Universal optimal block designs under hub correlation structure

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Model 3G

pp. 1–6 (col. fig: nil)

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Universal optimal block designs under hub correlation structure R. Khodsiani *, S. Pooladsaz Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

article

info

Article history: Received 25 February 2017 Received in revised form 23 June 2017 Accepted 26 June 2017 Available online xxxx Keywords: Optimal design Binary block design Generalized binary block design Completely symmetric matrix Semibalanced array

a b s t r a c t Universal optimal block designs under general correlation structures are usually difficult to specify theoretically or algorithmically. However, they can sometimes be found for a specific correlation and a particular parameter value. In this paper, a wide class of block designs, binary and non-binary with v treatments and b blocks each of size k is considered and we present a method to construct the universal optimal block designs under the hub correlation when, in this method, b is less than (or equal to) the number of blocks in semibalanced array methods. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The block experiments have been widely used in sciences and engineering, such as agriculture, electric, textile and etc. There is much interest in the optimality of designs of block experiments for different field trials such as interference effects, carryover effects and correlated observations (see Ai et al. (2009), Filipiak and Markiewicz (2014), Zheng (2013)). There are some criteria for assessment of optimality of designs which are used in experimental designs. Kiefer (1975) defined the universal optimality of designs such that the design is optimal under all optimality criteria. Constructing of the universal optimal designs, if exist, is difficult specially when the observations are correlated. The class of block designs is considered as Ω (v, b, k) where v , b and k are the numbers of treatments, blocks and plots per blocks, respectively. Let k = hv + s where h is a nonnegative integer and 0 ≤ s < v . There are different correlation structures for the observations which are considered by many researchers. The first-order autoregressive correlation (AR(1)) is the most usual correlation structure that was considered in many researches. Gill and Shukla (1985) showed that nearest neighbour balanced block designs (NNBDs) are universal optimal in the class of binary and equireplicate block designs under the AR(1) with positive correlations (ρ > 0). Kunert (1987) proved that the designs which are determined by Gill and Shukla (1985), are optimal over all possible block designs when k < v . Das and Dey (1989) introduced the generalized binary block designs (GBDs) for k > v and Pooladsaz and Martin (2005) showed that these designs are universally optimal under the AR(1) with ρ > 0. The circulant correlation is another correlation structure that was introduced by Zhu et al. (2003). They also obtained the D-optimal design for simple linear regression under the circulant correlation with 0 < ρ < 0.5. Under the nearest neighbour correlation structure, the optimal block designs for k ≤ v and v < k ≤ 2v were given by Martin et al. (1993) and Martin (1998), respectively. Rao (1961) introduced the orthogonal array of type II and semibalanced array (SBA). For any k and v , Chai and Majumdar (2000) showed that the universally optimal block designs can be constructed

*

Corresponding author. E-mail address: [email protected] (R. Khodsiani).

http://dx.doi.org/10.1016/j.spl.2017.06.024 0167-7152/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.

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2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

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Table 1 Different values of ρ1 for some k.

1 2 3 4 5 6 7 8 9 10 11 12 13

k

4

5

6

7

8

9

10

...

100

ρ1

0.5774

0.5

0.4472

0.4082

0.378

0.3536

0.3333

...

0.1005

from a SBA under the nearest neighbour correlation structure. Cheng (1988) used SBAs to obtain universally optimal binary block designs for the stationary correlation structures. In some network and other similar experiments, a special correlation structure namely hub correlation has widely application (see Zhu and Coster (2004)). Chang(2013) developed an efficient modified simulated annealing algorithm to solve the D-optimal design problem for 2-way polynomial regression under hub correlated observation. Chang and Coster (2015), based on balanced incomplete block designs (BIBDs), constructed weakly universal optimal block designs under the hub correlation structure. According to their results, under the hub correlation when v > k and a BIBD(v, b, k) exists, a weakly universal optimal block design can be constructed with kb blocks which is usually very large. We assume that all treatment contrasts are equally of interest. In this case, Kiefer (1975) showed that a design is universally optimal over all competing designs if its information matrix (C-matrix) is completely symmetric with maximal trace. Martin and Eccleston (1991) showed that any design constructed by a SBA have completely symmetric C-matrix when the within-block correlation matrix is symmetric. However, there are some limitations for constructing a SBA. Using a SBA with v levels, k constraints and index λ, the design has, b=λ

14

v (v − 1) 2

.

