true for N ≤ 256 http://www.research.att.com/∼njas/hadamard/ 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 (genertor) to the left 10 times. – add a row of −’s. • Appendix 7A (p. 330). – cyclically shift the first row (genertor) 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). Table 7.5 Generating Row Vectors for Plackett-Bruman Designs of Run Size N N Vector 12 + +−+ + +−−−+− 20 + +−−+ + + +−+−+−−−−+ +− 24 + + + + +−+−+ +−−+ +−−+−+−−−− 36 −+−+ + +−−−+ + + + +−+ + +−−+−−−−+−+−+ +−−+− 44 + +−−+−+−−+ + +−+ + + + +−−−+−+ + +−−−−−+−−− + +−+−+ +− 7 Appendix. Optimal Choice of Nonregular Designs Question: How to compare nonregular designs? A.1 Generalized minimum aberration criterion For a design D of n factors and N runs, consider the (ANOVA) model Y = Iα0 + X1α1 + · · ·+ Xnαn + ε, • Y is the vector of N observations • αj is the vector of all j-factor interactions • Xj is the matrix of orthonormal coefficients for αj Define, if Xj = [x (j) ik ], Aj = N −2‖ITXj‖2 = N−2 ∑ k ∣∣∣∣∣∑ i x (j) ik ∣∣∣∣∣ 2 . The GMA criterion (Xu and Wu, 2001, Annals of Statistics) • to sequentially minimize A1, A2, A3, . . .. Example: Two 2-Level Designs • Design 1 (One-factor-at-a-time design) X1 X2 X3 1 2 3 12 13 23 123 1 + + + + + + + 2 − + + − − + − 3 − − + + − − + 4 − − − + + + − Sum -2 0 2 2 0 2 0 – A1 = [(−2)2 + 02 + 22]/42 = 0.5, – A2 = [2 2 + 02 + 22]/42 = 0.5, 10 – A3 = 0 2/42 = 0. • Design 2 (23−1 with I = 123) X1 X2 X3 1 2 3 12 13 23 123 1 + + + + + + + 2 + − − − − + + 3 − + − − + − + 4 − − + + − − + Sum 0 0 0 0 0 0 4 – A1 = (0 2 + 02 + 02)/42 = 0, – A2 = (0 2 + 02 + 02)/42 = 0, – A3 = 4 2/42 = 1. • The 2nd design has less aberration than the 1st design. • The 2nd design is preferred to the 1st design. Example: A 3-Level Design (with C = A + B (mod 3)) With orthogonal polynomial contrasts X1 X2 X3 AB C A B C A×B A× C B × C A×B × C 0 0 0 −1 1−1 1−1 1 1−1−1 1 1−1−1 1 1−1−1 1−1 1 1 −1 1 −1 −1 1 0 1 1 −1 1 0−2 0−2 0 2 0−2 0 2 0−2 0 0 0 4 0 0 0 −4 0 0 0 4 0 2 2 −1 1 1 1 1 1−1−1 1 1−1−1 1 1 1 1 1 1−1 −1 −1 −1 1 1 1 1 1 0 1 0−2−1 1 0−2 0 0 2−2 0 0 0 4 0 2 0−2 0 0 0 0 0 −4 0 4 1 1 2 0−2 0−2 1 1 0 0 0 4 0 0−2−2 0 0−2−2 0 0 0 0 0 0 4 4 1 2 0 0−2 1 1−1 1 0 0−2−2 0 0 2−2−1 1−1 1 0 0 0 0 2 −2 2−2 2 0 2 1 1−1 1 1 1−1 1−1 1 1 1 1 1−1−1 1 1−1 −1 1 1 −1 −1 1 1 2 1 0 1 1 0−2−1 1 0−2 0−2−1 1−1 1 0 0 2−2 0 0 2 −2 0 0 2−2 2 2 1 1 1 1 1 0−2 1 1 1 1 0−2 0−2 0−2 0−2 0 −2 0 −2 0 −2 0−2 Sum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0−3 −3 3 −9 3 −9 9 9 Scale a b a b a b a2 a b a b b2 a2 a b a b b2 a2 a b a b b2 a3 a2 ba2 ba b2 a2 ba b2 a b2 b3 a = √ 3/2 and b = 1/ √ 2 • A1 = [02 + 02 + 02 + 02 + 02 + 02]/92 = 0, • A2 = [02 + 02 + 02 + 02 + 02 + 02 + 02 + 02 + 02 + 02 + 02 + 02]/92 = 0, • A3 = [(−3a3)2 + (−3a2b)2 + (3a2b)2 + (−9ab2)2 + (3a2b)2 + (−9ab2)2 + (9ab2)2 + (9b3)2]/92 = 2. 11 Properties of Generalized Minimum Aberration • A1 = A2 = . . . = At = 0 if and only if D is an OA of strength t. • The definition of Aj is independent of the choice of orthonormal con- trasts. • For a regular design the generalized wordlength pattern is the same as the wordlength pattern. • The GMA reduces to MA for regular designs. A.2 Minimum moment aberration criterion For an N × n matrix D = [xij], define the tth power moment to be Kt(D) = Ei<j [δij(D)] t = [N(N − 1)/2]−1 ∑ 1≤i<j≤N [δij(D)] t , where δij(D) is the number of coincidences between the ith and jth rows, that is, number of k’s such that xik = xjk. • The power moments measure the similarity among the rows. • A good design should have small power moments. The minimum moment aberration criterion (Xu, 2003, Statistica Sinica) • to sequentially minimize K1, K2, K3, . . .. 12