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Construction of optimal fractional Order-of-Addition designs via block designs
Journal Pre-proof Construction of optimal fractional Order-of-Addition designs via block designs Jianbin Chen, Rahul Mukerjee, Dennis K.J. Lin
PII: DOI: Reference:
S0167-7152(20)30031-6 https://doi.org/10.1016/j.spl.2020.108728 STAPRO 108728
To appear in:
Statistics and Probability Letters
Received date : 29 October 2019 Revised date : 29 December 2019 Accepted date : 2 February 2020 Please cite this article as: J. Chen, R. Mukerjee and D.K.J. Lin, Construction of optimal fractional Order-of-Addition designs via block designs. Statistics and Probability Letters (2020), doi: https://doi.org/10.1016/j.spl.2020.108728. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Elsevier B.V. All rights reserved.
Journal Pre-proof
Construction of Optimal Fractional
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Order-of-Addition Designs via Block Designs Jianbin Chena , Rahul Mukerjeeb , and Dennis K.J. Linc a
School of Statistics and Data Science, LPMC, Nankai University, Tianjin 300071, China
b
c
Indian Institute of Management Calcutta, Joka, Diamond Harbour Road, Kolkata 700104, India Department of Statistics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A
Abstract
Order of addition (OofA) experiments have found wide applications. Each order (run) of an OofA experiment is a permutation of m (≥ 2) components. It is typically infeasible to compare all the m! possible runs, especially when m is large. This calls for experimentation with a subset or fraction of these m! runs. However, only a few systematic results are available
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on the construction of such fractions ensuring optimality. We employ block designs to propose a systematic combinatorial construction method for optimal fractional OofA designs, and extend the method to construct highly efficient OofA designs, both in much smaller run sizes than the currently available optimal fractions.
Keywords: Balanced incomplete block design, Fractional OofA design, Optimality, Pairwise order model, Systematic combinatorial construction. MSC: 62K15, 62K10
1
Introduction
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Many physical phenomena encountered in science and engineering are affected by the addition order of m materials or components. In chemical experiments, the performance or the amount of reaction production is often determined by the order of adding reagents. For example, Song et al. (2014) conducted experiments with the 3! orderings of three reagents, namely, carbon dots, F e2+ and H2 O2 , and found that addition of F e2+ and H2 O2 before carbon dots leads to stronger photoluminescence intensities. Jiang and Ng (2014) reported chemical experiments where the output is influenced by the order of addition of four components, Hg 2+ , Cu2+ , S 2− and light. Order of addition (OofA) experiments have found wide application also in bio-chemistry
1
Journal Pre-proof (Shinohara and Ogawa, 1998), nutritional science (Karim, et al., 2000) and NP-hard ordering problem (Chen, et al., 2019), just to name a few. With m components, there are m! possible orders or runs, each a permutation of {1, 2, . . . , m}.
It is infeasible to include all of these in an experiment, especially when m is large. This warrants experimentation with a subset or fraction of these m! runs. The problem of selecting such a
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fraction has received attention in the literature. For instance, Van Nostrand (1995) considered the design with “pseudo factors” or pairwise ordering (PWO) factors. Lin and Peng (2019) reviewed the latest work on the design and model of OofA experiments, and introduced some new thoughts. Voelkel (2019) provided some smaller OofA orthogonal arrays under some proposed design criteria. Peng, et al. (2019a) considered different types of optimality criteria and proposed a systematic method to construct a class of robust optimal fractional OofA designs. Zhao, et al. (2018) considered the minimal-point OofA designs. Yang, et al. (2018) obtained a number of OofA designs called component orthogonal arrays that are optimal under their component-position model. Chen, et al. (2019) introduced a statistical method to speculate solutions of NP-hard ordering problem by making use of design for OofA experiment. The current literature on OofA designs focuses on computer search. A notable exception is Peng, et al. (2019a) who present a systematic construction of optimal fractional OofA designs in m!/s! runs, where m = 2s for even m and m = 2s + 1 for odd m. However, these are quite large for larger m. Thus, with m = 10, their construction requires 30240 runs which, despite
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being less than 1% of the run size (= 10!) of the full design, is still rather large. With regard to smaller fractions, Peng, et al. (2019a) hinted at the possibility of using block designs, but did not pursue the idea any further. We develop and formalize this approach here, and show in the subsequent sections how it can yield optimal or highly efficient fractional OofA designs in considerably smaller run sizes. The proofs appear in the appendix.
2
Optimality of fractional OofA designs
Denote a typical order by π = (π1 , . . . , πm ), which is a permutation of {1, . . . , m}. Based on a
fraction Π of size N from the m! possible orders, consider the pairwise order (PWO) model
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y(π) = β0 +
X
βij Iij (π) + ε,
i
π ∈ Π,
(1)
where Iij = +1, if component i precedes component j; and Iij = −1 otherwise; y is the response of interest; and ε is the observational error. As usual, such errors are assumed to be uncorrelated,
all with mean zero and the same variance. There are p = 1 + m(m − 1)/2 parameters of
interest as given by the elements of β = (β0 , β12 , . . . , β(m−1)m )T , where the superscript denotes
transpose. Let X = [X0 , X12 , . . . , X(m−1)m ] be the model matrix of Π, where X0 corresponds to β0 and Xij corresponds to βij . The fraction Π has moment matrix M = N −1 X T X and D-value DΠ = {det(M )}1/p , which captures the generalized variance of the estimator of β arising from 2
Journal Pre-proof Π. The PWO model has been widely studied as a reasonable model for OofA experiments (see Jiang and Ng (2014) for an example of a practical situation, and Peng, et al. (2019a) for more details). While we mainly focus on the D-criterion here, other design criteria, such as the Aand E-criteria can be considered similarly. Let d0 be the full design that includes all the m! possible orders. Peng, et al. (2019a)
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proved the optimality of d0 under various criteria, including the D-, A- and E-criteria. Hence the D-efficiency of any fraction Π equals DΠ /Df ull , where Df ull is the D-value of d0 . Obviously, a fraction is D-, A- and E-optimal if its moment matrix equals that of d0 . Peng, et al. (2019a) obtained the moment matrix of d0 as M0 =diag{1, I + (1/3)V }, where I is the identity matrix of
order q = m(m − 1)/2, and V is a q × q matrix, with rows and columns indexed by the elements of S = {ij, 1 ≤ i < j ≤ m}, such that the (ij, kl)th element of V is
1, if i = k, j 6= l or i 6= k, j = l, V (ij, kl) = −1, if i = l or j = k, 0, otherwise.
