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Partially replicated block designs for two-level factorial experiments
Journal Pre-proof Partially replicated block designs for two-level factorial experiments Chen-Tuo Liao
PII: DOI: Reference:
S0167-7152(19)30299-8 https://doi.org/10.1016/j.spl.2019.108653 STAPRO 108653
To appear in:
Statistics and Probability Letters
Received date : 8 August 2018 Revised date : 13 August 2019 Accepted date : 6 October 2019 Please cite this article as: C.-T. Liao, Partially replicated block designs for two-level factorial experiments. Statistics and Probability Letters (2019), doi: https://doi.org/10.1016/j.spl.2019.108653. 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.
© 2019 Published by Elsevier B.V.
Journal Pre-proof
Chen-Tuo Liao
of
Partially Replicated Block Designs for Two-Level Factorial Experiments
p ro
Division of Biometry, Institute of Agronomy, National Taiwan University, Taipei, Taiwan
Abstract
Pr e-
In a two-level factorial experiment, we consider blocking in the parallel-flats design with identical flats within or between blocks. We propose three classes of optimal designs with equal or unequal block sizes. Keywords: Optimum design; Orthogonal blocking; Random block effects. 1. Introduction
Jo
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The theory and application of partially replicated parallel-flats designs have drawn more attention, due to their flexible run sizes, well-structured information matrices, and pure replicates. They have been used to identify the location and dispersion effects in two-level factorial experiments. Some real-world examples can be found in Tsai et al. (2015) and Tsai and Liao (2019). Some application of parallel flats designs in biomedical research was presented in Li (1998). For a given requirement set of specified effects, Li and Srivastava (1997) showed that the designs obtained by adding any single flat to a two-level orthogonal parallel-flats design are D- and E-optimal designs. Based on this result, Liao and Chai (2009) proposed a set of sufficient conditions for a parallel-flats design with two identical flats to be D-optimal for the specified effects. In this study, we focus on the extension to the blocking in the twolevel parallel-flats designs with partial replication.
Preprint submitted to Statistics and Probability Letters
August 13, 2019
Journal Pre-proof
2. Notation and Definitions
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2.1. Parallel-flats block design Let di be a two-level single-flat design consisting of all the solutions t satisfying At = ci over GF(2), where A denotes a p × n alias matrix of rank p; ci represents a p × 1 coset indicator vector and GF(2) stands for the Galois field of order two. The juxtaposition of f single-flat designs d1 , d2 , . . . , df is defined as an f parallel-flats design (PFD). Suppose that the treatment combinations determined by At = cjh , where h = 1, 2, . . . , gj , are assigned to the experimental units in the jth block, j = 1, 2, . . . , b. Namely, the PFD determined by (A, Cj ) is allocated in the jth block, where Cj = [cj1 cj2 · · · cjgj ] and the block size kj = gj ×2n−p . By combining such b blocks, a parallel-flats block design, abbreviated as f -PFBD, be determined by (A, C), where Pcan b C = [C1 C2 · · · Cb ]. Note that f = j=1 gj and the run size N = f × 2n−p . 2.2. Orthogonal blocking Suppose that an N -run block design is used to estimate a requirement set β with v potentially active effects. The N runs are allocated to b blocks of sizes k1 , k2 , . . . , kb . Let Y be the responses vector, then Y can be described as Y = 1N µ + Xβ + W γ + ,
(1)
urn
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where 1N denotes the vector of length N with all elements equal to 1; µ the constant term; X the model matrix whose entries are either equal to 1 or −1; W the block indicator matrix; γ the block effects vector and the vector of random errors assumed to be uncorrelated variables with common mean 0 and variance σe2 . The information matrix for β is given by h −1 T i M = X T IN − W W T W W X,
Jo
where IN denotes the identity matrix of order N . A factorial design is said to be orthogonally blocked with respect to β, if W T X = 0, where 0 denotes a zero matrix. This condition requires that both levels of each element in β occur equally often within every block.
