Feasible blocked multi-factor designs of unequal block sizes

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Model 3G

pp. 1–8 (col. fig: nil)

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Feasible blocked multi-factor designs of unequal block sizes Xiaodi Wang a, *, Xueping Chen b , Yingshan Zhang c a b c

School of Statistics and mathematics, Central University of Finance and Economics, Beijing, China Department of Mathematics, Jiangsu University of Technology, Changzhou, China School of Statistics, East China Normal University, Shanghai, China

article

info

Article history: Received 5 January 2017 Accepted 3 December 2017 Available online xxxx Keywords: Feasible Unequal block sizes Orthogonal meeting balanced

a b s t r a c t A block design is called feasible if all the factor effects in it are estimable. This paper studies the design structures of feasible block designs when the block sizes are unequal. Based on a necessary and sufficient condition, we find two structure features which respectively lead to unfeasible and feasible designs. According to these results, an effective way to find feasible designs is provided. © 2017 Elsevier B.V. All rights reserved.

1. Introduction In design of experiments, to reduce the variability of a noise source that is not of primary interest to the experimenter, blocking is to arrange the experimental units in groups (blocks) that are similar to one another. How to settle the factor’s levels in different blocks motivates the issue of selecting good block designs. There is a large body of literature on the construction of block designs. Many of the results deal with one factor problem. Related studies include balanced incomplete block design (BIBD), pairwise balanced design (PBD) and partially balanced incomplete block design (PBIBD), see Raghavarao and Padgett (2005) for reference. Recently, construction of block design that arranges multiple factors is studied. Sitter et al. (1997), Chen and Cheng (1999), Cheng and Wu (2002) and Dey (2010) gave minimum-aberration criterions for blocked fractional factorial designs. Then the construction of blocked optimal designs is proposed, including regular orthogonal block designs (Cheng and Tsai, 2009) and non-regular blocked two-level designs (Cheng et al., 2004; Das and Dey, 2004). These results deal primarily with orthogonal designs. Several results are also available for the construction of blocked non-orthogonal designs, such as Bagchi (2010), Jacroux (2013) and Chen et al. (2015). However, all such results given are under the assumption that all blocks have equal sizes, with the constrain of two-level or three-level factors. When designing a blocked multi-factor design, one should consider the situation that the blocks have unequal sizes for practical limitation. This paper considers experimental situations in which m factors are to be studied in n runs which are partitioned into b blocks of unequal sizes and where only the main effects of the factors are to be used. A good introduction to such types of models can be seen in John and Williams (1995). The number of the levels of each factor is set to arbitrary integer larger than 1. This complicates the issue of selecting a proper design. In this study, we aim to exploring the structure of feasible block designs, wherein all the factors’ main effects are estimable. By imposing a projection matrix Bt , we obtain a necessary and sufficient condition for feasibility of a block design. Then both classes of designs which are unfeasible and feasible are found. Based on the results, an effective way for finding feasible designs is provided. In Section 2, we give our basic notations and definitions. In Section 3, we propose our main results concerning feasibility and the algorithms for selecting feasible block designs are given in Section 4. Conclusions are drawn in Section 5.

*

Corresponding author. E-mail address: [email protected] (X. Wang).

https://doi.org/10.1016/j.spl.2017.12.001 0167-7152/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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2. Basic notation and definitions Throughout this paper, we use H to denote a block design having m factors occur in n runs arranged in b blocks of sizes k1 , . . . , kb . We shall represent H by H = (h0 ; h1 , . . . , hm ) = (hpt ), p = 1, . . . , n, t = 0, 1, . . . , m, which is an n × (m + 1) matrix whose pth row corresponds to run p, the first column indicates the blocks equaling (1′k , 21′k , . . . , b1′k )′ , where 1ki b 1 2 is the ki × 1 vector of ones, and (t + 1)th column (t ≥ 1) corresponds to factor t whose entries hpt ∈ {1, . . . , vt }, where vt is the number of levels of factor t . The model to be used for analyzing the data under a given block design H is Y = C0 α + C1 β1 + · · · + Cm βm + ε,

where Y is an n × 1 vector of observations, C0 is the block incidence matrix which is an n × b matrix with (p, q)th element being 1 or 0, p = 1, . . . , n, q = 1, . . . , b, depending on whether the pth run occurs in the qth block, α′ = (α1 , . . . , αb ) is the vector of block parameters, Ct is the incidence matrix (Zhang et al., 1999; Raghavarao and Padgett, 2005; Chen et al., 2015) of factor t , t = 1, . . . , m, which is an n × vt matrix with (p, q)th element being 1 or 0, p = 1, . . . , n, q = 1, . . . , vt , depending on whether hpt = q, β′t = (βt1 , . . . , βt vt ) is the vector of main effects of factor t which has a constraint