We are now in a position to initiate the construction of optimal fractions in smaller run sizes, using binary block designs where every symbol appears at most once in each block. Let d be such a design in symbols 1, . . . , m (m ≥ 4), and b blocks C1 , . . . , Cb , each of size h (< m),
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such that symbol i appears in ri blocks, and symbols i and j occur together in λij blocks (i, j = 1, . . . , m; i 6= j). Let C¯u = {1, . . . , m} \ Cu , u = 1, . . . , b. Also, let A and A¯ be OofA
orthogonal array designs, involving h and m − h symbols, respectively, and both having R runs, i.e., A and A¯ have the same moment matrices, under the PWO model, as the corresponding ¯u for the A and A¯ with the symbols in Cu and C¯u , full designs (Voelkel, 2019). Write Bu and B respectively. Form the (2bR) × m array D = (D1 T , D2 T , · · · , Db T )T , where ! ¯u Bu B Du = , u = 1, . . . , b, ¯ u Bu ∼B with ∼ representing the operator of column reversal of any matrix. For example, if m = 6, h = 3, and C1 = {1, 2, 3} is the first block of d, then C¯1 = {4, 5, 6}, and with both A and A¯ runs, one obtains 4 5 3 4 6 2 5 4 3 ¯1 = , B 5 6 1 2 6 4 6 5 1
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taken as the full design in 3! 1 2 1 3 2 1 B1 = 2 3 3 1 3 2
6
5 6 , 4 5 4
¯1 = ∼B
¯1 has the same columns as B ¯1 but in a reversed order. where ∼ B 3
6 5 4
5 6 4 6 4 5 , 4 6 5 5 4 6 4 5 6
Journal Pre-proof Theorem 1 The m-component fractional OofA design d∗ in N = 2bR runs, as given by the rows of D, has moment matrix M0 under the PWO model and hence is D-, A- and E-optimal among all designs with the same number of runs, provided for every distinct i, j, k in {1, . . . , m},
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λik − λjk = (ri − rj )/2.
(2)
Proposition 1 below addresses the crucial issue of choosing the block design d so as to meet condition (2).
Proposition 1 The following choices of d satisfy condition (2): (I) Take d as a balanced incomplete block (BIB) design;
(II) Let d˜ be a BIB design in symbols {2, . . . , m} and b blocks, where each symbol appears in r blocks, each pair of distinct symbols appear together in λ blocks, and b − 3r + 2λ = 0.
(3)
˜ Obtain d by adding a single symbol, say 1, to every block of d.
While (I) obviously entails condition (2), the truth of (2) for (II) follows noting that
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(i) if i, j, k ∈ {2, . . . , m} or k = 1 and i, j ∈ {2, . . . , m}, then λik − λjk = 0 = (ri − rj )/2; (ii) if i = 1 and j, k ∈ {2, . . . , m}, then λik − λjk = r − λ, (ri − rj )/2 = (b − r)/2, and they are equal by (3);
(iii) if j = 1 and i, k ∈ {2, . . . , m}, then λik − λjk = λ − r, (ri − rj )/2 = (r − b)/2, and the two are equal by (3).
In particular, condition (3) is met if the BIB design d˜ is constructed from a Hadamard matrix of order 4t, in which case it has b = 4t − 1, r = 2t − 1, λ = t − 1. It is also met if m(= 2s) is even, and the blocks of d˜ are taken as all (s − 1)-subsets of {2, . . . , m}. The construction in Peng, et al. (2019a) corresponds to a choice of d from such a d˜ via (II), along with A and A¯ chosen as
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the full design in s symbols. In contrast, we allow d to have much fewer blocks and take A and A¯ as OofA orthogonal arrays. This leads to a substantial reduction in run size as illustrated in the next section.
3 3.1
Construction details Optimal fractions
Theorem 1 and Proposition 1 lead to a systematic construction which is conveniently described in Algorithm 1 below.
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Journal Pre-proof Algorithm 1 Construction method of optimal fractional OofA designs Step 1: Start with OofA orthogonal arrays A and A¯ of orders R×h and R×(m−h), respectively. Step 2: As in (I) or (II) of Proposition 1, obtain a block design d in m symbols and b blocks C1 , . . . , Cb , each of size h (< m). For u = 1, . . . , b, let C¯u = {1, . . . , m} \ Cu , and write Bu
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¯u for A and A¯ with the symbols in Cu and C¯u , respectively. and B
Step 3: Obtain an m-component D-, A- and E-optimal fractional OofA design d∗ in N = 2bR runs, as given by the rows of D = (D1 T , D2 T , · · · , Db T )T , where ! ¯u Bu B , u = 1, . . . , b. Du = ¯ u Bu ∼B
(4)
Remark 1 Let A3 denote the full OofA design for m = 3, and A4 , A5 , A6 denote OofA orthogonal arrays in 12, 12 and 24 runs for m = 4, 5, 6, respectively (Voelkel, 2019). In Algorithm 1, A or A¯ may, in particular, be chosen as these arrays. An illustrative example follows. Example 1
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(a) For m = 8, obtain d as in (II) above with d˜ constructed from a Hadamard matrix of order 8, and take A = A¯ = A4 . Then b = 7, R = 12, and d∗ = A8 has 168 runs. (b) For m = 10, obtain d as in (II) above with d˜ taken as a BIB design having parameters b = 18, r = 8, λ = 3, and take A = A¯ = A5 . Then b = 18, R = 12, and d∗ = A10 has 432 runs.
(c) For m = 12, obtain d as in (II) above with d˜ constructed from a Hadamard matrix of order 12, and take A = A¯ = A6 . Then b = 11, R = 24, and d∗ = A12 has 528 runs. (d) For m = 7, obtain d as in (a) above. Take A = A4 , A¯ as a two-fold repetition of A3 . Then b = 7, R = 12, and d∗ = A7 has 168 runs.
(e) For m = 9, obtain d as in (b) above. Take A = A5 , A¯ = A4 . Then b = 18, R = 12, and
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d∗ = A9 has 432 runs.