(1)

18

Note that λ is even when v is even. In this paper, we present a method to construct the universally optimal block designs λv (v−1) under the hub correlation without using SBA with b ≤ . Also, we will specify the value of b when v and k are known. 2 In Section 2, some preliminaries about the hub correlation is presented. In Section 3 the universally optimal block designs in Ω (v, b, k) are found under the hub correlation for any v and k.

19

2. Preliminaries

15 16 17

20 21

Let y′ = (y11 , y12 , . . . , y1k , y21 , . . . , ybk ) be the n-vector of observations where n = bk and yji is the observation on plot i of block j. Consider the usual block-treatment additive model in matrix form as below, y = Xd τ + Bβ + ε

22

23 24 25 26

where Xd is the n × v treatment design matrix; B = Ib 1k is the n × b block design matrix; Ib is the b × b identity matrix; 1k ⨂ is the k-vector of ones; is the Kronecker product; τ is the v -vector of treatment effects; β is the b-vector of block effects, and ε is the n-vector of errors such that E(ε ) = 0 and v ar(ε ) = V σ 2 where V is a positive-definite n × n matrix and σ 2 is the variance of errors. Let, ′ Xd′ = Xd,1

29 30 31

Xd′ ,2

(

27 28

1 Λ= ρ 1k−1

32

33

35 36 37

...

Xd′ ,b

)

where Xd,j is the treatment design matrix for block j of design d. We assume that observations in different blocks are uncorrelated but that observations within blocks are correlated with ⨂ the same correlation structure in each block i.e. V = Ib Λ, where Λ is the k × k within-block correlation matrix. For the hub correlation structure, Λ is as below (see Zhu and Coster (2004)),

[

34

(2)

⨂

ρ 1′k−1 Ik−1

]

.

In fact, under hub correlation, in each block only plot 1 has correlation with the remaining plots and the plots 2, 3, . . . k are uncorrelated. It is easy to show that det(Λ) = 1 − (k − 1)ρ 2 . Since Λ must be positive-definite, it can be shown that |ρ| < ρ1 where ρ1 is shown in Table 1. For special cases k = 4 and 0.5 < |ρ| < 0.5774, the optimality of designs is not considered in this paper. By generalized least-squares equations, the C-matrix for treatment effects of design d in the model (2) is, Cd =

38

b ∑

Xd′ ,j Λ∗ Xd,j

(3)

j=1 39 40

41

where Λ∗ = Λ−1 − 1′k Λ−1 1k

(

)−1

Λ−1 1[k 1′k Λ−1 . 0 1k−1

For simplicity, we introduce A1 = A∗4 =

[

0

0k−1

0′k−1 Jk−1

]

, w1 =

−1 1+(k−1)(1−2ρ )

,

1′k−1 Ok−1

]

, A2 =

w2 = (1 − k)w1 ,

[

1 0k−1

0′k−1 Ok−1

w3 = 1 +

]

, A3 =

[

ρ 2 +w1 (1−ρ )2 1−(k−1)ρ 2

0 0k−1

0′k−1 Ik−1

]

, A4 =

[

0 0k−1

0′k−1 Jk−1 − Ik−1

]

,

and w4 = (1 − 2ρ )w1 where Jk = 1k 1′k ,

Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.

