An approach to the selection of multistratum fractional factorial designs
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1 An approach to the selection of multistratum fractional factorial designs Ching-Shui Cheng & Pi-Wen Tsai 11 August, 2008, DEMA2008 at Isaac Newton Institute Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 1 / 32
2 Multistratum fractional factorial designs Split-plot designs Huang, Chen & Voelkel (1998) Bingham & Sitter (1999a,b; 2001) Bingham, Schoen & Sitter (2004) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 2 / 32
3 Multistratum fractional factorial designs Split-plot designs Huang, Chen & Voelkel (1998) Bingham & Sitter (1999a,b; 2001) Bingham, Schoen & Sitter (2004) Blocked split-plot designs McLeod & Brewster (2004) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 2 / 32
4 Multistratum fractional factorial designs Split-plot designs Huang, Chen & Voelkel (1998) Bingham & Sitter (1999a,b; 2001) Bingham, Schoen & Sitter (2004) Blocked split-plot designs McLeod & Brewster (2004) Strip-plot designs (two processing stages) Miller (1997) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 2 / 32
5 Multistratum fractional factorial designs Experiments with multiple processing stages Split-lot designs: Mee & Bates (1998) Bingham, Sitter, Kelly, Moore & Olivas (2008) Ranjan, Bingham & Dean (2008) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 3 / 32
6 Multistratum fractional factorial designs Treatment structure factorial structure Unit (block, plot) structure simple orthogonal block structure (Nelder, 1965) Tjur block structure (Tjur, 1984) orthogonal block structure (Speed and Bailey, 1982; Bailey, 1985) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 4 / 32
7 Simple orthogonal block structures (Nelder 1965) Nesting: block/unit Crossing: row column Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 5 / 32
8 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 6 / 32
9 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 6 / 32
10 Strip-plot designs: Miller (1997) Block structure: 32 runs, 2 block/(4 rows 4 columns) Treatment structure: 2 10 A, B, C, D, E, F: rst-stage factors a,b,c,d: second-stage factors Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 7 / 32
11 Strip-plot designs: Miller (1997) Block structure: 32 runs, 2 block/(4 rows 4 columns) Treatment structure: 2 10 A, B, C, D, E, F: rst-stage factors a,b,c,d: second-stage factors Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 7 / 32
12 Multiple processing stages Split-lot designs: Mee & Bates (1998) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 8 / 32
13 Multiple processing stages Split-lot designs: Mee & Bates (1998) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 8 / 32
14 Multiple processing stages Split-lot designs: Mee & Bates (1998) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 8 / 32
15 Multiple processing stages Split-lot designs: Mee & Bates (1998) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 8 / 32
16 Multiple processing stages Bingham et al (2008): 32 runs; two processing stages First-stage randomization: 8/4 Second-stage randomization: 8/4 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 9 / 32
17 Multiple processing stages Bingham et al (2008): 32 runs; two processing stages First-stage randomization: 8/4 Second-stage randomization: 8/ v v v v 2 v v v v 3 v v v v 4 v v v v 5 v v v v 6 v v v v 7 v v v v 8 v v v v Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 9 / 32
18 Multiple processing stages Bingham et al (2008): 32 runs; two processing stages First-stage randomization: 8/4 Second-stage randomization: 8/ v v v v 2 v v v v 3 v v v v 4 v v v v 5 v v v v 6 v v v v 7 v v v v 8 v v v v Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 9 / 32
19 Multistratum fractional factorial designs E(y) = Xα, cov(y) = s ξ i S i, i=0 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 10 / 32
20 Multistratum fractional factorial designs s E(y) = Xα, cov(y) = ξ i S i, where ξ i is an eigenvalue of cov(y), and S i is the orthogonal projection matrix onto the eigenspace S i of cov(y) with eigenvalue ξ i. i=0 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 10 / 32
21 Multistratum fractional factorial designs s E(y) = Xα, cov(y) = ξ i S i, where ξ i is an eigenvalue of cov(y), and S i is the orthogonal projection matrix onto the eigenspace S i of cov(y) with eigenvalue ξ i. i=0 S 0 = 1 N J N=run size and J is the matrix of all 1's. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 10 / 32