(f) For m = 11, obtain d as in (c) above. Take A = A6 , A¯ as a two-fold repetition of A5 . Then b = 11, R = 24, and d∗ = A11 has 528 runs. For m = 7, 9 and 11, optimal fractional OofA designs with the same run sizes as above could also be obtained by deleting components 8, 10 and 12 from the constructions in (a), (b) and (c), respectively. The optimal fractions in this example are much smaller than their counterparts arising from the systematic construction in Peng, et al. (2019a). For m = 7, 8, 9, 10, 11 and 12, their method entails run sizes 840, 1680, 15120, 30240, 332640 and 665280, respectively. 5
Journal Pre-proof 3.2
Highly efficient fractions
The OofA orthogonal arrays A and A¯ are major building blocks in Algorithm 1. These arrays, each with R rows, lead to an m-component optimal fraction d∗ in 2bR runs. If R is small then d∗ also has a relatively small run size. For larger m, however, it is hard to find A and A¯ in
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small R. This happens, for example, with m = 16, when either A or A¯ must have at least eight columns. In this case, one can take A = A¯ = A8 , where A8 is as in Example 1(a), and obtain d via (II) of Proposition 1, with d˜ constructed from a Hadamard matrix of order 16. Then Algorithm 1 yields an optimal fraction d∗ in 5040 runs, because A8 is an OofA orthogonal array, and b = 15, R = 168. This d∗ , although much smaller than the full design (m! = 16! ≈ 2 × 1013 ), is still large due to a relatively large R.
With a view to overcoming the above difficulty, we consider a modification of Algorithm 1, where Steps 2 and 3 are left unchanged while in Step 1, instead of demanding A and A¯ to be OofA orthogonal arrays, it is only required that these arrays, of orders R × h and R × (m − h), should be efficient OofA designs, in h and m−h components, respectively. This modified version is, hereafter, referred to as Algorithm 1(a).
In Step 1 of Algorithm 1(a), the arrays A and A¯ may, for example, be chosen following Peng, et al. (2019b) or Winker, et al. (2019). While the final design d∗ in Step 3 may no longer be optimal, it is anticipated to have high efficiency as a result of A and A¯ being efficient
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OofA designs. Quite pleasantly, our computations show that the efficiency of d∗ can considerably ¯ thus signifying that the role of these arrays in the construction exceed the efficiencies of A and A, is outweighed by that of the block design d. An illustrative example follows. Example 2
(a) For m = 16, obtain d as in (II) of Proposition 1, with d˜ constructed from a Hadamard 0 0 matrix of order 16. Take A = A¯ = A , where A is the 29-run, 8-component OofA design 8
8
in Winker, et al. (2019). Then b = 15, R = 29, and d∗ has 870 runs, much smaller than 5040 runs that A = A¯ = A8 leads to. The d∗ as obtained here has D-efficiency 0.97, 0
which far exceeds the D-efficiency, 0.75, of A8 as an 8-component OofA design. Deletion of three, two or one components from this d∗ yields OofA designs for m = 13, 14 and 15
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in 870 runs and each having D-efficiency 0.98.
(b) Even for smaller m, Algorithm 1(a) can considerably reduce the run size while maintaining high efficiency. For instance, with m = 8, if one obtains d as in Example 1(a) and takes 0 0 A = A¯ = A , where A is the 7-run, 4-component OofA design in Winker, et al. (2019), 4
4
0
then b = R = 7, and d∗ has 98 runs. This d∗ has D-efficiency 0.98, whereas A4 has D-efficiency 0.89 as a 4-component OofA design. Note that d∗ is almost as efficient as its counterpart in Example 1(a) but has a significantly smaller run size. (c) In the spirit of (b) above, let m = 12. If one obtains d as in Example 1(c) and takes 0 0 A = A¯ = A6 , where A6 is the 16-run, 6-component OofA design in Winker, et al. (2019), 6
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then b = 11, R = 16, and d∗ has 352 runs. This d∗ has D-efficiency 0.98, while A6 has D-efficiency 0.88 as a 6-component OofA design. Again, d∗ is almost as efficient as its counterpart in Example 1(c) but has only two-thirds run size.
Concluding remarks
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4
Block designs were employed in this paper to obtain a systematic construction method leading to optimal and highly efficient fractional OofA designs in much smaller run sizes than the currently available optimal fractions. Future work on further reduction in run size in a systematic manner with theoretical assurance of optimality will be welcome.
While we worked in the framework of the PWO model (1), computations show that our designs remain highly efficient under variations of this model that allow tapering (Peng, et al., 2019a) of the impact of any pairwise order with an increase in the distance between the components in the pair. Our designs are also anticipated to perform well under other OofA models such as the component-position model (Yang, et al., 2018) and the triplet order model (Mee, 2019), and it will be of interest to examine this in some detail.
It is hoped that the present work will generate more interest in the above and related issues.
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Acknowledgements
We thank the referees for their very constructive suggestions. The work of Jianbin Chen was supported by the National Natural Science Foundation of China (Grant Nos. 11771220). The work of Rahul Mukerjee was supported by the J.C. Bose National Fellowship of Government of India and a grant from Indian Institute of Management Calcutta. The work of Dennis Lin was supported by the National Science Foundation via Grant DMS-18102925.
Appendix A: Proofs
Proof of Theorem 1. Let X = [X0 , X12 , . . . , X(m−1)m ] be the model matrix of d∗ as introduced below equation (1). Write N = 2bR for the run size of d∗ . Because d∗ has moment matrix
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(1/N )X T X, the result follows from the lemma below. Although this lemma looks similar to a result in Peng, et al. (2019a), a major difference lies in its proof which exploits condition (2) on the block design d that does not arise in their setup. Lemma 1 If condition (2) is met for every distinct i, j, k in {1, . . . , m}, then the following hold. (a) For ij ∈ S, X0T Xij = 0 and XijT Xij = N . Also, X0T X0 = N . TX T (b) For i, j, k satisfying 1 ≤ i < j < k ≤ m, XijT Xik = N/3, Xik jk = N/3 and Xij Xjk =
−N/3.
7
Journal Pre-proof (c) For ij, kl ∈ S, if the sets {i, j} and {k, l} are disjoint, then XijT Xkl = 0. Proof. (a) Clearly, XijT Xij = X0T X0 = N for ij ∈ S, as X has elements ±1. Let Gu (ij) denote ¯u , the contribution of Du to X T Xij . Then Gu (ij) = 0, by (4) and the definitions of Bu and B 0
irrespective of whether i is in Cu or C¯u , and j is in Cu or C¯u . Hence X0T Xij = 0, ij ∈ S.