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0k is the k-vector of zeros and Ok = 0k 0′k . It can be shown that, 1

Λ−1 =

1 − (k − 1) ρ 2

1

[ ] −ρ A1 + A2 + (1 − (k − 2) ρ 2 )A3 + ρ 2 A4

2

and therefore,

3

Λ∗ = w1 A1 + w2 A2 + w3 A3 + w4 A4 .

(4)

We consider the end-design of d as the design for plots 2, 3, . . . , k of each block of d. In fact, the end-design of d consists of v treatments and b blocks each of size k − 1. Let rd,j,t denote the number of times treatment t occurs in block j of d, ∑b j = 1, 2, . . . b, and rd,t = j=1 rd,j,t is the replication of treatment t. Also, Rd = diag {rd,1 , . . . , rd,v }. The corresponding values for the end-design of d are considered as rde,j,t , rde,t and Red . Let rd1,j,t be the number of times treatment t occurs in the first plot

∑b

e e 1 1 1 of block j and R1d = diag {rd1,1 , . . . , rd1,v } where rd1,t = j=1 rd,j,t . Then we have rd,t = rd,t + rd,t and Rd = Rd + Rd . 1 The self-concurrence of treatment t is denoted by λd,t , in which,

λ1d,t =

b k ∑ ∑ ∑

(Xd,j )u,t (Xd,j )l,t

4

5 6 7 8 9 10

(5)

11

j=1 u=1 l̸ =u

where (Xd,j )u,t is the element (u, t)-th of Xd,j .

12

Definition 1. For hub correlation, the concurrence-correlation for treatments t1 and t2 is the number of times one is in plot 1 and the another one is in plots 2, 3, . . . , k of the same block which is defined by, md,t1 ,t2 =

∑

rde,j,t1

+

j∈j∗ t

where jt = { j ;

rde,j,t2

15

1

(Xd,j )1,t = 1}.

16

By Definition 1, the concurrence-correlation matrix for design d is obtained by Md =

∑

′

j Xd,j A1 Xd,j .

17

3. The universal optimal designs

18

By Kiefer (1975), for constructing a universal optimal design in Ω (v, b, k), first we should determine the subclass which consists of designs in Ω (v, b, k) with completely symmetric C-matrix (denote this subclass by Ω ∗ (v, b, k)). Then, we need to find d∗ in Ω ∗ (v, b, k) such that tr(Cd∗ ) = maxd∈Ω ∗ (v,b,k) tr(Cd ). The following two lemmas are useful for determining of Ω ∗ (v, b, k) and d∗ . Lemma 2. Under hub correlation, the C-matrix of design d in Ω (v, b, k) is completely symmetric if both following conditions hold, (i) The end-design of d is equiconcurrence and, (ii) all off-diagonal elements of Md are equal.

′

(6)

where = j Xd,j A4 Xd,j . Under generalized least squares estimation, the C-matrix is completely symmetric if it has equal off-diagonal elements. From (6) and by definition of Md and S4∗,d , the off-diagonal elements of Cd are, b ∑

rde,j,t1 rde,j,t2

21 22

23

26

∗

(Cd )t1 ,t2 = w1 md,t1 ,t2 + w4

20

25

Cd = w1 Md + w2 R1d + Red + w4 S4∗,d

∑

19

24

Proof. Since w3 − w4 = 1, by (3) and (4), it can be shown that,

S4∗,d

14

j∈j∗ t

2

∗

∑

13

t1 ̸ = t2 .

(7)

27

28 29

30

j=1

It is easy to show that (7) is the first order polynomial of ρ and it does not depend on the values of ρ if both the following conditions hold for all t1 and t2 , t1 ̸ = t2 , b ∑

rde,j,t1 rde,j,t2 = λed

31 32

(8)

33

(9)

34

j=1

md,t1 ,t2 = md where λ and md are integer values. The conditions (8) and (9) are equivalent to (i) and (ii), respectively. □ e d

Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.

