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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
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. 2 + − + + + − − − + − + 4. 3 − + + + − − − + − + + 4. 4 + + + − − − + − + + − 5. 5 + + − − − + − + + − + 7. 6 + − − − + − + + − + + 5. 7 − − − + − + + − + + + 5. 8 − − + − + + − + + + − 6. 9 − + − + + − + + + − − 5. 10 + − + + − + + + − − − 5. 11 − + + − + + + − − − + 5. 12 − − − − − − − − − − − 4.
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. C. caras H 2 O level (ml) 20 28 35 D. centrifuge RPM 2100 2300 2500 E. centrifuge time (min) 1.75 3 4. F. (sensitivity, absorption) (0.10,2.5) (0.25,2) (0.50,1.5) G. temperature (^0 C) 25 30 37 H. dilution ratio 1:51 1:101 1:
- OA(18, 2137 ) in Table 7.4.
- 2 k−p, 3k−p^ and Latin squares are (regular) OAs.
- A 2kR− pdesign is an OA(N = 2k−p, 2 k, t = R − 1).
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 m^1 · · · sγ mγ^ ) has strength t = 2.
7.2 Some Advantages of Nonregular Designs Over the
2 k−p^ and 3k−p^ Series of Designs
Two advantages:
- run size economy
- flexibility
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.
To study 7 two-level factors, can use
- (^27) IV− 3 (16 runs)
- (^27) III−^4 (8 runs, saturated, no df for error estimation)
To study 8-11 two-level factors
- A regular design needs at least 16 runs (2^8 −^4 , 2^11 −^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 (3^7 −^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, s 1 m^1 · · · sγ mγ^ , t), its run size N must be divisible by the least common multiple (l.c.m.) of s 1 k^1 s 2 k^2 · · · sγ kγ^ , for all possible combinations of ki with ki ≤ mi and k 1 + k 2 + · · · + kγ = t.
Examples
- OA(N, 2137 , 2): N is a multiple of l.c.m.(2^131 , 32 ) = 18.
- OA(N, 2237 , 2): N is a multiple of l.c.m.(2^2 , 2131 , 32 ) = 36.
- OA(N, 2137 , 3): N is a multiple of l.c.m.(2^132 , 33 ) = 54.
- OA(N, 4131210 , 2): N is a multiple of l.c.m.(4^131 , 4121 , 3121 , 22 ) = 24.
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
H 2 N =
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, 2 N^ −^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 + + − − + − + − − + + + − + + + + + − − − + − + + + − − − − − + − − −
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 2^15 −^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 N 1 2 4 8 12 16 20 24 28 32 36
1 1 1 1 1 5 3 60 487??
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).
7.5 A Collection of Useful Mixed-Level OAs
Appendix 7C gives a collection of mixed-level OAs with 12–54 runs and 2– levels.
- Table 7C.1 OA( 12 , 3124 ) and OA′(12, 3126 )
- For OA(N, 3124 ), N is a multiple of l.c.m.(3^121 , 22 ) = 12.
- There is no OA(12, 312 k) 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.
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 = Iα 0 + X 1 α 1 + · · · + 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( ikj) ],
Aj = N −^2 ‖IT^ Xj ‖^2 = N −^2
k
i
x( ikj)
2 .
The GMA criterion (Xu and Wu, 2001, Annals of Statistics)
- to sequentially minimize A 1 , A 2 , A 3 ,.. ..
Example: Two 2-Level Designs
- Design 1 (One-factor-at-a-time design)
X 1 X 2 X 3 1 2 3 12 13 23 123 1 + + + + + + + 2 − + + − − + − 3 − − + + − − + 4 − − − + + + − Sum -2 0 2 2 0 2 0
- A 1 = [(−2)^2 + 0^2 + 2^2 ]/ 42 = 0.5,
- A 2 = [2^2 + 0^2 + 2^2 ]/ 42 = 0.5,
– A 3 = 0^2 / 42 = 0.
- Design 2 (2^3 −^1 with I = 123)
X 1 X 2 X 3 1 2 3 12 13 23 123 1 + + + + + + + 2 + − − − − + + 3 − + − − + − + 4 − − + + − − + Sum 0 0 0 0 0 0 4
- A 1 = (0^2 + 0^2 + 0^2 )/ 42 = 0,
- A 2 = (0^2 + 0^2 + 0^2 )/ 42 = 0,
- A 3 = 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
X 1 X 2 X 3 A B 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 a^2 a b a b b^2 a^2 a b a b b^2 a^2 a b a b b^2 a^3 a^2 b a^2 b a b^2 a^2 b a b^2 a b^2 b^3 a =
√ 3 /2 and b = 1/
√ 2
- A 1 = [0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 ]/ 92 = 0,
- A 2 = [0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 ]/ 92 = 0,
- A 3 = [(− 3 a^3 )^2 + (− 3 a^2 b)^2 + (3a^2 b)^2 + (− 9 ab^2 )^2 + (3a^2 b)^2 + (− 9 ab^2 )^2 + (9ab^2 )^2 + (9b^3 )^2 ]/ 92 = 2.
Example: Two 2-Level Designs
Design Matrix Coincidence Matrix Moments (δij ) Kt = Ei 0
- Advantages of MMA over GMA:
- MMA is easier to understand for non-statisticians.
- MMA is easier to implement.
- MMA is cheaper to compute.
- MMA is easier to deal with.
Learn More?
- Stats 296: Advanced Topics in Experimental Designs
- http://www.stat.ucla.edu/∼hqxu/