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(b)Write Gu for the row vector with elements Gu (ij, ik), Gu (ik, jk) and Gu (ij, jk), where Gu (ij, ik) is the contribution of Du to XijT Xik , and Gu (ik, jk), Gu (ij, jk) are similarly defined. ¯u , the following hold. By (4) and the definitions of Bu and B (i) Gu = (2R/3)(1, 1, −1), if either i, j, k ∈ Cu , or i, j, k ∈ C¯u ;
(ii) Gu = 2R(1, 0, 0), if either j, k ∈ Cu , i ∈ C¯u , or i ∈ Cu , j, k ∈ C¯u ;
(iii) Gu = 2R(0, 1, 0), if either i, j ∈ Cu , k ∈ C¯u , or k ∈ Cu , i, j ∈ C¯u ;
(iv) Gu = 2R(0, 0, −1), if either i, k ∈ Cu , j ∈ C¯u , or j ∈ Cu , i, k ∈ C¯u .
If i, j and k occur together in λijk blocks of d, then the situations (i), (ii), (iii) and (iv) correspond to
λijk + (b − ri − rj − rk + λij + λik + λjk − λijk ) = b − ri − rj − rk + λij + λik + λjk = a0 , (λjk − λijk ) + (ri − λij − λik + λijk ) = ri − λij − λik + λjk = a1 ,
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(λij − λijk ) + (rk − λik − λjk + λijk ) = rk − λik − λjk + λij = a2 , (λik − λijk ) + (rj − λij − λjk + λijk ) = rj − λij − λjk + λik = a3
choices of u, respectively. Hence
XijT Xik = (2R/3)(a0 + 3a1 ) = (2R/3) (b + 2ri − rj − rk − 2λij − 2λik + 4λjk ) and similarly, we have
T Xik Xjk = (2R/3)(b + 2rk − ri − rj − 2λik − 2λjk + 4λij ),
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XijT Xjk = −(2R/3) (b + 2rj − ri − rk − 2λij − 2λjk + 4λik ) .
Under condition (2), these three scalar products equal N/3, N/3 and −N/3, respectively, with N = 2bR.
(c) Let Gu (ij, kl) be the contribution of Du to XijT Xkl . By (4) and the definitions of Bu and ¯u , B
(i) Gu (ij, kl) = 2R, if either i, k ∈ Cu , j, l ∈ C¯u , or j, l ∈ Cu , i, k ∈ C¯u ; (ii) Gu (ij, kl) = −2R, if either i, l ∈ Cu , j, k ∈ C¯u , or j, k ∈ Cu , i, l ∈ C¯u ; 8
Journal Pre-proof and Gu (ij, kl) = 0 in all other situations. Hence, as in (b) above, if i, j, k and l occur together in λijkl blocks of d, then we obtain under (2), XijT Xkl = 2R[{(λik − λijk − λikl + λijkl ) + (λjl − λijl − λjkl + λijkl )} −{(λil − λijl − λikl + λijkl ) + (λjk − λijk − λjkl + λijkl )}]
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The proof is completed.
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= 2R{(λik − λjk ) − (λil − λjl )} = R{(ri − rj ) − (ri − rj )} = 0.
9
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Appendix B: Proposed Designs for m = 7, 8, . . . , 16 Table 1 shows some designs obtained by the methods proposed here. For each such design, the number of runs, N , and D-efficiency are displayed. As before, D-efficiency of any fraction Π is defined as DΠ /Df ull , and any design with D-efficiency 1 is D-optimal. We also indicate the
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construction of these designs. For even m, this is done by specifying the algorithm employed, along with the corresponding BIB(b, r, λ) design d˜ in Proposition 1(II) as well as A, A¯ or A0 , A¯0 . In the table, the arrays A4 , A5 , A6 are as in Voelkel (2019), A8 is design (1) for m = 8, and A04 , A05 , A06 , A08 are as in Winker, et al. (2019). For comparison, Table 1 also shows the number of runs, N ∗ (= m!/s!), in the optimal fraction that the systematic construction in Peng, et al. (2019a) yields.
m 7
Table 1: Some designs obtained by the proposed methods m!/s!) N Construction 840 168 Delete component 8 from design (1) for m = 8 98 Delete component 8 from design (2) for m = 8
N ∗ (=
D-efficiency 1 0.98
1,680
168 (1) Algorithm 1, BIB(7, 3, 1), A = A¯ = A4 98 (2) Algorithm 1(a), BIB(7, 3, 1), A0 = A¯0 = A04
1 0.98
9
15,120
432 Delete component 10 from design (1) for m = 10 396 Delete component 10 from design (2) for m = 10
1 0.99
10
30,240
432 (1) Algorithm 1, BIB(18, 8, 3), A = A¯ = A5 396 (2) Algorithm 1(a), BIB(18, 8, 3), A0 = A¯0 = A05
1 0.99
11
332,640
528 Delete component 12 from design (1) for m = 12 352 Delete component 12 from design (2) for m = 12
1 0.99
12
665,280
528 (1) Algorithm 1, BIB(11, 5, 2), A = A¯ = A6 352 (2) Algorithm 1(a), BIB(11, 5, 2), A0 = A¯0 = A06
1 0.98
13
8,648,640
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8
5,040 Delete components 14, 15, 16 from design (1) for m = 16 870 Delete components 14, 15, 16 from design (2) for m = 16
1 0.98 1 0.98
15 259,459,200 5,040 Delete components 16 from design (1) for m = 16 870 Delete components 16 from design (2) for m = 16
1 0.98
16 518,918,400 5,040 (1) Algorithm 1, BIB(15, 7, 3), A = A¯ = A8 870 (2) Algorithm 1(a), BIB(15,7, 3), A0 = A¯0 = A08
1 0.97
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14 17,297,280 5,040 Delete components 15, 16 from design (1) for m = 16 870 Delete components 15, 16 from design (2) for m = 16
10
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230–241.
[12] Winker, P., Chen, J.B. and Lin, D.K.J. (2019). The construction of optimal design for order-of-addition experiment via threshold accepting. Submitted manuscript. [13] Yang, J.F., Sun, F.S. and Xu H. (2018). Component orthogonal arrays for order-of-addition experiments. Submitted manuscript. [14] Zhao, Y.N., Lin, D.K.J. and Liu, M.Q. (2018). Minimal-point design for order of addition experiment. Submitted manuscript.