βt1 + · · · + βt vt = 0,

22 23 24 25 26

27 28

29

(2)

and ε is a vector of random error terms whose entries are assumed to be uncorrelated with the zero mean and constant variance σ 2 . According to Eq. (2),

βt =

(

Ivt −1

)

−1′vt −1

β∗t

where β∗t = (βt1 , . . . , βt vt −1 )′ . Thus we shall interchangeably represent the model as Y = C0 α + D1 β∗1 + · · · + Dm β∗m + ε,

( 21

(1)

where Dt = Ct

Ivt −1

−1′v −1 t

)

(3)

.

Definition 2.1. A block design H is called feasible if all the factor effects are estimable. That is, for each βt , there is a matrix Ft such that E(Ft Y ) = βt . Let rit (x) denote the frequency that level x of factor t occurs in the ith block. Then the meeting times of arbitrary two different levels x and y in block i should be rit (x)rit (y). Definition 2.2. Denote λt (x, y) = balanced if for t = 1, . . . , m,

∑b

rit (x)rit (y)

i=1

ki

as the meeting degree of levels x and y for factor t . Then H is called meeting

λt (x, y) = λt

30

for arbitrary x and y, where λt is a constant not 0.

31

3. Feasibility results

(4)

34

In this section, we give our feasible results for H under the meeting balanced condition. We first establish a necessary and sufficient condition for feasibility by some projection matrices Bt , and then obtain the feasible design structures from the properties of Bt .

35

3.1. Preliminary

32 33

36

37

38 39

First we give and prove three lemmas. Lemma 3.1. Let PC0 = C0 (C0′ C0 )− C0′ , τ = In − PC0 , Pvt = Ct′ τ Ct = λt vt τvt

1

vt

1vt 1′vt and τvt = Ivt − Pvt . Then Eq. (4) is equivalent to (5)

for t = 1, . . . , m. Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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Proof. Notice that

1

Ct τ Ct = Ct (In − PC0 )Ct = Ct (In − block(Pk1 , . . . , Pkb ))Ct . ′

′

′

2

∑b

t ′ The (x, y)th element of Ct′ Ct is i=1 ri (x) if x = y or 0 if x ̸ = y. The (x, y)th element of Ct block(Pk1 , . . . , Pkb )Ct is ∑b 1 t t ′ i=1 k ri (x)ri (y). Thus, the (x, y)th element of Ct τ Ct is i

3 4

⎧ b ∑ ⎪ 1 ⎪ ⎪ ⎪ (rit (x) − rit (x)rit (x)), if x = y, ⎪ ⎨ ki i=1

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−

5

b ∑ 1 i=1

,

if x ̸ = y.

rit (x)rit (y)

ki

Given λt (x, y) = λt for arbitrary x and y, we have b ∑

(rit (x) −

i=1

1 ki

rit (x)rit (x)) =

b ∑

rit (x)(

b ∑∑ 1 y̸ =x i=1

∑b

1 t t i=1 ki ri (x)ri (y)

and −

ki − rit (x) ki

i=1

=

6

ki

b ∑ ∑1

)=

i=1 y̸ =x

rit (x)rit (y) =

∑

ki

rit (x)rit (y)

λt = λt (vt − 1)

7

8

y ̸ =x

= −λt . Thus,

9

Ct τ Ct = [λt (vt − 1) + λt ]Ivt − λt Jvt vt = λt vt (Ivt − Pvt ) = λt vt τvt . ′

where Jvt vt is the vt × vt matrix of ones. ∑b Conversely, if Ct′ τ Ct = λt vt τvt , then − i=1 k1 rit (x)rit (y) = −λt , for any x ̸ = y, and Eq. (4) holds. □ i

10 11 12

Lemma 3.2. Denote D = (D1 , . . . , Dm ), then H is feasible if and only if τ D has full column rank.