2
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3. Partially Replicated f -PFBD
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Let β (e) represent the factorial effect with defining vector e, where e denotes an 1 × n vector whose qth element equals 1 if factor q appears in this effect; otherwise equals 0. Two factorial effects β(e1 ) and β(e2 ) are said to be in the same alias set determined by the alias matrix A, if the row vector e1 + e2 appears in the row space of A, i.e., e1 + e2 = wT A for some vector w over GF(2). In particular, factorial effect β(e) is said to be aliased with µ, if e = wT A. So the effects confounded with block effects must be aliased with µ. Suppose that an f -PFBD is orthogonally blocked with respect to a requirement set β. The best linear unbiased estimator (BLUE) for β is given by βˆ = (X T X)−1 X T Y . The BLUE is not affected by the block effects. The corresponding information matrix M is thus reduced to be X T X of the unblocked design. From Liao et al. (1996), M can be expressed as a block diagonal matrix. Let Gi be the alias set with vi elements of β corresponding to the P ith diagonal matrix of M , for i = 0, 1, . . . , m, where m = 2n−p − 1. Then, m i=0 vi = v and G0 consists of the effects aliased with µ.
urn
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3.1. Equal block size design We first consider a specific class of f -PFBDs with equal number of flats within each block. That is, g1 = · · · = gb = g, hence k1 = · · · = kb = k = g × 2n−p , f = b × g and N = f × 2n−p . We impose the constraint that there are at most two identical flats among the g flats within each block on this class of f -PFBDs, and call it as the α0 -family of f -PFBDs. For given β, we seek orthogonal designs for estimating β over the α0 -family. A vector over GF(2) is said to be an OA(1), orthogonal array of strength one, if it consists of equal occurrences of 0 and 1. The orthogonal array was introduced by Rao (1947) and its definition in the context of general GF(s) can be found in Srivastava and Li (1996). Theorem 1. An f -PFBD d∗ determined by the paired matrices (A∗ , C ∗ ), where C ∗ = [C1∗ C2∗ · · · Cb∗ ], is orthogonal for β over the α0 -family, if it satisfies the following conditions.
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(i) For β(e) ∈ G0 and e = wT A∗ , then wT Cj∗ is an OA(1) for j = 1, 2, . . . , b. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C ∗ is an OA(1) for all 0 ≤ i ≤ m, where m = 2n−p − 1. 3
Journal Pre-proof
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For an f -PFBD, the effects of β confounded with the block effects must belong to alias set G0 . Condition (i) makes those confounded effects clear from the block effects, hence d∗ has orthogonal blocking for β. Condition (ii) makes the information matrix equal to the diagonal matrix N Iv , showing that d∗ is orthogonal (A-optimal, D-optimal and E-optimal) for β. We refer to Liao et al. (1996) for a proof of the conditions. We further provide a lower bound for the number of flats in each block for an orthogonal design over the α0 -family. A proof of Proposition 1 can be found in Supplementary material online.