22 Multistratum fractional factorial designs s E(y) = Xα, cov(y) = ξ i S i, where ξ i is an eigenvalue of cov(y), and S i is the orthogonal projection matrix onto the eigenspace S i of cov(y) with eigenvalue ξ i. i=0 S 0 = 1 N J N=run size and J is the matrix of all 1's. Each of the eigenspaces with i 1 is called a stratum. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 10 / 32
23 Multistratum fractional factorial designs s E(y) = Xα, cov(y) = ξ i S i, where ξ i is an eigenvalue of cov(y), and S i is the orthogonal projection matrix onto the eigenspace S i of cov(y) with eigenvalue ξ i. i=0 S 0 = 1 N J N=run size and J is the matrix of all 1's. Each of the eigenspaces with i 1 is called a stratum. ξ i : the ith stratum variance. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 10 / 32
24 Multistratum fractional factorial designs c S i var(c y) = ξ i c 2. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 11 / 32
25 Multistratum fractional factorial designs c S i var(c y) = ξ i c 2. c S i, d S i, i i cov(c y, d y) = 0. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 11 / 32
26 Multistratum fractional factorial designs c S i var(c y) = ξ i c 2. c S i, d S i, i i cov(c y, d y) = 0. Orthogonal designs: Im(X) = j T j, where each T j S i for some i. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 11 / 32
27 Multistratum fractional factorial designs c S i var(c y) = ξ i c 2. c S i, d S i, i i cov(c y, d y) = 0. Orthogonal designs: Im(X) = j T j, where each T j S i for some i. Chapter 10 of Bailey (2008) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 11 / 32
28 Multistratum fractional factorial designs Each stratum corresponds to a partition of the units into subsets of equal size. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 12 / 32
29 Multistratum fractional factorial designs Each stratum corresponds to a partition of the units into subsets of equal size. If the partition corresponding to stratum i is coarser than that of stratum i, then we say that the latter is nested in the former. We assume that in this case, ξ i ξ i. This is true for the usual mixed-eect models. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 12 / 32
30 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
31 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
32 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Null ANOVA: Three strata block whole-plot subplots (units) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
33 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Null ANOVA: Three strata block whole-plot subplots (units) U B W S Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
34 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Null ANOVA: Three strata block whole-plot subplots (units) U 1 B 3 W 12 S 16 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
35 Blocked split-plot designs: McLeod & Brewster (2004) Block structure: 32 runs, 4 weeks/4 days/2 runs. block/whole-plot/subplot Treatment structure: A B p q r s. Each of the six factors has two levels. A, B: hard to vary (whole-plot factors) p, q, r, s: easy to vary (subplot factors) Null ANOVA: Three strata block whole-plot subplots (units) U 1 B 3 W 12 S 16 ξ B ξ W ξ S Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 13 / 32
36 Strip-plot designs: Miller (1997) Block structure: 32 runs, 2 block/(4 rows 4 columns) Treatment structure: 2 10 A, B, C, D, E, F: rst-stage factors a,b,c,d: second-stage factors Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 14 / 32
37 Strip-plot designs: Miller (1997) Block structure: 32 runs, 2 block/(4 rows 4 columns) Treatment structure: 2 10 A, B, C, D, E, F: rst-stage factors a,b,c,d: second-stage factors U 1 B 1 6 R C E 18 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 14 / 32
38 Multiple processing stages Split-lot designs: Mee & Bates (1998) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 15 / 32
39 Multiple processing stages Split-lot designs: Mee & Bates (1998) U 1 3 S 1 S S 3 S 4 3 E 3 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 15 / 32
40 Multiple processing stages Bingham et al (2008): 32 runs; two processing stages First-stage randomization: 8/4 Second-stage randomization: 8/ v v v v 2 v v v v 3 v v v v 4 v v v v 5 v v v v 6 v v v v 7 v v v v 8 v v v v Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 16 / 32
41 Multiple processing stages Bingham et al (2008): 32 runs; two processing stages First-stage randomization: 8/4 Second-stage randomization: 8/ v v v v 2 v v v v 3 v v v v 4 v v v v 5 v v v v 6 v v v v 7 v v v v 8 v v v v U 1 S 1 S S 1 S 2 6 E 18 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 16 / 32
