General Minimum Lower-Order Confounding Designs with Multi-Block Variables

Blocking the inhomogeneous units of experiments into groups is an efficient way to reduce the influence of systematic sources on the estimations of treatment effects. In practice, there are two types of blocking problems. One considers only a single block variable and the other considers multi-block variables..e present paper considers the blocking problem of multi-block variables. .eoretical results and systematical construction methods of optimal blocked 2 m designs with (N/4) + 1≤ n≤ 5N/16 are developed under the prevalent general minimum lower-order confounding (GMC) criterion, where N � 2 .


Introduction
e regular 2 n− m factorial experiment has played an important role in engineering, manufacturing industry, agriculture and medicine, and so on. It allows efficient and economic experimentation to estimate treatment effects. When the size of the experimental units is large, the inhomogeneity will cause unwanted variance to the estimations of treatment effects. To reduce such bad influence, a crucial way is to partition the experimental units into blocks.
ere are two kinds of blocking problems as pointed out in [1]. One is called the single block variable problem which considers only a single block variable, and the other is called the multi-block variable problem which considers two or more block variables. In the last decades, choosing optimal blocked 2 n− m designs with a single block variable has been well investigated; for example, the authors in [2][3][4][5][6][7][8][9] studied the blocked 2 n− m designs under different minimum aberration criteria; Chen et al. and Zhao et al. [10,11] explored the blocked 2 n− m designs under the clear effects criterion; Zhang and Mukerjee, Zhao et al., and Zhao and Zhao [12][13][14] explored the constructions of the blocked 2 n− m designs under the general minimum lower-order confounding (GMC) criterion proposed in [15]; and Zhao et al. [16][17][18] gave construction methods of the blocked 2 n− m designs under another GMC criterion proposed in [19].
Compared to the large body of work on the blocking problem of single block variable, the studies on choosing optimal blocked 2 n− m designs with multi-block variables are relatively rare. However, it has been recognized that the blocking problem of multi-block variables can arise quite naturally in many practical situations. For example, in the agricultural context, Bisgaard [1] pointed that when designs are laid out in rectangular schemes, both row and column inhomogeneity effects probably exist in the soil. Another example of multi-block variables is from [20]. Considering the comparison of two gasoline additives by testing them on two cars with two drivers over two days, the "cars," "drivers," and "days" are three block variables which should be taken into account when performing experiments.
Under the clear effects criterion, Zhao and Zhao [21] proposed an algorithm for finding optimal blocked 2 n− m designs with multi-block variables. Under the minimum aberration criterion, Zhao and Zhao [22] developed some rules for constructing optimal blocked 2 n− m designs with multi-block variables. Zhang et al. [23] extended the idea of GMC criterion to the case of multi-block variable problem and developed the blocked GMC criterion, called the B 2 -GMC criterion. Inheriting the advantage of the GMC criterion, the B 2 -GMC designs are particularly preferable when some prior information on the importance ordering of treatment effects is present. By computer search, Zhang et al. [23] tabulated some B 2 -GMC deigns with small n and N, where N � 2 n− m . When n or N is large, computer search becomes computationally expensive. Zhao et al. [24] and Zhao and Zhao [25] completed the constructions of B 2 -GMC designs with 5N/16 + 1 ≤ n ≤ N − 1. is paper aims at providing theories and systematical construction methods of the B 2 -GMC designs with N/4 + 1 ≤ n ≤ 5N/16. e rest of the paper is organized as follows. Section 2 reviews doubling theory and B 2 -GMC criterion. Section 3 provides theoretical results and construction methods of B 2 -GMC designs. Section 4 gives concluding remarks. Some useful lemmas are deferred to Appendix.

Preliminaries
as a double of X, where ⊗ is the Kronecker product. Let D q (X) denote the design obtained by repeatedly doubling Xq times, i.e., D q (X) � D(D q− 1 (X)). When X � 1, we write D q (1) � (I 2 q , 1 2 q , 2 2 q , 1 2 q 2 2 q , . . . , 1 2 q 2 2 q 3 2 q , . . . , q 2 q ), where the subscript 2 q means the dimension of a column, I 2 q is a column of 1's, are q independent columns, and the other columns are the component-wise products of some of these q independent columns. For example, 1 2 q 2 2 q is the component-wise product of the columns 1 2 q and 2 2 q . Write D q (1) � (I 2 q , D q (·)); then, D q (·) is just the regular two-level saturated design with columns arranged in Yates order. As some where ∅ denotes the empty set, the superscript q of H q r refers to that H q r is a subdesign of D q (·), and r 2 q H q r− 1 � (r 2 q d 2 q : d 2 q ∈ H q r− 1 ), i.e., H q r consists of the first 2 r − 1 columns of D q (·). Especially, H q q � D q (·).