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Proposition 1. A design d∗ over the α0 -family is orthogonal for β with v possibly active effects, then the number of flats in each block g must satisfy the inequality 2(v + 1)2 ≤ g. 2n−p (bv + 1) Example 1. In a 25 factorial experiment, we consider an orthogonal design for the requirement set β = {1, 2, 3, 4, 5, 13, 14, 23, 24}. The design d1 determined by (A, C), where 1 1 1 1 1 A= 1 1 1 0 0 0 0 1 1 0 and
with
C1 C2 C3 C4
0 0 0 0 0 0 0 0 C1 = 0 0 , C 2 = 0 0 , C 3 = 1 1 , C 4 = 1 1 0 0 1 1 0 0 1 1
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C=
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is orthogonal for the β over the α0 -family of 8-PFBDs. The details of checking the example that satisfies Theorem 1 are presented in Supplementary material. By Proposition 1, the number of flats in each block g for d1 must satisfy the inequality 2(9 + 1)2 ≤ g. 25−3 (4 × 9 + 1) Equivalently, 50/37 ≤ g. So g = 2 is the minimal number satisfying the inequality, meaning that d1 with N = 32 is a minimal run-size orthogonal design for the β over the α0 -family. 4
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3.2. Unequal block size design By Proposition 1, an orthogonal design over the α0 -family can turn out to be a large run size design. We thus consider another class of f -PFBDs with g1 = g 0 + 1 and g2 = · · · = gb = g 0 , hence k1 = (g 0 + 1) × 2n−p and k2 = · · · = kb = k = g 0 × 2n−p , f = b × g 0 + 1 and N = f × 2n−p . The constraint that there are two identical flats in block 1 only is imposed on this class of designs. We call it as the α1 -family of f -PFBDs. A design in the α1 -family can be expressed by (A, C), where C = [C1 C2 · · · Cb ]. Here C1 = [c10 c11 · · · c1g0 ] and Cj = [cj1 cj2 · · · cjg0 ] for j = 2, · · · , b. WLOG, let c10 = c11 = 0, forcing the 2n−p treatment combinations satisfying At = 0 to appear twice in block 1. For given β, we seek D-optimal designs for estimating β over the α1 -family.
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Theorem 2. An f -PFBD d∗ is determined by the paired matrices (A∗ , C ∗ ). Let vi∗ denote the number of effects in Gi determined by the alias matrix A∗ for i = 0, 1, . . . , m, where m = 2n−p − 1. Then d∗ is D-optimal for β over the α1 -family, if it satisfies the following conditions.
al
(i) v0∗ = 0. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C0∗ is an OA(1) for all 1 ≤ i ≤ m, where C0∗ is obtained by removing c∗10 from C ∗. Pm Q ∗ (iii) The v1∗ , . . . , vm maximize m i=1 vi = i=1 vi (= v1 ×v2 · · ·×vm ) subject to v.
urn
Condition (i) ensures that d∗ has orthogonal blocking for β. From Theorem 1, if β(e) ∈ G0 with e = wT A, then it is not possible to have that wT Cj is an OA(1) for j = 1, . . . , b, since either g 0 + 1 or g 0 is an odd number. Thus, G0 must be empty, so v0∗ = 0. The information for β now is reduced to be X T X of the unblocked design, a proof for Conditions (ii) and (iii) can be found in Liao and Chai (2009).
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Example 2. Let d2 be the design obtained from d1 in Example 1 by removing one coset indicator vector from each of C2 , C3 and C4 . Then d2 is D-optimal for the same β over the α1 -family of 5-PFBDs. We modify the α1 -family by adding a zero vector as c20 to C2 , and call this as α2 -family of f -PFBDs. Note that c10 = c11 = c20 = 0, this forces the 2n−p treatment combinations satisfying At = 0 to appear twice in block 1 and once in block 2. Theorem 3 is straightforward from Theorem 2. 5
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Theorem 3. An f -PFBD d∗ is determined by the paired matrices (A∗ , C ∗ ). Let vi∗ denote the number of effects in Gi determined by the alias matrix A∗ for i = 0, 1, . . . , m, where m = 2n−p − 1. Then d∗ is D-optimal for β over the α2 -family, if it satisfies the following conditions.
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(i) v0∗ = 0. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C0∗ is an OA(1) for all 1 ≤ i ≤ m, where C0∗ is obtained by removing c∗10 and c∗20 from C ∗ . Q Pm ∗ (iii) The v1∗ , . . . , vm maximize m i=1 vi subject to i=1 vi = v.
4. Discussion
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Example 3. Let d3 be the design by adding 0 to C2 of d2 in Example 2. So d3 is D-optimal for the same β over the α2 -family of 6-PFDBs.