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Jianbin Chen performed writing-original draft, methodology and conceptualization. Rahul Mukerjee and Dennis K.J. Lin performed writing-review, methodology, conceptualization and editing.
PII: DOI: Reference:
S0167-7152(20)30031-6 https://doi.org/10.1016/j.spl.2020.108728 STAPRO 108728
To appear in:
Statistics and Probability Letters
Received date : 29 October 2019 Revised date : 29 December 2019 Accepted date : 2 February 2020 Please cite this article as: J. Chen, R. Mukerjee and D.K.J. Lin, Construction of optimal fractional Order-of-Addition designs via block designs. Statistics and Probability Letters (2020), doi: https://doi.org/10.1016/j.spl.2020.108728. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Elsevier B.V. All rights reserved.
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Construction of Optimal Fractional
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Order-of-Addition Designs via Block Designs Jianbin Chena , Rahul Mukerjeeb , and Dennis K.J. Linc a
School of Statistics and Data Science, LPMC, Nankai University, Tianjin 300071, China
b
c
Indian Institute of Management Calcutta, Joka, Diamond Harbour Road, Kolkata 700104, India Department of Statistics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A
Abstract
Order of addition (OofA) experiments have found wide applications. Each order (run) of an OofA experiment is a permutation of m (≥ 2) components. It is typically infeasible to compare all the m! possible runs, especially when m is large. This calls for experimentation with a subset or fraction of these m! runs. However, only a few systematic results are available
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on the construction of such fractions ensuring optimality. We employ block designs to propose a systematic combinatorial construction method for optimal fractional OofA designs, and extend the method to construct highly efficient OofA designs, both in much smaller run sizes than the currently available optimal fractions.
Keywords: Balanced incomplete block design, Fractional OofA design, Optimality, Pairwise order model, Systematic combinatorial construction. MSC: 62K15, 62K10
1
Introduction
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Many physical phenomena encountered in science and engineering are affected by the addition order of m materials or components. In chemical experiments, the performance or the amount of reaction production is often determined by the order of adding reagents. For example, Song et al. (2014) conducted experiments with the 3! orderings of three reagents, namely, carbon dots, F e2+ and H2 O2 , and found that addition of F e2+ and H2 O2 before carbon dots leads to stronger photoluminescence intensities. Jiang and Ng (2014) reported chemical experiments where the output is influenced by the order of addition of four components, Hg 2+ , Cu2+ , S 2− and light. Order of addition (OofA) experiments have found wide application also in bio-chemistry
1
Journal Pre-proof (Shinohara and Ogawa, 1998), nutritional science (Karim, et al., 2000) and NP-hard ordering problem (Chen, et al., 2019), just to name a few. With m components, there are m! possible orders or runs, each a permutation of {1, 2, . . . , m}.
It is infeasible to include all of these in an experiment, especially when m is large. This warrants experimentation with a subset or fraction of these m! runs. The problem of selecting such a
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fraction has received attention in the literature. For instance, Van Nostrand (1995) considered the design with “pseudo factors” or pairwise ordering (PWO) factors. Lin and Peng (2019) reviewed the latest work on the design and model of OofA experiments, and introduced some new thoughts. Voelkel (2019) provided some smaller OofA orthogonal arrays under some proposed design criteria. Peng, et al. (2019a) considered different types of optimality criteria and proposed a systematic method to construct a class of robust optimal fractional OofA designs. Zhao, et al. (2018) considered the minimal-point OofA designs. Yang, et al. (2018) obtained a number of OofA designs called component orthogonal arrays that are optimal under their component-position model. Chen, et al. (2019) introduced a statistical method to speculate solutions of NP-hard ordering problem by making use of design for OofA experiment. The current literature on OofA designs focuses on computer search. A notable exception is Peng, et al. (2019a) who present a systematic construction of optimal fractional OofA designs in m!/s! runs, where m = 2s for even m and m = 2s + 1 for odd m. However, these are quite large for larger m. Thus, with m = 10, their construction requires 30240 runs which, despite
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being less than 1% of the run size (= 10!) of the full design, is still rather large. With regard to smaller fractions, Peng, et al. (2019a) hinted at the possibility of using block designs, but did not pursue the idea any further. We develop and formalize this approach here, and show in the subsequent sections how it can yield optimal or highly efficient fractional OofA designs in considerably smaller run sizes. The proofs appear in the appendix.
2
Optimality of fractional OofA designs
Denote a typical order by π = (π1 , . . . , πm ), which is a permutation of {1, . . . , m}. Based on a
fraction Π of size N from the m! possible orders, consider the pairwise order (PWO) model
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y(π) = β0 +
X
βij Iij (π) + ε,
i
π ∈ Π,
(1)
where Iij = +1, if component i precedes component j; and Iij = −1 otherwise; y is the response of interest; and ε is the observational error. As usual, such errors are assumed to be uncorrelated,
all with mean zero and the same variance. There are p = 1 + m(m − 1)/2 parameters of
interest as given by the elements of β = (β0 , β12 , . . . , β(m−1)m )T , where the superscript denotes
transpose. Let X = [X0 , X12 , . . . , X(m−1)m ] be the model matrix of Π, where X0 corresponds to β0 and Xij corresponds to βij . The fraction Π has moment matrix M = N −1 X T X and D-value DΠ = {det(M )}1/p , which captures the generalized variance of the estimator of β arising from 2
Journal Pre-proof Π. The PWO model has been widely studied as a reasonable model for OofA experiments (see Jiang and Ng (2014) for an example of a practical situation, and Peng, et al. (2019a) for more details). While we mainly focus on the D-criterion here, other design criteria, such as the Aand E-criteria can be considered similarly. Let d0 be the full design that includes all the m! possible orders. Peng, et al. (2019a)
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proved the optimality of d0 under various criteria, including the D-, A- and E-criteria. Hence the D-efficiency of any fraction Π equals DΠ /Df ull , where Df ull is the D-value of d0 . Obviously, a fraction is D-, A- and E-optimal if its moment matrix equals that of d0 . Peng, et al. (2019a) obtained the moment matrix of d0 as M0 =diag{1, I + (1/3)V }, where I is the identity matrix of
order q = m(m − 1)/2, and V is a q × q matrix, with rows and columns indexed by the elements of S = {ij, 1 ≤ i < j ≤ m}, such that the (ij, kl)th element of V is
1, if i = k, j 6= l or i 6= k, j = l, V (ij, kl) = −1, if i = l or j = k, 0, otherwise.