42 Optimality criteria Existing works use modications of the minimum aberration criterion. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 17 / 32
43 Optimality criteria Existing works use modications of the minimum aberration criterion. Block eects are treated as xed. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 17 / 32
44 Optimality criteria Existing works use modications of the minimum aberration criterion. Block eects are treated as xed. For split-plot designs, the usual minimum aberration criterion for unstructured units is used. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 17 / 32
45 Optimality criteria Existing works use modications of the minimum aberration criterion. Block eects are treated as xed. For split-plot designs, the usual minimum aberration criterion for unstructured units is used. Stratum variances are not taken into account. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 17 / 32
46 New criterion Cheng & Tsai (2008) propose a unied approach to the selection of blocked and split-plot fractional factorial designs. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 18 / 32
47 New criterion Cheng & Tsai (2008) propose a unied approach to the selection of blocked and split-plot fractional factorial designs. This approach can be extended to experiments with more complicated block structures. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 18 / 32
48 Assumptions The three-factor and higher-order interactions are negligible. The main eects must be estimated. The two-factor interactions are equally important. Only two-level regular and orthogonal designs will be considered. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 19 / 32
49 Information capacity In justifying the minimum aberration criterion for designs with unstructured units, Cheng, Steinberg and Sun (1999) showed that it is a good surrogate for maximizing the number of estimable models containing all the main eects and a given number of two-factor interactions. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 20 / 32
50 Information capacity In justifying the minimum aberration criterion for designs with unstructured units, Cheng, Steinberg and Sun (1999) showed that it is a good surrogate for maximizing the number of estimable models containing all the main eects and a given number of two-factor interactions. In the multistratum situation, dierent estimable models may have dierent eciencies. Thus we take the average eciency over the candidate models. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 20 / 32
51 Information capacity Suppose the information for the main eects is specied to lie in certain strata. Then it is enough to consider the eciencies of the estimators of two-factor interactions, measured by D 1/k, where D is the determinant of the information matrix for the two-factor interactions, and k is the number of two-factor interactions in the model. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 21 / 32
52 Information capacity Suppose the information for the main eects is specied to lie in certain strata. Then it is enough to consider the eciencies of the estimators of two-factor interactions, measured by D 1/k, where D is the determinant of the information matrix for the two-factor interactions, and k is the number of two-factor interactions in the model. For a given design d, dene the information capacity I k (d) as the average of D 1/k over the set of models containing all the main eects and k two-factor interactions. The objective is to maximize I k (d). Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 21 / 32
53 Information capacity Suppose the information for the main eects is specied to lie in certain strata. Then it is enough to consider the eciencies of the estimators of two-factor interactions, measured by D 1/k, where D is the determinant of the information matrix for the two-factor interactions, and k is the number of two-factor interactions in the model. For a given design d, dene the information capacity I k (d) as the average of D 1/k over the set of models containing all the main eects and k two-factor interactions. The objective is to maximize I k (d). Sun (1993), Tsai, Gilmour and Mead (2000) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 21 / 32
54 Information capacity I k (d) is time-consuming to calculate and will be replaced by a good surrogate. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 22 / 32
55 Information capacity I k (d) is time-consuming to calculate and will be replaced by a good surrogate. I k (d), or its surrogate, depends on the stratum variances ξ 1,..., ξ s. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 22 / 32
