#### Abstract

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. The present paper considers the blocking problem of multi-block variables. Theoretical results and systematical construction methods of optimal blocked designs with are developed under the prevalent general minimum lower-order confounding (GMC) criterion, where .

#### 1. Introduction

The regular 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.

There 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 designs with a single block variable has been well investigated; for example, the authors in [2–9] studied the blocked designs under different minimum aberration criteria; Chen et al. and Zhao et al. [10, 11] explored the blocked designs under the clear effects criterion; Zhang and Mukerjee, Zhao et al., and Zhao and Zhao [12–14] explored the constructions of the blocked designs under the general minimum lower-order confounding (GMC) criterion proposed in [15]; and Zhao et al. [16–18] gave construction methods of the blocked 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 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 designs with multi-block variables. Under the minimum aberration criterion, Zhao and Zhao [22] developed some rules for constructing optimal blocked 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 -GMC criterion. Inheriting the advantage of the GMC criterion, the -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 -GMC deigns with small and , where . When or is large, computer search becomes computationally expensive. Zhao et al. [24] and Zhao and Zhao [25] completed the constructions of -GMC designs with . This paper aims at providing theories and systematical construction methods of the -GMC designs with .

The rest of the paper is organized as follows. Section 2 reviews doubling theory and -GMC criterion. Section 3 provides theoretical results and construction methods of -GMC designs. Section 4 gives concluding remarks. Some useful lemmas are deferred to Appendix.

#### 2. Preliminaries

##### 2.1. Doubling Theory

Let be a matrix with entries 1 or . Denote and , where ′ denotes transpose. Defineas a *double* of , where is the Kronecker product. Let denote the design obtained by repeatedly doubling times, i.e., . When , we write , where the subscript means the dimension of a column, is a column of 1’s,are independent columns, and the other columns are the component-wise products of some of these independent columns. For example, is the component-wise product of the columns and . Write ; then, is just the regular two-level saturated design with columns arranged in Yates order. As some subdesigns in , denote , for , where denotes the empty set, the superscript of refers to that is a subdesign of , and , i.e., consists of the first columns of . Especially, .

##### 2.2. -GMC Criterion

Before introducing the -GMC criterion, we first review some principles in the multi-block variable problem. Let denote the block variables which cause the inhomogeneity of the experimental units. Suppose that the block variable partitions the experimental units into blocks; then, independent columns are needed to carry out this blocking plan. Denote as the set of the independent columns related to the block variable . The block columns should follow the following rules:(i)The block columns in are independent of each other.(ii)A block column from is not necessarily independent of the block columns from with .

In this paper, we focus on the case where each block variable is at two levels, i.e., .

The effect hierarchy principle for blocked designs with multi-block variables is as follows (see [23]):(i)The 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)The 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 -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. Thus, the confounding between the main effects of treatment factors and any potentially significant block effect is not allowed.

Denote as a design, where consists of treatment factors corresponding to a regular design and consists of block factors each of which can partition the runs into 2 blocks. Denote as the number of main treatment effects which are aliased with fi’s but not with any potentially significant block effects, where and . Similarly, denotes the number of 2fi’s which are aliased with the other fi’s but not with any potentially significant block effects, where . Denote

A blocked design is called a -GMC design if it sequentially maximizes (4). Let be the number of main effects which are aliased with 2fi’s of and be the number of 2fi’s which are aliased with the other 2fi’s of . Let

A design is called a GMC design if sequentially maximizes (5).

Let . Constructing a -GMC design is to choose and from such that (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 . For example, we use , , and instead of , , and , respectively. Denoteand then 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. This requires , and thus for .

For and , definewhere denotes the cardinality of a set and stands for the two-factor interaction of and . Thus, is the number of 2fi’s of appearing in the alias set that contains .

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 is a one-to-one mapping from to such that for every . The designs and are isomorphic if there exists an isomorphism that maps onto . The designs and are isomorphic if there exists an isomorphism that maps onto and onto .