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The significance test for an effect in β is typically performed by Student’s t-test, and the estimate of error variance σe2 serves as the basis of the test. To obtain a replication-based estimate for the error variance, we estimate σe2 using real replications within blocks. All the optimal designs over α0 -, α1 and α2 -family can provide sufficient degrees of freedom for the t-test. The effects originally confounded with the block effects under the fixed block effects situation become estimable if the block effects are random (Cheng, 2014. p.98). Under the random blocks situation, it is assumed that the effects in γ and those in of Model (1) are mutually independent random variables distributed as normal distributions with common mean 0, and variances σb2 and σe2 , respectively. For example, suppose that effect β is confounded with block effects of γ1 and γ2 . Also, the estimate of β is given by βˆ = y 1 − y 2 , where y 1 and y 2 separately denote the averages of n1 and n2 responses in blocks 1 and 2. From Model (1), we have that βˆ = β + (γ1 − γ2 ) + (1 − 2 ). Clearly, β is confounded with γ1 − γ2 under the fixed block effects situation. However, under the random block effects situation, the expectation and variance of βˆ are equal to β and 2σb2 +(1/n1 +1/n2 )σe2 , respectively. Therefore, β is free from the block effects and the inference about β involves both the estimates of σb2 and σe2 . Unbiased estimators for the two variance components using the D-optimal design over the α2 -family can be found in Supplementary material online.
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Acknowledgements
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The author thanks the reviewer for her/his constructive comments. This research was supported by the Ministry of Science and Technology Taiwan (grant number: MOST 106-2118-M-002-002-MY2).
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References
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Cheng, C. S. (2014). Theory of factorial design: single- and multi-stratum experiments. CRC Press, Taylor & Francis Group. Boca Raton, FL. Li, J. (1998). Orthogonal factorial design of parallel flats type and its application in biomedical research. Journal of Statistical Planning and Inference, 73, 61-75. Li, J. and Srivastava, J. N. (1997). Optimal 2n factorial designs of parallel flats type. Communications in Statistics- Theory and Methods, 26, 24732488. Liao, C. T. and Chai, F. S. (2009). Design and analysis of two-level factorial experiments with partial replication. Technometrics, 51, 66-74. Liao, C. T., Iyer, H. K. and Vecchia, D. F. (1996). Construction of orthogonal two-level designs of user-specified resolution where N 6= 2k . Technometrics, 38, 342-353. Rao, C. R. (1947). Factorial experiments derivable from combinatorial arrangements of arrays. Supplement to the Journal of the Royal Statistical Society, 9, 128-139. Srivastava, J. N. and Li, J. (1996). Orthogonal designs of parallel flats type. Journal of Statistical Planning and Inference, 53, 261-283. Tsai, S. F. and Liao, C. T. (2019). Detection of location and dispersion effects from partially replicated two-level factorial designs. Journal of Quality Technology, https://doi.org/10.1080/00224065.2019.1571349. Tsai, S. F., Liao, C. T. and Chai, F. S. (2015). Identification of dispersion effects from partially replicated two-level factorial designs. Journal of Quality Technology, 47, 43-53.
7
PII: DOI: Reference:
S0167-7152(19)30299-8 https://doi.org/10.1016/j.spl.2019.108653 STAPRO 108653
To appear in:
Statistics and Probability Letters
Received date : 8 August 2018 Revised date : 13 August 2019 Accepted date : 6 October 2019 Please cite this article as: C.-T. Liao, Partially replicated block designs for two-level factorial experiments. Statistics and Probability Letters (2019), doi: https://doi.org/10.1016/j.spl.2019.108653. 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.
© 2019 Published by Elsevier B.V.
Journal Pre-proof
Chen-Tuo Liao
of
Partially Replicated Block Designs for Two-Level Factorial Experiments
p ro
Division of Biometry, Institute of Agronomy, National Taiwan University, Taipei, Taiwan
Abstract
Pr e-
In a two-level factorial experiment, we consider blocking in the parallel-flats design with identical flats within or between blocks. We propose three classes of optimal designs with equal or unequal block sizes. Keywords: Optimum design; Orthogonal blocking; Random block effects. 1. Introduction
Jo
urn
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The theory and application of partially replicated parallel-flats designs have drawn more attention, due to their flexible run sizes, well-structured information matrices, and pure replicates. They have been used to identify the location and dispersion effects in two-level factorial experiments. Some real-world examples can be found in Tsai et al. (2015) and Tsai and Liao (2019). Some application of parallel flats designs in biomedical research was presented in Li (1998). For a given requirement set of specified effects, Li and Srivastava (1997) showed that the designs obtained by adding any single flat to a two-level orthogonal parallel-flats design are D- and E-optimal designs. Based on this result, Liao and Chai (2009) proposed a set of sufficient conditions for a parallel-flats design with two identical flats to be D-optimal for the specified effects. In this study, we focus on the extension to the blocking in the twolevel parallel-flats designs with partial replication.