We are now in a position to initiate the construction of optimal fractions in smaller run sizes, using binary block designs where every symbol appears at most once in each block. Let d be such a design in symbols 1, . . . , m (m ≥ 4), and b blocks C1 , . . . , Cb , each of size h (< m),
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such that symbol i appears in ri blocks, and symbols i and j occur together in λij blocks (i, j = 1, . . . , m; i 6= j). Let C¯u = {1, . . . , m} \ Cu , u = 1, . . . , b. Also, let A and A¯ be OofA
orthogonal array designs, involving h and m − h symbols, respectively, and both having R runs, i.e., A and A¯ have the same moment matrices, under the PWO model, as the corresponding ¯u for the A and A¯ with the symbols in Cu and C¯u , full designs (Voelkel, 2019). Write Bu and B respectively. Form the (2bR) × m array D = (D1 T , D2 T , · · · , Db T )T , where ! ¯u Bu B Du = , u = 1, . . . , b, ¯ u Bu ∼B with ∼ representing the operator of column reversal of any matrix. For example, if m = 6, h = 3, and C1 = {1, 2, 3} is the first block of d, then C¯1 = {4, 5, 6}, and with both A and A¯ runs, one obtains 4 5 3 4 6 2 5 4 3 ¯1 = , B 5 6 1 2 6 4 6 5 1
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taken as the full design in 3! 1 2 1 3 2 1 B1 = 2 3 3 1 3 2
6
5 6 , 4 5 4
¯1 = ∼B
¯1 has the same columns as B ¯1 but in a reversed order. where ∼ B 3
6 5 4
5 6 4 6 4 5 , 4 6 5 5 4 6 4 5 6
Journal Pre-proof Theorem 1 The m-component fractional OofA design d∗ in N = 2bR runs, as given by the rows of D, has moment matrix M0 under the PWO model and hence is D-, A- and E-optimal among all designs with the same number of runs, provided for every distinct i, j, k in {1, . . . , m},
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λik − λjk = (ri − rj )/2.
(2)
Proposition 1 below addresses the crucial issue of choosing the block design d so as to meet condition (2).
Proposition 1 The following choices of d satisfy condition (2): (I) Take d as a balanced incomplete block (BIB) design;
(II) Let d˜ be a BIB design in symbols {2, . . . , m} and b blocks, where each symbol appears in r blocks, each pair of distinct symbols appear together in λ blocks, and b − 3r + 2λ = 0.
(3)
˜ Obtain d by adding a single symbol, say 1, to every block of d.
While (I) obviously entails condition (2), the truth of (2) for (II) follows noting that
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(i) if i, j, k ∈ {2, . . . , m} or k = 1 and i, j ∈ {2, . . . , m}, then λik − λjk = 0 = (ri − rj )/2; (ii) if i = 1 and j, k ∈ {2, . . . , m}, then λik − λjk = r − λ, (ri − rj )/2 = (b − r)/2, and they are equal by (3);
(iii) if j = 1 and i, k ∈ {2, . . . , m}, then λik − λjk = λ − r, (ri − rj )/2 = (r − b)/2, and the two are equal by (3).
In particular, condition (3) is met if the BIB design d˜ is constructed from a Hadamard matrix of order 4t, in which case it has b = 4t − 1, r = 2t − 1, λ = t − 1. It is also met if m(= 2s) is even, and the blocks of d˜ are taken as all (s − 1)-subsets of {2, . . . , m}. The construction in Peng, et al. (2019a) corresponds to a choice of d from such a d˜ via (II), along with A and A¯ chosen as
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the full design in s symbols. In contrast, we allow d to have much fewer blocks and take A and A¯ as OofA orthogonal arrays. This leads to a substantial reduction in run size as illustrated in the next section.
3 3.1
Construction details Optimal fractions
Theorem 1 and Proposition 1 lead to a systematic construction which is conveniently described in Algorithm 1 below.
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Journal Pre-proof Algorithm 1 Construction method of optimal fractional OofA designs Step 1: Start with OofA orthogonal arrays A and A¯ of orders R×h and R×(m−h), respectively. Step 2: As in (I) or (II) of Proposition 1, obtain a block design d in m symbols and b blocks C1 , . . . , Cb , each of size h (< m). For u = 1, . . . , b, let C¯u = {1, . . . , m} \ Cu , and write Bu
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¯u for A and A¯ with the symbols in Cu and C¯u , respectively. and B
Step 3: Obtain an m-component D-, A- and E-optimal fractional OofA design d∗ in N = 2bR runs, as given by the rows of D = (D1 T , D2 T , · · · , Db T )T , where ! ¯u Bu B , u = 1, . . . , b. Du = ¯ u Bu ∼B
(4)
Remark 1 Let A3 denote the full OofA design for m = 3, and A4 , A5 , A6 denote OofA orthogonal arrays in 12, 12 and 24 runs for m = 4, 5, 6, respectively (Voelkel, 2019). In Algorithm 1, A or A¯ may, in particular, be chosen as these arrays. An illustrative example follows. Example 1
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(a) For m = 8, obtain d as in (II) above with d˜ constructed from a Hadamard matrix of order 8, and take A = A¯ = A4 . Then b = 7, R = 12, and d∗ = A8 has 168 runs. (b) For m = 10, obtain d as in (II) above with d˜ taken as a BIB design having parameters b = 18, r = 8, λ = 3, and take A = A¯ = A5 . Then b = 18, R = 12, and d∗ = A10 has 432 runs.
(c) For m = 12, obtain d as in (II) above with d˜ constructed from a Hadamard matrix of order 12, and take A = A¯ = A6 . Then b = 11, R = 24, and d∗ = A12 has 528 runs. (d) For m = 7, obtain d as in (a) above. Take A = A4 , A¯ as a two-fold repetition of A3 . Then b = 7, R = 12, and d∗ = A7 has 168 runs.
(e) For m = 9, obtain d as in (b) above. Take A = A5 , A¯ = A4 . Then b = 18, R = 12, and
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d∗ = A9 has 432 runs.