B 2 -GMC Criterion.
Before introducing the B 2 -GMC criterion, we first review some principles in the multi-block variable problem. Let b 1 , b 2 , . . . , b s denote the s block variables which cause the inhomogeneity of the experimental units. Suppose that the block variable b j partitions the N(� 2 n− m ) experimental units into 2 l j blocks; then, l j independent columns are needed to carry out this blocking plan.
Denote S j as the set of the l j independent columns related to the block variable b j . e block columns should follow the following rules: (i) e l j block columns in S j (j � 1, 2, . . . , s) are independent of each other.
(ii) A block column from S j is not necessarily independent of the block columns from S i with j ≠ i.
In this paper, we focus on the case where each block variable is at two levels, i.e., l j � 1. e effect hierarchy principle for blocked designs with multi-block variables is as follows (see [23]): (i) e lower-order treatment factorial effects are more likely to be important than the higher-order ones, and the treatment factorial effects of the same order are equally likely to be important. (ii) e lower-order block factorial effects are more likely to be important than the higher-order ones, and the block factorial effects of the same order are equally likely to be important. (iii) All the interactions between treatment factors and block factors are negligible.
Since each variable or factor is assigned to one column of the design matrix when an experiment is carried out, we do not differentiate the variable, factor, and column. Based on the effect hierarchy principle and weak assumption that the effects involving three or more factors are usually not important and negligible, Zhang et al. [23] proposed the B 2 -GMC criterion which pays attention to only the confounding among main treatment effects and the two-factor interactions of treatment factors (2fi's for short). For the same reason, a common assumption in blocking problem is that only the main effects of block variables and the interactions of any two block factors are potentially significant, and if a treatment effect is confounded with a potentially significant block effect, the treatment effect cannot be estimated. us, the confounding between the main effects of treatment factors and any potentially significant block effect is not allowed. 2 (D) as the number of main treatment effects which are aliased with p2fi's but not with any potentially significant block effects, where p � 0, 1, 2, . . . , P and P � n(n − 1)/2. Similarly, # 2 C (p) 2 (D) denotes the number of 2fi's which are aliased with the other p2fi's but not with any potentially significant block effects, where p � 0, 1, . . . , P. Denote ,  (4) is sequentially maximized. In the following, without causing confusions, we omit the subscript of a column and the superscript of a design when they are taken from D q (·). For example, we use a, H r , and H q instead of a 2 q , H q r , and H q q , respectively. Denote (6) and then U(D b ) consists of all the potentially significant block effects. As previously stated, the confounding between main treatment effects and potentially significant block effects is not allowed. is requires where # denotes the cardinality of a set and d 1 d 2 stands for the two-factor interaction of d 1 and d 2 . us, B 2 (D t , γ) is the number of 2fi's of D t appearing in the alias set that contains γ. Isomorphism introduced by Tang and Wu [26] is a useful concept which helps narrow down the search of the optimal blocked designs here. An isomorphism ϕ is a one-toone mapping from H q to H q such that ϕ(xy) � ϕ(x)ϕ(y) for every x ≠ y ∈ H q . e 2 n− m designs D t and D * t are isomorphic if there exists an isomorphism ϕ that maps D t onto [27]) if it has resolution IV and maximum number of clear 2fi's, where a resolution R design has no c-factor interaction confounded with any other interaction involving less than R − c factors (see [28]), and a 2fi is called clear if it is not aliased with any main treatment effect and other 2fi's. Cheng and Zhang [29] showed that a 2 n− m design with n � N/4 + 1 is a MaxC2 design if and only if it is a GMC design. ey also pointed out that, up to isomorphism, the GMC 2 n− m design with n � N/4 + 1 can be uniquely expressed as S N/4+1 � (q − 1, q, q(q − 1)H q− 2 ). It is easy to obtain that