Preprint submitted to Statistics and Probability Letters
August 13, 2019
Journal Pre-proof
2. Notation and Definitions
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2.1. Parallel-flats block design Let di be a two-level single-flat design consisting of all the solutions t satisfying At = ci over GF(2), where A denotes a p × n alias matrix of rank p; ci represents a p × 1 coset indicator vector and GF(2) stands for the Galois field of order two. The juxtaposition of f single-flat designs d1 , d2 , . . . , df is defined as an f parallel-flats design (PFD). Suppose that the treatment combinations determined by At = cjh , where h = 1, 2, . . . , gj , are assigned to the experimental units in the jth block, j = 1, 2, . . . , b. Namely, the PFD determined by (A, Cj ) is allocated in the jth block, where Cj = [cj1 cj2 · · · cjgj ] and the block size kj = gj ×2n−p . By combining such b blocks, a parallel-flats block design, abbreviated as f -PFBD, be determined by (A, C), where Pcan b C = [C1 C2 · · · Cb ]. Note that f = j=1 gj and the run size N = f × 2n−p . 2.2. Orthogonal blocking Suppose that an N -run block design is used to estimate a requirement set β with v potentially active effects. The N runs are allocated to b blocks of sizes k1 , k2 , . . . , kb . Let Y be the responses vector, then Y can be described as Y = 1N µ + Xβ + W γ + ,
(1)
urn
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where 1N denotes the vector of length N with all elements equal to 1; µ the constant term; X the model matrix whose entries are either equal to 1 or −1; W the block indicator matrix; γ the block effects vector and the vector of random errors assumed to be uncorrelated variables with common mean 0 and variance σe2 . The information matrix for β is given by h −1 T i M = X T IN − W W T W W X,
Jo
where IN denotes the identity matrix of order N . A factorial design is said to be orthogonally blocked with respect to β, if W T X = 0, where 0 denotes a zero matrix. This condition requires that both levels of each element in β occur equally often within every block.
2
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3. Partially Replicated f -PFBD
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Let β (e) represent the factorial effect with defining vector e, where e denotes an 1 × n vector whose qth element equals 1 if factor q appears in this effect; otherwise equals 0. Two factorial effects β(e1 ) and β(e2 ) are said to be in the same alias set determined by the alias matrix A, if the row vector e1 + e2 appears in the row space of A, i.e., e1 + e2 = wT A for some vector w over GF(2). In particular, factorial effect β(e) is said to be aliased with µ, if e = wT A. So the effects confounded with block effects must be aliased with µ. Suppose that an f -PFBD is orthogonally blocked with respect to a requirement set β. The best linear unbiased estimator (BLUE) for β is given by βˆ = (X T X)−1 X T Y . The BLUE is not affected by the block effects. The corresponding information matrix M is thus reduced to be X T X of the unblocked design. From Liao et al. (1996), M can be expressed as a block diagonal matrix. Let Gi be the alias set with vi elements of β corresponding to the P ith diagonal matrix of M , for i = 0, 1, . . . , m, where m = 2n−p − 1. Then, m i=0 vi = v and G0 consists of the effects aliased with µ.