13

Proof. Given C0 and D, Model (1) can be represented as

14

Y = C0 α + Dβ + ε, ∗

∗′

15

∗′

where β∗ = (β1 , . . . , βm )′ . By expressing α in terms of β∗ , the above model reduces to Q = (In − PC0 )Dβ + τ ε = τ Dβ + ε , ∗

∗

∗

17

where Q = (In − PC0 )Y . H is feasible if and only if β is estimable. According to the linear model theory, τ D is full column rank. □ ∗

Lemma 3.3. Consider the designs which have the column structure of the form given in Eq. (4). Denote wt = τ Ct as the generalized incidence matrix of factor t and let At = Ct (wt′ wt )− wt′ . Then for t = 1, . . . , m, we have At Dt = Dt .

At Ct βt = Ct (wt′ wt )− wt′ Ct βt = 1

1

λt vt

Ct τvt (τ Ct )′ Ct βt

where the penultimate equality holds because the sum of all entries of βt is 0, such that Pvt βt = 0. The above equation is equivalent to At Dt βt = Dt βt , ∗

for arbitrary βt . Therefore, ∗

At Dt = Dt .

19

20 21

23

24

1

Ct τvt (Ct′ τ Ct )βt = Ct λt vt τvt τvt βt λt vt λt vt = Ct τvt βt = Ct (Ivt − Pvt )βt = Ct (βt − 0) = Ct β t ,

∗

18

22

Proof. By the result of Lemma 3.1, we obtain

=

16

□

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3.2. Main theorems Now we show the main result. Theorem 3.1. Consider the designs whose columns have the structure of the form given in Eq. (4). Then H is feasible if and only if rk(

4

m ∑

Bt ) =

m ∑

(vt − 1),

t =1

(6)

t =1

5

where Bt = τ At and rk(A) means the rank of matrix A.

6

Proof. From Lemma 3.3, we have R(Dt ) ⊂ R(At ),

7 8 9

and vt − 1 = rk(Dt ) ≤ rk(At ). On the other hand, we have from the definition that At has vt different rows. Furthermore, given

wt′ Jnn = = = =

10 11 12 13 14

21

rk(At ) = vt − 1 = rk(Dt ). Notice that Bt τ Dt = τ At τ Dt = τ Ct (wt′ wt )− (Ct′ τ )τ Dt = τ Ct (wt′ wt )− Ct′ τ Dt = τ At Dt = τ Dt , then R(τ Dt ) ⊂ R(Bt ). On the other hand, from Eq. (5), we have Dt τ Dt = λt vt Ut τvt Ut , where Ut = ′

24

27 28 29 30 31 32

33

(

Ivt −1

−1′v −1 t

) , and

= rk(Ut′ Ut ) = rk(Ut ) = vt − 1 = rk(Dt ).

23

26

′

rk(τ Dt ) = rk((τ Dt )′ (τ Dt )) = rk(D′t τ Dt ) = rk(Ut′ τvt Ut )

22

25

Ct′ (Jnn − Jnn ) = 0,

That is, the linear combination of At ’s rows is 0. Thus we have

19 20

Ct′ (Jnn − diag(Pk1 , . . . , Pkb )Jnn )

Jnn τ At = Jnn τ Ct (wt′ wt )− wt′ = Jnn wt (wt′ wt )− wt′ = 0.

17 18

Ct′ (Jnn − C0 (C0′ C0 )− C0′ )Jnn

we obtain

15 16

(τ Ct )′ Jnn = Ct′ (In − PC0 )Jnn

Thus, rk(Bt ) = rk(τ At ) ≤ rk(At ) = vt − 1 = rk(τ Dt ) and, R(Bt ) = R(τ Dt ). Since Pτ Dt = (τ Dt )((τ Dt )′ (τ Dt ))−1 (τ Dt )′ and Bt are the projection matrices of R(τ Dt ), from the uniqueness of projection matrices, we obtain Pτ Dt = Bt . Now we prove the result that ‘‘if’’: Since H is a feasible design, that is, β∗ are estimable. According to Lemma 3.2, τ D has full column rank, i.e. rk(τ D) =

m ∑

(vt − 1).

t =1 34

35

Furthermore, m ∑ t =1

36 37

Pτ Dt =

m ∑

(τ Dt )((τ Dt )′ (τ Dt ))−1 (τ Dt )′

t =1

= (τ D1 , . . . , τ Dm )diag(((τ D1 )′ (τ D1 ))−1 , . . . , ((τ Dm )′ (τ Dm ))−1 )(τ D1 , . . . , τ Dm )′ = (τ D)K (τ D)′ . Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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Notice that K = diag(((τ D1 )′ (τ D1 ))−1 , . . . , ((τ Dm )′ (τ Dm ))−1 ) is a positive definite matrix and there exists an invertible matrix L such that K = LL′ . Then, we have m ∑

Pτ Dt = (τ D)LL′ (τ D)′ = (τ DL)(τ DL)′ .