(f) For m = 11, obtain d as in (c) above. Take A = A6 , A¯ as a two-fold repetition of A5 . Then b = 11, R = 24, and d∗ = A11 has 528 runs. For m = 7, 9 and 11, optimal fractional OofA designs with the same run sizes as above could also be obtained by deleting components 8, 10 and 12 from the constructions in (a), (b) and (c), respectively. The optimal fractions in this example are much smaller than their counterparts arising from the systematic construction in Peng, et al. (2019a). For m = 7, 8, 9, 10, 11 and 12, their method entails run sizes 840, 1680, 15120, 30240, 332640 and 665280, respectively. 5
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Highly efficient fractions
The OofA orthogonal arrays A and A¯ are major building blocks in Algorithm 1. These arrays, each with R rows, lead to an m-component optimal fraction d∗ in 2bR runs. If R is small then d∗ also has a relatively small run size. For larger m, however, it is hard to find A and A¯ in
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small R. This happens, for example, with m = 16, when either A or A¯ must have at least eight columns. In this case, one can take A = A¯ = A8 , where A8 is as in Example 1(a), and obtain d via (II) of Proposition 1, with d˜ constructed from a Hadamard matrix of order 16. Then Algorithm 1 yields an optimal fraction d∗ in 5040 runs, because A8 is an OofA orthogonal array, and b = 15, R = 168. This d∗ , although much smaller than the full design (m! = 16! ≈ 2 × 1013 ), is still large due to a relatively large R.
With a view to overcoming the above difficulty, we consider a modification of Algorithm 1, where Steps 2 and 3 are left unchanged while in Step 1, instead of demanding A and A¯ to be OofA orthogonal arrays, it is only required that these arrays, of orders R × h and R × (m − h), should be efficient OofA designs, in h and m−h components, respectively. This modified version is, hereafter, referred to as Algorithm 1(a).
In Step 1 of Algorithm 1(a), the arrays A and A¯ may, for example, be chosen following Peng, et al. (2019b) or Winker, et al. (2019). While the final design d∗ in Step 3 may no longer be optimal, it is anticipated to have high efficiency as a result of A and A¯ being efficient
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OofA designs. Quite pleasantly, our computations show that the efficiency of d∗ can considerably ¯ thus signifying that the role of these arrays in the construction exceed the efficiencies of A and A, is outweighed by that of the block design d. An illustrative example follows. Example 2
(a) For m = 16, obtain d as in (II) of Proposition 1, with d˜ constructed from a Hadamard 0 0 matrix of order 16. Take A = A¯ = A , where A is the 29-run, 8-component OofA design 8
8
in Winker, et al. (2019). Then b = 15, R = 29, and d∗ has 870 runs, much smaller than 5040 runs that A = A¯ = A8 leads to. The d∗ as obtained here has D-efficiency 0.97, 0
which far exceeds the D-efficiency, 0.75, of A8 as an 8-component OofA design. Deletion of three, two or one components from this d∗ yields OofA designs for m = 13, 14 and 15
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in 870 runs and each having D-efficiency 0.98.
(b) Even for smaller m, Algorithm 1(a) can considerably reduce the run size while maintaining high efficiency. For instance, with m = 8, if one obtains d as in Example 1(a) and takes 0 0 A = A¯ = A , where A is the 7-run, 4-component OofA design in Winker, et al. (2019), 4
4
0
then b = R = 7, and d∗ has 98 runs. This d∗ has D-efficiency 0.98, whereas A4 has D-efficiency 0.89 as a 4-component OofA design. Note that d∗ is almost as efficient as its counterpart in Example 1(a) but has a significantly smaller run size. (c) In the spirit of (b) above, let m = 12. If one obtains d as in Example 1(c) and takes 0 0 A = A¯ = A6 , where A6 is the 16-run, 6-component OofA design in Winker, et al. (2019), 6
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then b = 11, R = 16, and d∗ has 352 runs. This d∗ has D-efficiency 0.98, while A6 has D-efficiency 0.88 as a 6-component OofA design. Again, d∗ is almost as efficient as its counterpart in Example 1(c) but has only two-thirds run size.
Concluding remarks
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4
Block designs were employed in this paper to obtain a systematic construction method leading to optimal and highly efficient fractional OofA designs in much smaller run sizes than the currently available optimal fractions. Future work on further reduction in run size in a systematic manner with theoretical assurance of optimality will be welcome.
While we worked in the framework of the PWO model (1), computations show that our designs remain highly efficient under variations of this model that allow tapering (Peng, et al., 2019a) of the impact of any pairwise order with an increase in the distance between the components in the pair. Our designs are also anticipated to perform well under other OofA models such as the component-position model (Yang, et al., 2018) and the triplet order model (Mee, 2019), and it will be of interest to examine this in some detail.
It is hoped that the present work will generate more interest in the above and related issues.
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Acknowledgements
We thank the referees for their very constructive suggestions. The work of Jianbin Chen was supported by the National Natural Science Foundation of China (Grant Nos. 11771220). The work of Rahul Mukerjee was supported by the J.C. Bose National Fellowship of Government of India and a grant from Indian Institute of Management Calcutta. The work of Dennis Lin was supported by the National Science Foundation via Grant DMS-18102925.
Appendix A: Proofs
Proof of Theorem 1. Let X = [X0 , X12 , . . . , X(m−1)m ] be the model matrix of d∗ as introduced below equation (1). Write N = 2bR for the run size of d∗ . Because d∗ has moment matrix
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(1/N )X T X, the result follows from the lemma below. Although this lemma looks similar to a result in Peng, et al. (2019a), a major difference lies in its proof which exploits condition (2) on the block design d that does not arise in their setup. Lemma 1 If condition (2) is met for every distinct i, j, k in {1, . . . , m}, then the following hold. (a) For ij ∈ S, X0T Xij = 0 and XijT Xij = N . Also, X0T X0 = N . TX T (b) For i, j, k satisfying 1 ≤ i < j < k ≤ m, XijT Xik = N/3, Xik jk = N/3 and Xij Xjk =
−N/3.