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3.1. Equal block size design We first consider a specific class of f -PFBDs with equal number of flats within each block. That is, g1 = · · · = gb = g, hence k1 = · · · = kb = k = g × 2n−p , f = b × g and N = f × 2n−p . We impose the constraint that there are at most two identical flats among the g flats within each block on this class of f -PFBDs, and call it as the α0 -family of f -PFBDs. For given β, we seek orthogonal designs for estimating β over the α0 -family. A vector over GF(2) is said to be an OA(1), orthogonal array of strength one, if it consists of equal occurrences of 0 and 1. The orthogonal array was introduced by Rao (1947) and its definition in the context of general GF(s) can be found in Srivastava and Li (1996). Theorem 1. An f -PFBD d∗ determined by the paired matrices (A∗ , C ∗ ), where C ∗ = [C1∗ C2∗ · · · Cb∗ ], is orthogonal for β over the α0 -family, if it satisfies the following conditions.
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(i) For β(e) ∈ G0 and e = wT A∗ , then wT Cj∗ is an OA(1) for j = 1, 2, . . . , b. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C ∗ is an OA(1) for all 0 ≤ i ≤ m, where m = 2n−p − 1. 3
Journal Pre-proof
p ro
of
For an f -PFBD, the effects of β confounded with the block effects must belong to alias set G0 . Condition (i) makes those confounded effects clear from the block effects, hence d∗ has orthogonal blocking for β. Condition (ii) makes the information matrix equal to the diagonal matrix N Iv , showing that d∗ is orthogonal (A-optimal, D-optimal and E-optimal) for β. We refer to Liao et al. (1996) for a proof of the conditions. We further provide a lower bound for the number of flats in each block for an orthogonal design over the α0 -family. A proof of Proposition 1 can be found in Supplementary material online.
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Proposition 1. A design d∗ over the α0 -family is orthogonal for β with v possibly active effects, then the number of flats in each block g must satisfy the inequality 2(v + 1)2 ≤ g. 2n−p (bv + 1) Example 1. In a 25 factorial experiment, we consider an orthogonal design for the requirement set β = {1, 2, 3, 4, 5, 13, 14, 23, 24}. The design d1 determined by (A, C), where 1 1 1 1 1 A= 1 1 1 0 0 0 0 1 1 0 and
with
C1 C2 C3 C4
0 0 0 0 0 0 0 0 C1 = 0 0 , C 2 = 0 0 , C 3 = 1 1 , C 4 = 1 1 0 0 1 1 0 0 1 1
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C=
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is orthogonal for the β over the α0 -family of 8-PFBDs. The details of checking the example that satisfies Theorem 1 are presented in Supplementary material. By Proposition 1, the number of flats in each block g for d1 must satisfy the inequality 2(9 + 1)2 ≤ g. 25−3 (4 × 9 + 1) Equivalently, 50/37 ≤ g. So g = 2 is the minimal number satisfying the inequality, meaning that d1 with N = 32 is a minimal run-size orthogonal design for the β over the α0 -family. 4
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3.2. Unequal block size design By Proposition 1, an orthogonal design over the α0 -family can turn out to be a large run size design. We thus consider another class of f -PFBDs with g1 = g 0 + 1 and g2 = · · · = gb = g 0 , hence k1 = (g 0 + 1) × 2n−p and k2 = · · · = kb = k = g 0 × 2n−p , f = b × g 0 + 1 and N = f × 2n−p . The constraint that there are two identical flats in block 1 only is imposed on this class of designs. We call it as the α1 -family of f -PFBDs. A design in the α1 -family can be expressed by (A, C), where C = [C1 C2 · · · Cb ]. Here C1 = [c10 c11 · · · c1g0 ] and Cj = [cj1 cj2 · · · cjg0 ] for j = 2, · · · , b. WLOG, let c10 = c11 = 0, forcing the 2n−p treatment combinations satisfying At = 0 to appear twice in block 1. For given β, we seek D-optimal designs for estimating β over the α1 -family.