1 2

3

t =1

Thus,

4

m

rk(

∑

m

Bt ) = rk(

t =1

∑

Pτ Dt ) = rk((τ DL)(τ DL)′ )

5

t =1

= rk(τ DL) = rk(τ D) m ∑ = (vt − 1).

6

7

t =1

‘‘only if’’: Notice that m

8

m

∑

Bt =

t =1

m

∑

Pτ Dt =

t =1 m

=

∑

(τ Dt )((τ Dt )′ (τ Dt ))−1 (τ Dt )′

9

t =1

∑

′ ′ (τ Dt )Et = (τ D1 , . . . , τ Dm )(E1′ , . . . , Em )

10

t =1

= τ DE ,

11

where E = (E1 , . .∑ . , Em ) . ∑m m Thus, from rk( t =1 Bt ) = t =1 (vt − 1), we have ′

′ ′

m

rk(τ D) ≥ rk(

∑

12 13

m

Bt ) =

t =1

∑

(vt − 1).

14

t =1

On the other hand, the number of the columns of τ D is

∑m

v − 1), and thus rk(τ D) ≤

t =1 ( t

∑m

v − 1). Hence,

t =1 ( t

15

m

rk(τ D) =

∑

(vt − 1),

16

t =1

that is, τ D has full column rank. From Lemma 3.2, H is feasible. □

17

Corollary 3.1. Consider H as the design which has the column structure given in Eq. (4). Then H is unfeasible if there are two columns j and k such that their different level combinations is less than vj + vk − 1. Proof. From the definition of At , we have that there are vt different rows in At , depending on different levels located in ht . Let l be the number of different level combinations of columns j and k. Then according to the property of At above, Aj + Ak has at most l different rows. Furthermore, based on the proof of Theorem 3.1, we have Jnn τ (Aj + Ak ) = Jnn τ Aj + Jnn τ Ak = 0,

18 19

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and

24

rk(Aj + Ak ) ≤ l − 1.

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Furthermore, given l < vj + vk − 1, we obtain

26

m

rk(

∑

Bt ) ≤ rk(Bj + Bk ) +

∑

rk(Bd ) = rk(τ (Aj + Ak )) +

d̸ =j,k

t =1

≤ rk(Aj + Ak ) +

∑

∑

rk(Bd )

27

d̸ =j,k

(vd − 1) ≤ (l − 1) +

d̸ =j,k

< (vj − 1) + (vk − 1) +

∑

(vd − 1)

28

d̸ =j,k

∑

(vd − 1)

29

d̸ =j,k

=

m ∑

(vt − 1).

30

t =1

From Theorem 3.1, H is unfeasible. □ Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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Corollary 3.2. Consider H as the design which has the column structure given in Eq. (4), then H is feasible if Bj Bk = 0

(7)

3

for any j ̸ = k

4

Proof. Given Bj Bk = 0 for j ̸ = k, we have

5

(

m ∑

Bt )′ =

t =1 6

7

That is,

m ∑

B′t =

t =1

∑m

rk(

m ∑

t =1 Bt

m ∑

Bt and (

t =1

m ∑

Bt )2 =

t =1

m ∑

B2t =

t =1

m ∑

Bt .

t =1

is a projection matrix. Therefore,

Bt ) = tr(

t =1

m ∑

Bt ) =

t =1

m ∑

tr(Bt ) =

t =1

m ∑

rk(Bt ) =

m ∑

(vt − 1).

t =1

t =1

8

According to Theorem 3.1, H is feasible. □

9

Corollary 3.3. Consider H as the design which has the column structure given in Eq. (4), then H is feasible if

10

b ∑

jt

ri (x, y) =

i=1

b j ∑ r (x)r t (y) i

i=1

i

(8)

ki jt

12

for any j ̸ = t , x ∈ {1, . . . , vj }, y ∈ {1, . . . , vt }, where ri (x, y) denotes the repetition times that level combination (x, y) of factor j and factor t occurs in the ith block.