Constructions of B 2 -GMC Designs
Lemma 1 is a straightforward extension of Lemma A.1, in Appendix, introduced from [24,25].
and the equality holds when D b has k + 1 independent columns.
and the equality holds when D b has k + 1 independent columns.
and the equality holds when D b has k + 1 independent columns.
Lemma 2 provides a necessary condition for a 2 n− m : (8)-(10), we can obtain which is sequentially maximized by D, and # 2 C (4). is leads to and and Wu and Wu [30] showed that a 2 n− m design with n � N/4 + 1 is a MaxC2 design if and only if this design has N/2 − 1 clear 2fi's. is obtains that D * t � S N/4+1 up to isomorphism.
and u 2 � # U(D b ) ∩ H q− 2 ; then, from (10), we have If D is a B 2 -GMC design, then D must sequentially minimize (u 1 , u 2 ). ere are two different ways to choose D b from H q \S N/4+1 : It is not hard to verify that D b in case (ii) results in is completes the proof. □ e following example illustrates the construction method in eorem 1.
3.2. B 2 -GMC Designs with N/4 + 1 < n ≤ 5N/16. To sequentially maximize (4), the first part # 1 C 2 (D) of (4) should be first maximized. Recall that If D t has resolution IV, then # 1 C 2 (D) must be maximized. According to [31], when N/4 + 1 < n ≤ 5N/16, the D t with resolution IV must be an n-projection of some second-order saturated (SOS) designs. In the following, we first review the concept of SOS design.
A 2 n− m design is called an SOS design if all of its degrees of freedom can be used to estimate only the main treatment effects and 2fi's. In terms of coding theory and projective geometry, Davydov and Tombak [32] showed that, given N, only the SOS designs of N/4 + 1, N(2 w− 2 + 1)/2 w with w ≥ 4 and N/2 factors exist. Block and Mee [31] further showed that an SOS design of N(2 w− 2 + 1)/2 w factors can be obtained by doubling some smaller SOS 2 (2 w− 2 +1)− (2 w− 2 +1− w) designs q − w times. Zhang and Cheng [33] showed that the SOS design of N/2 factors can be uniquely represented by S N/2 � (q, qH q− 1 ) up to isomorphism. Let denote the collection of all the SOS designs obtained by doubling some SOS 2 (2 w− 2 +1)− (2 w− 2 +1− w) designs q − w times. Especially, in L(w), we denote the SOS design obtained by doubling the MaxC2
Proof. As discussed in the first paragraph of this section, if Let P 1 be an n-projection of S N/2 . According to eorem 3 in [33], we obtain Denote p i as the number of clear 2fi's of X v involving a i for i � 1, 2, . . . , 2 v− 2 + 1 and c as the total number of clear 2fi's of X v . Let P v be an n-projection of (23) and (24) in the proof of eorem 3.1 of [29], among all the n-pro- , where a j is the column such that p j � max p 1 , p 2 , . . . , p 2 v− 2 +1 . For a P v with P v ⊂ D q− v (a j ), according to equation (24) in the proof of eorem 3.1 of [29], it is obtained that or in a re-changed Yates order.
Applying equations (23) and (24) in the proof of eorem 3.1 of [29], among all the In the following, for which the following analysis and final conclusion are the same as that (18) and (23), it is obtained that where the second equality can be easily verified by noting that erefore, (24) and (26), we obtain Mathematical Problems in Engineering Comparing (27) with (19), (20) for 4 ≤ v ≤ w, and (26) for is completes the proof.   (16) and (18), for any γ ∈ H q− w ,

Proof. By Lemma 3 and its proof, if
Let and then Mathematical Problems in Engineering 7 erefore, D * sequentially maximizes # 2 C 2 among all the possible 2 n− m : 2 s designs, only if D * sequentially minimizes For ease of presenting, let chosen as (i). In the following, we consider only and for p ≥ b, From (33), it is obtained that Note that D t is a GMC design [29]; then, D maximizes among all the possible 2 n− m : 2 s designs. Suppose D is not a B 2 -GMC design; then, there should be a D * � (D * t : D * b ) which outperforms D � (D t : D b ) in terms of (3). is implies that there exists some p 1 ≥ b − 1 such that 8 Mathematical Problems in Engineering Recalling the definitions of # 2 C (p) 2 (D) and B 2 (D t , γ), we have where the second and third equalities are due to B 2 (D t , γ) � 0 for any γ ∈ S N(2 w− 2 +1)/2 w . Similarly, From (33) and (38), we have By (36) and (37), it is obtained that Mathematical Problems in Engineering en, according to (39)-(41), we obtain #U(D * b ) < 2 k − 1 which contradicts Lemma 1 (i) and (ii).
For (b), similar to (a), when k ≤ q − w − 1, there are two different ways to choose With a similar argument to (a), case (i) results in u 1 � u 2 � u 3 � 0 while case (ii) results in should be chosen according to case (i), i.e., Lemma 1 (iii). e remainder of the proof is similar to that of (a).
For (c), when k > q − w − 1, there are two different ways With a similar argument to (a), it is not hard to verify that case (ii) results in u 1 > 0 or u 2 > 0 while case (i) results in }. e remainder of the proof is similar to that of (a).
In the following, an example is provided to illustrate the construction method in eorem 2 and Remark 1.

Concluding Remarks
e regular 2 n− m designs have wide applications in engineering, manufacturing industry, agriculture and medicine, and so on. When the size of experimental units is large, the inhomogeneity of experimental units results in unwanted variances of estimations of treatment effects. An essential way to solve this problem is to partition the experimental units into blocks.
ere are two kinds of blocking problems as pointed out in [1]. One is called the single block variable problem which considers only a single block variable, and the other is called multi-block variable problem which considers two or more block variables. As stated in Section 1, experiments which involve multi-block variables are more widely concerned in practice than those which involve only a single block variable. However, due to the complexity of multi-block variable problem, the studies on constructing optimal designs with multi-block variables are relatively rare.
In this paper, we aim at exploring the theories and constructions of optimal blocked 2 n− m designs with multiblock variables. e prevalent B 2 -GMC criterion is adopted.
is criterion is preferable when there is some prior knowledge on the importance ordering of the treatment effects.
e systematical construction methods of the B 2 -GMC 2 n− m : 2 s designs with N/4 + 1 ≤ n ≤ 5N/16 are developed. e construction methods are concise and easy to implement as indicated by the examples provided.

Conflicts of Interest
e author declares that there are no conflicts of interest.