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Theorem 2. An f -PFBD d∗ is determined by the paired matrices (A∗ , C ∗ ). Let vi∗ denote the number of effects in Gi determined by the alias matrix A∗ for i = 0, 1, . . . , m, where m = 2n−p − 1. Then d∗ is D-optimal for β over the α1 -family, if it satisfies the following conditions.
al
(i) v0∗ = 0. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C0∗ is an OA(1) for all 1 ≤ i ≤ m, where C0∗ is obtained by removing c∗10 from C ∗. Pm Q ∗ (iii) The v1∗ , . . . , vm maximize m i=1 vi = i=1 vi (= v1 ×v2 · · ·×vm ) subject to v.
urn
Condition (i) ensures that d∗ has orthogonal blocking for β. From Theorem 1, if β(e) ∈ G0 with e = wT A, then it is not possible to have that wT Cj is an OA(1) for j = 1, . . . , b, since either g 0 + 1 or g 0 is an odd number. Thus, G0 must be empty, so v0∗ = 0. The information for β now is reduced to be X T X of the unblocked design, a proof for Conditions (ii) and (iii) can be found in Liao and Chai (2009).
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Example 2. Let d2 be the design obtained from d1 in Example 1 by removing one coset indicator vector from each of C2 , C3 and C4 . Then d2 is D-optimal for the same β over the α1 -family of 5-PFBDs. We modify the α1 -family by adding a zero vector as c20 to C2 , and call this as α2 -family of f -PFBDs. Note that c10 = c11 = c20 = 0, this forces the 2n−p treatment combinations satisfying At = 0 to appear twice in block 1 and once in block 2. Theorem 3 is straightforward from Theorem 2. 5
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Theorem 3. An f -PFBD d∗ is determined by the paired matrices (A∗ , C ∗ ). Let vi∗ denote the number of effects in Gi determined by the alias matrix A∗ for i = 0, 1, . . . , m, where m = 2n−p − 1. Then d∗ is D-optimal for β over the α2 -family, if it satisfies the following conditions.
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(i) v0∗ = 0. (ii) For β(e1 ) and β(e2 ) ∈ Gi and e1 + e2 = wT A∗ , then wT C0∗ is an OA(1) for all 1 ≤ i ≤ m, where C0∗ is obtained by removing c∗10 and c∗20 from C ∗ . Q Pm ∗ (iii) The v1∗ , . . . , vm maximize m i=1 vi subject to i=1 vi = v.
4. Discussion
Pr e-
Example 3. Let d3 be the design by adding 0 to C2 of d2 in Example 2. So d3 is D-optimal for the same β over the α2 -family of 6-PFDBs.
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The significance test for an effect in β is typically performed by Student’s t-test, and the estimate of error variance σe2 serves as the basis of the test. To obtain a replication-based estimate for the error variance, we estimate σe2 using real replications within blocks. All the optimal designs over α0 -, α1 and α2 -family can provide sufficient degrees of freedom for the t-test. The effects originally confounded with the block effects under the fixed block effects situation become estimable if the block effects are random (Cheng, 2014. p.98). Under the random blocks situation, it is assumed that the effects in γ and those in of Model (1) are mutually independent random variables distributed as normal distributions with common mean 0, and variances σb2 and σe2 , respectively. For example, suppose that effect β is confounded with block effects of γ1 and γ2 . Also, the estimate of β is given by βˆ = y 1 − y 2 , where y 1 and y 2 separately denote the averages of n1 and n2 responses in blocks 1 and 2. From Model (1), we have that βˆ = β + (γ1 − γ2 ) + (1 − 2 ). Clearly, β is confounded with γ1 − γ2 under the fixed block effects situation. However, under the random block effects situation, the expectation and variance of βˆ are equal to β and 2σb2 +(1/n1 +1/n2 )σe2 , respectively. Therefore, β is free from the block effects and the inference about β involves both the estimates of σb2 and σe2 . Unbiased estimators for the two variance components using the D-optimal design over the α2 -family can be found in Supplementary material online.
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Acknowledgements
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The author thanks the reviewer for her/his constructive comments. This research was supported by the Ministry of Science and Technology Taiwan (grant number: MOST 106-2118-M-002-002-MY2).
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