56 Information capacity I k (d) is time-consuming to calculate and will be replaced by a good surrogate. I k (d), or its surrogate, depends on the stratum variances ξ 1,..., ξ s. The partial order among the ξ's can be used to simplify conditions for optimality. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 22 / 32
57 A surrogate Suppose the run size is 2 n p. Then there are a total of 2 n p 1 alias sets. Let f = 2 n p 1 n. Then f is the number of alias sets that do not contain main eects. Suppose that h i of these alias sets fall in the i th stratum. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 23 / 32
58 A surrogate Suppose the run size is 2 n p. Then there are a total of 2 n p 1 alias sets. Let f = 2 n p 1 n. Then f is the number of alias sets that do not contain main eects. Suppose that h i of these alias sets fall in the i th stratum. For each 1 j h i, let m (i) j contained in the jth of these alias sets. be the number of two-factor interactions Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 23 / 32
59 A surrogate Suppose the run size is 2 n p. Then there are a total of 2 n p 1 alias sets. Let f = 2 n p 1 n. Then f is the number of alias sets that do not contain main eects. Suppose that h i of these alias sets fall in the i th stratum. For each 1 j h i, let m (i) j contained in the jth of these alias sets. Let m be the vector (ξ 1/k m (1),..., ξ 1/k m (1) h 1,..., ξs 1/k be the number of two-factor interactions m (s),..., ξ 1/k 1 s m (s) h s ). Then it can be shown that I k (d) is a Schur concave function of m, and is nondecreasing in each component of m. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 23 / 32
60 A surrogate It follws that a good surrogate for maximizing I k (d) is to Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 24 / 32
61 A surrogate It follws that a good surrogate for maximizing I k (d) is to (a) maximize s i=1 ξ 1/k i h i j=1 m (i) j, and then Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 24 / 32
62 A surrogate It follws that a good surrogate for maximizing I k (d) is to (a) maximize (b) minimize s i=1 s i=1 ξ 1/k i ξ 2/k i h i j=1 h i j=1 m (i) j, and then (m (i) j ) 2 among those which maximize (a) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 24 / 32
63 A useful theorem The following is a sucient condition for a design d to be optimal under (a) and (b) with respect to all k and all ξ 1,..., ξ s such that ξ i ξ i if the partition corresponding to stratum i is nested in that of stratum i: Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 25 / 32
64 A useful theorem The following is a sucient condition for a design d to be optimal under (a) and (b) with respect to all k and all ξ 1,..., ξ s such that ξ i ξ i if the partition corresponding to stratum i is nested in that of stratum i: (i) d maximizes i T hi m(i) j=1 j, and Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 25 / 32
65 A useful theorem The following is a sucient condition for a design d to be optimal under (a) and (b) with respect to all k and all ξ 1,..., ξ s such that ξ i ξ i if the partition corresponding to stratum i is nested in that of stratum i: (i) d maximizes i T hi (ii) minimizes hi i T hi i T m(i) j=1 j. m(i) j=1 j (m(i) j=1 j, and ) 2 among those which maximize for all subsets T of {1,..., s} such that if i T, then i T for all i such that the partition corresponding to stratum i is nested in that of stratum i. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 25 / 32
66 A useful theorem The following is a sucient condition for a design d to be optimal under (a) and (b) with respect to all k and all ξ 1,..., ξ s such that ξ i ξ i if the partition corresponding to stratum i is nested in that of stratum i: (i) d maximizes i T hi (ii) minimizes hi i T hi i T m(i) j=1 j. m(i) j=1 j (m(i) j=1 j, and ) 2 among those which maximize for all subsets T of {1,..., s} such that if i T, then i T for all i such that the partition corresponding to stratum i is nested in that of stratum i. This result can be used to rule out inadmissible designs. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 25 / 32
67 Blocked split-plot designs (McLeod & Brewster, 2004) I = pqrs, δ 1 = ABpq, δ 2 = Bpr Block Stratum Source of variation df Bpr 1 Aqr 1 ABpq 1 WP Stratum A 1 B 1 AB 1 qr = ps 1 pr = qs 1 pq = rs 1 Residuals 6 Unit Stratum p 1 q 1 r 1 s 1 Ap 1 Aq 1 Ar 1 As 1 Bp 1 Bq 1 Br 1 Bs 1 Residuals 4 B 3 W 12 S 16 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 26 / 32