7
Journal Pre-proof (c) For ij, kl ∈ S, if the sets {i, j} and {k, l} are disjoint, then XijT Xkl = 0. Proof. (a) Clearly, XijT Xij = X0T X0 = N for ij ∈ S, as X has elements ±1. Let Gu (ij) denote ¯u , the contribution of Du to X T Xij . Then Gu (ij) = 0, by (4) and the definitions of Bu and B 0
irrespective of whether i is in Cu or C¯u , and j is in Cu or C¯u . Hence X0T Xij = 0, ij ∈ S.
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(b)Write Gu for the row vector with elements Gu (ij, ik), Gu (ik, jk) and Gu (ij, jk), where Gu (ij, ik) is the contribution of Du to XijT Xik , and Gu (ik, jk), Gu (ij, jk) are similarly defined. ¯u , the following hold. By (4) and the definitions of Bu and B (i) Gu = (2R/3)(1, 1, −1), if either i, j, k ∈ Cu , or i, j, k ∈ C¯u ;
(ii) Gu = 2R(1, 0, 0), if either j, k ∈ Cu , i ∈ C¯u , or i ∈ Cu , j, k ∈ C¯u ;
(iii) Gu = 2R(0, 1, 0), if either i, j ∈ Cu , k ∈ C¯u , or k ∈ Cu , i, j ∈ C¯u ;
(iv) Gu = 2R(0, 0, −1), if either i, k ∈ Cu , j ∈ C¯u , or j ∈ Cu , i, k ∈ C¯u .
If i, j and k occur together in λijk blocks of d, then the situations (i), (ii), (iii) and (iv) correspond to
λijk + (b − ri − rj − rk + λij + λik + λjk − λijk ) = b − ri − rj − rk + λij + λik + λjk = a0 , (λjk − λijk ) + (ri − λij − λik + λijk ) = ri − λij − λik + λjk = a1 ,
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(λij − λijk ) + (rk − λik − λjk + λijk ) = rk − λik − λjk + λij = a2 , (λik − λijk ) + (rj − λij − λjk + λijk ) = rj − λij − λjk + λik = a3
choices of u, respectively. Hence
XijT Xik = (2R/3)(a0 + 3a1 ) = (2R/3) (b + 2ri − rj − rk − 2λij − 2λik + 4λjk ) and similarly, we have
T Xik Xjk = (2R/3)(b + 2rk − ri − rj − 2λik − 2λjk + 4λij ),
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XijT Xjk = −(2R/3) (b + 2rj − ri − rk − 2λij − 2λjk + 4λik ) .
Under condition (2), these three scalar products equal N/3, N/3 and −N/3, respectively, with N = 2bR.
(c) Let Gu (ij, kl) be the contribution of Du to XijT Xkl . By (4) and the definitions of Bu and ¯u , B
(i) Gu (ij, kl) = 2R, if either i, k ∈ Cu , j, l ∈ C¯u , or j, l ∈ Cu , i, k ∈ C¯u ; (ii) Gu (ij, kl) = −2R, if either i, l ∈ Cu , j, k ∈ C¯u , or j, k ∈ Cu , i, l ∈ C¯u ; 8
Journal Pre-proof and Gu (ij, kl) = 0 in all other situations. Hence, as in (b) above, if i, j, k and l occur together in λijkl blocks of d, then we obtain under (2), XijT Xkl = 2R[{(λik − λijk − λikl + λijkl ) + (λjl − λijl − λjkl + λijkl )} −{(λil − λijl − λikl + λijkl ) + (λjk − λijk − λjkl + λijkl )}]
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The proof is completed.
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= 2R{(λik − λjk ) − (λil − λjl )} = R{(ri − rj ) − (ri − rj )} = 0.
9
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Appendix B: Proposed Designs for m = 7, 8, . . . , 16 Table 1 shows some designs obtained by the methods proposed here. For each such design, the number of runs, N , and D-efficiency are displayed. As before, D-efficiency of any fraction Π is defined as DΠ /Df ull , and any design with D-efficiency 1 is D-optimal. We also indicate the
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construction of these designs. For even m, this is done by specifying the algorithm employed, along with the corresponding BIB(b, r, λ) design d˜ in Proposition 1(II) as well as A, A¯ or A0 , A¯0 . In the table, the arrays A4 , A5 , A6 are as in Voelkel (2019), A8 is design (1) for m = 8, and A04 , A05 , A06 , A08 are as in Winker, et al. (2019). For comparison, Table 1 also shows the number of runs, N ∗ (= m!/s!), in the optimal fraction that the systematic construction in Peng, et al. (2019a) yields.
m 7
Table 1: Some designs obtained by the proposed methods m!/s!) N Construction 840 168 Delete component 8 from design (1) for m = 8 98 Delete component 8 from design (2) for m = 8
N ∗ (=
D-efficiency 1 0.98
1,680
168 (1) Algorithm 1, BIB(7, 3, 1), A = A¯ = A4 98 (2) Algorithm 1(a), BIB(7, 3, 1), A0 = A¯0 = A04
1 0.98
9
15,120
432 Delete component 10 from design (1) for m = 10 396 Delete component 10 from design (2) for m = 10
1 0.99
10
30,240
432 (1) Algorithm 1, BIB(18, 8, 3), A = A¯ = A5 396 (2) Algorithm 1(a), BIB(18, 8, 3), A0 = A¯0 = A05
1 0.99
11
332,640
528 Delete component 12 from design (1) for m = 12 352 Delete component 12 from design (2) for m = 12
1 0.99
12
665,280
528 (1) Algorithm 1, BIB(11, 5, 2), A = A¯ = A6 352 (2) Algorithm 1(a), BIB(11, 5, 2), A0 = A¯0 = A06
1 0.98
13
8,648,640
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8
5,040 Delete components 14, 15, 16 from design (1) for m = 16 870 Delete components 14, 15, 16 from design (2) for m = 16
1 0.98 1 0.98
15 259,459,200 5,040 Delete components 16 from design (1) for m = 16 870 Delete components 16 from design (2) for m = 16
1 0.98
16 518,918,400 5,040 (1) Algorithm 1, BIB(15, 7, 3), A = A¯ = A8 870 (2) Algorithm 1(a), BIB(15,7, 3), A0 = A¯0 = A08
1 0.97
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14 17,297,280 5,040 Delete components 15, 16 from design (1) for m = 16 870 Delete components 15, 16 from design (2) for m = 16
10
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Jianbin Chen performed writing-original draft, methodology and conceptualization. Rahul Mukerjee and Dennis K.J. Lin performed writing-review, methodology, conceptualization and editing.