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2

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Let w (ρ ) = max (Λ∗ )ij

{

w (ρ ) =

{ w1 w4

i ̸ = j . Under hub correlation with |ρ| ≤ 0.5,

}

;

−0.5 ≤ ρ ≤ 0 . 0 < ρ ≤ 0.5

if if

(10)

Lemma 3. Under hub correlation with |ρ| ≤ 0.5, for any design d in Ω ∗ (v, b, k),

(

tr(Cd ) ≤ v w2 rd1,t + w3 rde,t + w (ρ )bh s +

)

(

v (h − 1) 2

)

.

5 6 7

8

9

• For h = 0, the equality holds if d is binary design, • For h > 0, the tr(Cd ) is the nearest value to this upper bound if d is GBD. Proof. For d in Ω ∗ (v, b, k), tr(Cd ) = v (Cd )t ,t . From (3), for t = 1, 2, . . . , v it is clear that, (Cd )t ,t =

b k ∑ ∑

(Xd,j )u,t (Λ∗ )u,u +

11 12

13

(Xd,j )u,t (Λ∗ )u,l (Xd,j )l,t .

j=1 u=1 l̸ =u

j=1 u=1 10

b k ∑ ∑ ∑

By (5) and definition of w (ρ ), it is easy to show that, (Cd )t ,t ≤ w2 rd1,t + w3 rde,t + w (ρ )λ1d,t . Therefore, tr(Cd ) ≤ v w2 rd1,t + w3 rde,t + w (ρ )

(

)

v ∑

λ1d,t .

(11)

t =1 14 15

16

17 18 19 20

21

∑v

By (10), w (ρ ) ≤ 0 for any |ρ| ≤ 0.5, then the tr(Cd ) is maximized when t =1 λ1d,t is minimum. If h = 0 then any binary block design db in Ω ∗ (v, b, k) has λ1d ,t = 0 for t = 1, 2, . . . , v . Therefore, b

tr(Cdb ) = v w

(

1 2 rdb ,t

+w

e 3 rdb ,t

)

.

If h > 0 then for any d in Ω (v, b, k) there exists at least one treatment t such that λ1d,t ̸ = 0. In this case, let dg be an arbitrary GBD in Ω ∗ (v, b, k). So, each block j consists of s treatments with rdg ,j,t = h + 1, and rdg ,j,t = h for other v − s treatments. h(h+1) h(h−1) Also, in any block of dg , the self-concurrence of s treatments is 2 , and it is equal to 2 for other v − s treatments. Therefore, for any j = 1, 2, . . . , b, ∗

v ∑ k ∑ ∑

(Xdg ,j )u,t (Xdg ,j )l,t = sh +

t =1 u=1 l̸ =u 22

23

25 26

27

v ∑

( ) v (h − 1) λ1dg ,t = bh s + .

29

Now, let d1 in Ω ∗ (v, b, k) is not a GBD. Also, d1 consists of blocks similar to the blocks of a GBD except block 1. Without loss of generality, suppose that treatment 1 in block 1 is replicated h + 2 and the replication of other treatments in block 1 is h or h + 1. Then, for block 1 of design d1 , we have, v ∑ k ∑ ∑

(Xd1 ,1 )u,t (Xd1 ,1 )l,t = sh +

31

v ∑

33

2

+ (h + 1).

( ) v (h − 1) λ1d1 ,t = bh s + + (h + 1). 2

Thus, by (13), v ∑ t =1

32

v h(h − 1)

By (12), it can be shown that,

t =1 30

(13)

2

t =1 u=1 l̸ =u 28

(12)

2

and then,

t =1 24

v h(h − 1)

λ1d1 ,t >

v ∑

λ1dg ,t .

(14)

t =1

Since dg is an arbitrary GBD, therefore by (11), any non-GBD in Ω ∗ (v, b, k) has,

( ) ( ) v (h − 1) 1 e tr(Cd ) < v w2 rd,t + w3 rd,t + w (ρ )bh s + . □ 2

Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.