13

Proof. Notice that

11

⎛ 14

⎞

Pk1 Pk2

⎜ ⎜ ⎝

Cj′ τ Ct = Cj′ (In − PC0 )Ct = Cj′ (In − block ⎜

..

⎟ ⎟ ⎟)Ct . ⎠

. Pkb

15

16

17

18 19

The (x, y)th element of Cj′ τ Ct is

jt i=1 ri (x

∑b

, y) −

∑b

i=1

j ri (x)rit (y)

ki

. Thus condition (8) is equivalent to

Cj′ τ Ct = 0. Furthermore, Bj Bt = wj (wj′ wj )− wj′ wt (wt′ wt )− wt′ = wj (wj′ wj )− (τ Cj )′ (τ Ct )(wt′ wt )− wt′

= wj (wj′ wj )− (Cj′ τ Ct )(wt′ wt )− wt′ = 0

20

for any j ̸ = t . From Corollary 3.2, H is feasible. □

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4. Algorithms for finding feasible designs

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We shall say that H is orthogonal balanced if Condition (8) holds. For b = 1, such designs reduce to the general orthogonal designs in Wang and Zhang (2016). For b = 1 and r t (1) = · · · = r t (vt ), such designs reduce to orthogonal arrays of strength two. If both Conditions (4) and (8) hold, we shall say that H is orthogonal meeting balanced. According to Corollary 3.3, such designs are feasible block designs. A simple method for finding orthogonal meeting balanced block designs is as follows:

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Algorithm 1.

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(1) Construct b general orthogonal designs H1 , . . . , Hb with run size k1 , . . . , kb through orthogonal arrays. (2) Merge the b GODs and add a column (1′k , . . . , b1′k )′ as the block, denoted as H ∗ . b 1 (3) Verify the meeting balanced condition of H ∗ by Eq. (5). Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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Example 4.1.

1

⎛

1 1 1 1 1

⎜1 ⎜ ⎜2 ⎜ ⎜ ⎜2 H1 = ⎜ ⎜2 ⎜ ⎜2 ⎜ ⎝3

⎞

2 2 2 2⎟ ⎟ ⎟ 1 1 2 2⎟

( ) ⎟ 1 1 1 2 2 ⎟ , H2 = 1 2 1 2⎟ 3 1 1 2 2 ⎟ 2 1 2 1⎟ ⎟ 1 2 2 1⎠ 2 2 1 1⎟

2

3 2 1 1 2 are two general orthogonal designs, then

⎛

3

⎞′

1 1 1 1 1 1 1 1 2 2

⎜1 ⎜ ⎜ ⎜1 ∗ H =⎜ ⎜1 ⎜ ⎝1

1 2 2 2 2 3 3 1 3⎟

⎟ ⎟ 2 1 2 1 2 1 2 1 1⎟ ⎟ 2 1 2 2 1 2 1 1 1⎟ ⎟ 2 2 1 1 2 2 1 2 2⎠

4

1 2 2 1 2 1 1 2 2 2 is an orthogonal meeting balanced block design for three factors in two blocks with k1 = 8 and k2 = 2.

5

Further Discussion on Algorithm 1: Steps 1 and 2 guarantee the orthogonal balanced condition of the result design but not the meeting balanced condition. Therefore, Condition (4) should be verified in Step 3. However, if we add some constraints for the general orthogonal designs in Step 1, then Condition (4) can also be guaranteed. (1) For the factors with vt > 2, let r t (x)r t (y) = c for any 1 ≤ x ̸ = y ≤ vt in one general orthogonal design, where c is a constant larger than 0 (that is, the tth factor of this design is balanced) and r t (x)r t (y) = 0 for 1 ≤ x ̸ = y ≤ vt in other general designs (that is, the tth factors of these designs have one level). For example, if b = 3, m = 2, v1 = 2, v3 = 3 are considered, then we can give

( H1 =

)

1 1 2 1

⎛

6 7 8 9 10 11 12 13

⎞

( ) 2 1 2 2 , , H2 = ⎝2 2⎠ , H3 =

14

1 2

2 3

and

15

⎞′

⎛

1 1 2 2 2 3 3 H ∗ = ⎝1 2 2 2 2 2 1⎠ .