68 Blocked split-plot designs (McLeod & Brewster, 2004) I = pqrs, δ 1 = ABpq, δ 2 = Bpr Block Stratum Source of variation df Bpr 1 Aqr 1 ABpq 1 WP Stratum A 1 B 1 AB 1 qr = ps 1 pr = qs 1 pq = rs 1 Residuals 6 Unit Stratum p 1 q 1 r 1 s 1 Ap 1 Aq 1 Ar 1 As 1 Bp 1 Bq 1 Br 1 Bs 1 Residuals 4 B 3 m (1) i : 000 W 12 m (2) i : S 16 m (3) 1 : Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 26 / 32
69 Blocked split-plot designs (McLeod & Brewster, 2004) I = pqrs, δ 1 = ABpq, δ 2 = Bpr Block Stratum Source of variation df Bpr 1 Aqr 1 ABpq 1 WP Stratum A 1 B 1 AB 1 qr = ps 1 pr = qs 1 pq = rs 1 Residuals 6 Unit Stratum p 1 q 1 r 1 s 1 Ap 1 Aq 1 Ar 1 As 1 Bp 1 Bq 1 Br 1 Bs 1 Residuals 4 B 3 m (1) i : 000 W 12 m (2) i : S 16 m (3) 1 : mi m 2 i (1)+(2)+(3) (2)+(3) (3) 8 8 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 26 / 32
70 Blocked split-plot designs: 4/4/2 Two optimal 4/4/2 designs by McLeod & Brewster (2004) The m i values for d 1 and d 2 : d 1 m i (1) : 000 d 2 m i (1) : 000 m i (2) : m i (2) : m i (3) : m i (3) : Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 27 / 32
71 Blocked split-plot designs: 4/4/2 Two optimal 4/4/2 designs by McLeod & Brewster (2004) The m i values for d 1 and d 2 : d 1 m i (1) : 000 d 2 m i (1) : 000 m i (2) : m i (2) : m i (3) : m i (3) : d 1 mi m 2 i (1)+(2)+(3) (2)+(3) (3) 8 8 d 2 mi m 2 i Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 27 / 32
72 Blocked split-plot designs: 4/4/2 Two optimal 4/4/2 designs by McLeod & Brewster (2004) The m i values for d 1 and d 2 : d 1 m i (1) : 000 d 2 m i (1) : 000 m i (2) : m i (2) : m i (3) : m i (3) : d 1 mi m 2 i (1)+(2)+(3) (2)+(3) (3) 8 8 d 2 mi m 2 i d 1 dominates d 2 Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 27 / 32
73 Blocked split-plot designs: 4/4/2 d 1 mi m 2 i (1)+(2)+(3) (2)+(3) (3) 8 8 d 2 mi m 2 i Another admissible design d 3 m i (1): 100 m i (2): m i (3): d 3 mi m 2 i Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 28 / 32
74 Blocked split-plot designs: 4/4/2 d 1 mi m 2 i (1)+(2)+(3) (2)+(3) (3) 8 8 d 2 mi m 2 i Another admissible design d 3 m i (1): 100 m i (2): m i (3): d 3 mi m 2 i The optimality between d 1 and d 3 depends on the variance ratios. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 28 / 32
75 Blocked split-plot designs A search of all 32-run designs with various numbers of factors and block structures shows that there is a single admissible design in most of the cases. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 29 / 32
76 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) B R C E Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
77 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) all strata B R C E Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
78 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) all strata without the Block stratum B R C E Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
79 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) all strata without the Block stratum without the Block & Column strata B R C E Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
80 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) all strata without the Block stratum without the Block & Column strata without the Block & Row strata B R C E Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
81 Strip-plot designs (Miller 97) Two processing-stage experiment block/(row column) all strata without the Block stratum without the Block & Column strata without the Block & Row strata Unit stratum (without Block, Row B R C E & Column strata) Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 30 / 32
82 Strip-plot designs (Miller 97) The design constructed by Miller is one of two admissible designs. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 31 / 32
83 Strip-plot designs (Miller 97) The design constructed by Miller is one of two admissible designs. There is a single admissible design in most of the 32-run cases. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 31 / 32
84 Summary Introduce a unied approach to the selection of multistratum factorial designs. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 32 / 32
85 Summary Introduce a unied approach to the selection of multistratum factorial designs. Provide a useful theorem for nding optimal designs. Cheng & Tsai (DEMA2008) Multi-stratum FF designs Isaac Newton Institute 32 / 32
Multistratum Fractional Factorial Designs. Ching-Shui Cheng and Pi-Wen Tsai. University of California, Berkeley and National Taiwan Normal University
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