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The main result is presented in the next theorem.

1

Theorem 4. Under hub correlation with |ρ| ≤ 0.5, the design d∗ in Ω (v, b, k) is universally optimal if,

2

(I) d is binary when h = 0, or d is GBD when h > 0, (II) the end-design of d∗ is equiconcurrence, and (III) Md∗ = md∗ (Jv − Iv ) + m(h)Iv where m(h) = 2rd1∗ (h − 1) if h > 0 or m(h) = 0 if h = 0. ∗

∗

3 4 5

Proof. The conditions (I) and (II) are satisfied by Lemma 3 and part (i) of Lemma 2, respectively. From (6) and the definition of Md and S4∗,d , each diagonal element of Cd is obtained as below, Cd,t ,t = w1 md,t ,t + w2 rd1,t + rde,t + w4

b ∑

(rde,j,t )2 .

6 7

(15)

8

j=1

Since w1 ≤ 0 for |ρ| ≤ 0.5, then the tr(Cd ) is maximized when md,t ,t has possible minimum value for t = 1, 2, . . . , v . If h = 0 then mdb ,t ,t = 0 for any binary block design db in Ω ∗ (v, b, k) and t = 1, 2, . . . , v . So, by (ii) in Lemma 2, for the universal optimal design d∗ in Ω (v, b, k), Md∗ = md∗ (Jv − Iv ). If h > 1 then for any GBD dg in Ω ∗ (v, b, k) there exist at least one treatment t such that mdg ,t ,t ̸ = 0. Without loss of generality, for h > 0, suppose that in each block of a GBD dg , there are h complete blocks in plots 1, 2, . . . , hv and an incomplete block in remaining plots. Consider dg ∗ is a GBD in Ω ∗ (v, b, k) such that (Xdg ∗ ,j )1,t = 1 if (Xdg ∗ ,j )u,t = 0 for u = hv + 1, . . . , hv + s and j = 1, 2, . . . , b. It is clear that mdg ∗ ,t ,t = 2rd1 ∗ ,t (h − 1) and mdg ∗ ,t ,t ≤ mdg ,t ,t for any dg in g Ω ∗ (v, b, k). According to (15), any design d in Ω ∗ (v, b, k) has rd1,t = rd1 for t = 1, 2, . . . , v where rd1 is an integer, and therefore the proof is completed by (ii) in Lemma 2. □ According to Theorem 4, when h > 0 and s = 0 the d is a block design with b = v where there are h complete blocks in each block and rd1∗ ,t = 1 for t = 1, 2, . . . , v . For example, the design with the blocks [1 2 3 4 5], [2 1 3 4 5], [3 1 2 4 5], [4 1 2 3 5] and [5 1 2 3 4] is universally optimal in Ω (5, 5, 5) under hub correlation with |ρ| < 0.5, Although, by (1), an optimal design with v = 5 constructed by a SBA, needs 10 blocks. For s ̸ = 0, we can use the following algorithm for constructing a universal optimal design in Ω (v, b, k) under hub correlation. ∗

Algorithm: Step 1.

Step 2.

Step 3.

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

i. When s = 1 and h > 0, construct an equiconcurrence binary block design d1 with v treatments and b blocks each of size k1 = 2. ii. When s > 1, construct an equiconcurrence binary block design d2 with v treatments and b blocks each of size k2 = k − 1 if h = 0 or k2 = s if h > 0. i. When s = 1 and h > 0, construct an array k × b such that the treatments on the first and second plots of block j (j = 1, 2, . . . , b) of d1 are considered as the first and kth elements of column j, respectively. ii. When s > 1, consider an array k × b and put each block of d2 at the end of each column (the elements 2, 3, . . . , k if h = 0 or the elements hv + 1, hv + 2, . . . , hv + s if h > 0). i. When s = 1 and h > 0, complete each column such that there are h complete blocks in the elements 1, 2, . . . , hv . Thus, the array is completed. ii. When s > 1, consider the same replications of each treatment at the first element of each column such that, (1) in filled array, the concurrence-correlation for any pair of treatments are the same and, (2) the treatment t (t = 1, 2, . . . , v ) can be at the first element of column j (j = 1, 2, . . . , b) if rd2 ,j,t = 0.