16

1 1 1 2 3 2 2

(v )

(v )

(2) For the factors with vt > 2, let the 2t level combinations (x, y) (1 ≤ x ̸ = y ≤ vt ) respectively appear in 2t general orthogonal designs, with their meeting degrees in each general orthogonal design being the same. As a simple case, the following case can achieve such condition: (i) H1 is an general orthogonal design with the tth column having two levels. (ii) (v H) 2 , . . . , Hb are designs which change the two levels of the tth column of H1 to the other level combinations, where b = 2t . For unequal block sizes, one can repeat any subblock of the above design integer times. It is easy to see that the orthogonality condition for orthogonal meeting balanced block design (Condition (8)) is stronger than Condition (6), thus orthogonal meeting balanced block designs are only part of the feasible designs. As shown in Corollary 3.1, the columns of feasible designs should have weaker orthogonality such that the row structures of any two Aj and Ak are not highly overlapped. Hence, an effective way to find other feasible designs is to remould the orthogonal meeting balanced block designs, such as follows. Algorithm 2.

17 18 19 20 21 22 23 24 25 26 27 28

29

∗

(1) Construct designs H that are orthogonal meeting balanced. (2) Derive designs based on H ∗ . Denote the new design as H ∗∗ . (3) Verify the feasibility of H ∗∗ by Eq. (6). Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.

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Example 4.2. Step 1: Construct an orthogonal meeting balanced block design:

⎞′

⎛

1 1 2 2 2 3 3

2

H ∗ = ⎝1 2 2 2 2 1 2⎠ . 2 2 1 2 3 2 2

3

Step 2: Remould H ∗ by adding a column of two levels, as

⎛ 4

⎞′

1 1 2 2 2 3 3

⎜1 2 2 2 2 1 2⎟ ⎟ H ∗∗ = ⎜ ⎝2 2 1 2 3 2 2⎠ . 1 2 2 1 1 1 1

5

6 7 8 9

∑3

Step 3: Verify that rk(

Further discussion on Algorithm 2: (1) One can let the foundation design H ∗ containing factors with vt > 2 and extend factors with vt = 2, because it is more easy to achieve Condition (4) for factors with vt > 2 by using general orthogonal designs. (2) Steps 1 and 2 cannot guarantee a feasible design. Consider

⎛ 10

= 4, and (v1 − 1) + (v2 − 1) + (v3 − 1) = 1 + 2 + 1 = 4. Then H ∗∗ is feasible.

t =1 Bt )

⎞′

1 1 2 2 2 3 3

⎜1 2 2 2 2 1 2⎟ ⎟ H ∗∗ = ⎜ ⎝2 2 1 2 3 2 2⎠ 2 2 2 1 1 1 1

14

for example, which is also an expanding of the above H ∗ in Example 4.4. The number of different level combinations of the second factor and the third factor is 3, less than v2 + v3 − 1 = 4, and the resulting design is unfeasible according to Corollary 3.1. Thus, Condition (6) should be verified in Step 3. However, we find by simulations that Condition (6) can be met in many cases.

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5. Conclusion

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In this paper, we study the problem of finding feasible designs for unequal block sizes. Theorem 3.1 gives the necessary and sufficient condition for the feasibility of design. Based on this condition, we further obtain two kinds of design structures which are unfeasible and feasible. Finally, two algorithms are given for efficiently finding feasible block designs according to the relation between the obtained feasible structure and general orthogonal designs. For different run sizes and blocks, plenty of feasible designs can be found. Even for a simple case that 5 runs are arranged in 2 blocks with k1 = 2 and k2 = 3, if two factors are considered with v1 = 3 and v2 = 2, then 16 feasible designs are selected. How to select optimal designs among these feasible designs is worth further study.

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Acknowledgments

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The authors are grateful to Professor Y. Xiao and the anonymous referee for their careful reading of this paper and valuable comments. The work was supported by National Natural Science Foundation of China (Nos. 11601538, 11601195), Natural Science Foundation of Jiangsu Province of China (No. BK20160289), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 16KJB110005).

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Please cite this article in press as: Wang X., et al., Feasible blocked multi-factor designs of unequal block sizes. Statistics and Probability Letters (2017), https://doi.org/10.1016/j.spl.2017.12.001.