If h = 0, the array is completed. Step 4. When s > 1 and h > 0, do the step 3-i and then the array is completed.

It is clear that b ≤

λv (v−1) 2

h=0

if

h > 0.

29 30 31 32 33 34 35 36 37 38

41 42 43

where r2 and λ are positive integers. Then, for constructing d2 , it is necessary that, if

28

40

λ(v − 1) = r2 (k2 − 1)

⎧ λv (v − 1) ⎪ ⎨ (k − 1)(k − 2) b= λv (v − 1) ⎪ ⎩ s(s − 1)

27

39

According to (15), any design d in Ω ∗ (v, b, k) has rde,t = rde for t = 1, 2, . . . , v , where rde is positive integer. So, for constructing d2 in step 1-ii, the following equations should be satisfied, r2 v = bk2 ,

26

44

(16)

for any h ≥ 0.

Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.

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R. Khodsiani, S. Pooladsaz / Statistics and Probability Letters xx (xxxx) xxx–xxx λv (v−1)

For constructing d1 in step 1-i we need b = . Therefore, for any 0 ≤ s < v and h ≥ 0 the universal optimal designs 2 in Theorem 4 have usually less number of blocks than the optimal designs which are constructed by SBAs. In the following examples, the algorithm is used to construct the universal optimal designs. Example 5. Let a binary design d∗1 in Ω (7, 7, 4) with the blocks [1 2 4 6] , [2 3 5 6] , [3 1 4 5] , [4 2 5 7] , [5 1 6 7] , [6 3 4 7] and [7 1 2 3]. The end-design of d∗1 i is equiconcurrence with λed∗ = 1 and also its concurrence-correlation matrix is 1

6 7

8 9 10

Md∗ = J7 − I7 . So, according to Theorem 4, d∗1 is universally optimal in Ω (7, 7, 4) under hub correlation with |ρ| ≤ 0.5. 1 Whereas, by (1), an optimal design with v = 7 which is constructed by a SBA has 21 blocks.

Example 6. Consider the design d∗2 with the blocks [1 2 3 4 5 6 7 3 5 6 ], [2 1 3 4 5 6 7 1 6 7 ], [3 1 2 4 5 6 7 2 5 7 ], [4 1 2 3 5 6 7 1 2 3 ], [5 1 2 3 4 6 7 2 4 6 ], [6 1 2 3 4 5 7 3 4 7 ] and [7 1 2 3 4 5 6 1 4 5 ] . The conditions of Theorem 4 are satisfied with λed∗ = 11 and Md∗ = 3 (J7 − I7 ). So, it is a universally optimal design in Ω (7, 7, 10) under hub correlation with |ρ| < 0.3333. 2

11 12 13

14

2

Example 7. The design d∗3 with the blocks [1 2 3 4 4], [2 1 3 4 1], [3 1 2 4 2], [4 1 2 3 3], [3 1 2 4 1], [4 1 2 3 2], [1 2 3 4 3], [2 1 3 4 4], [4 1 2 3 1], [1 2 3 4 2], [2 1 3 4 3] and [3 1 2 4 4] is universally optimal in Ω (4, 12, 5) under the hub correlation with |ρ| < 0.5. For this design, Md∗ = 8(J4 − I4 ) and λed∗ = 10. 3

3

Acknowledgements

16

This paper is a part of the first author’s Ph.D. thesis. The authors are grateful to the referees for valuable and helpful comments.

17

References

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Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024.