#### 3. Constructions of -GMC Designs

##### 3.1. -GMC Designs with

A design is of MaxC2 (see [27]) if it has resolution IV and maximum number of clear 2fi’s, where a *resolution* design has no -factor interaction confounded with any other interaction involving less than 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 design with is a MaxC2 design if and only if it is a GMC design. They also pointed out that, up to isomorphism, the GMC design with can be uniquely expressed as . It is easy to obtain that

Therefore,

Lemma 1 is a straightforward extension of Lemma A.1, in Appendix, introduced from [24, 25].

Lemma 1. *Suppose is any -projection of with for some , where is independent of the columns of . We have*(i)*If , then and the equality holds when has independent columns.*(ii)*If , then and the equality holds when has independent columns.*(iii)*If , then and the equality holds when has independent columns.*(iv)*If , then .*(v)*If , then .*

Lemma 2 provides a necessary condition for a design with to be a -GMC design.

Lemma 2. *Suppose is a design with and for some ; then, is a -GMC design only if .*

*Proof. *Let be a design and . From (8)–(10), we can obtainwhich is sequentially maximized by , and . Suppose that any is not a -GMC design for . Then, by the definition of -GMC criterion, there should be a design outperforming in terms of (4). This leads toand . Clearly,and noting that is the unblocked part of and has columns.

In fact, the inequality in the formula is not valid. Recall that the MaxC2 design has the largest number, , of clear 2fi’s among all the designs with . Therefore, has at most clear 2fi’s, i.e., . Wu and Wu [30] showed that a design with is a MaxC2 design if and only if this design has clear 2fi’s. This obtains that up to isomorphism.

With Lemma 2, Theorem 1 provides the constructions of -GMC designs with .

Theorem 1. *Suppose is a design with and for some ; then, is a -GMC design if and is any -projection of .*

*Proof. *By Lemma 2, if is a -GMC design with , then up to isomorphism. Thus, which is sequentially maximized by . Let and ; then, from (10), we haveIf is a -GMC design, then must sequentially minimize . There are two different ways to choose from :(i).(ii).It is not hard to verify that in case (ii) results in while in case (i) gives . Therefore, if is a -GMC design, then must be of case (i). Recall that , and we have . By Lemma 1 (i), if , then and the equality holds when up to isomorphism. This completes the proof.

The following example illustrates the construction method in Theorem 1.

*Example 1. *Consider the construction of the -GMC design. Here, , , and which leads to . According to Theorem 1, let and be any 5-projection of , say . Then, is a -GMC design.

##### 3.2. -GMC Designs with

To sequentially maximize (4), the first part of (4) should be first maximized. Recall that . If has resolution IV, then must be maximized. According to [31], when , the with resolution IV must be an -projection of some second-order saturated (SOS) designs. In the following, we first review the concept of SOS design.

A 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 , only the SOS designs of , with and factors exist. Block and Mee [31] further showed that an SOS design of factors can be obtained by doubling some smaller SOS designs times. Zhang and Cheng [33] showed that the SOS design of factors can be uniquely represented by up to isomorphism. Letdenote the collection of all the SOS designs obtained by doubling some SOS designs times. Especially, in , we denote the SOS design obtained by doubling the MaxC2 design as .

With a little algebra, it is easy to verify thatwhere , and . Then,

Therefore, can be expressed as

The following lemma provides a necessary condition for a design with to be a -GMC design.

Lemma 3. *Suppose is a design with for and for some ; then, is a -GMC design only if .*

*Proof. *As discussed in the first paragraph of this section, if is a -GMC design with , then must be an -projection of some SOS designs in with or .

Let be an -projection of . According to Theorem 3 in [33], we obtainLet be any SOS design but not a MaxC2 design; then, . Denote as the number of clear 2fi’s of involving for and as the total number of clear 2fi’s of . Let be an -projection of and . Then, . Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the -projections of , sequentially maximizes in (5), only if , where is the column such that . For a with , according to equation (24) in the proof of Theorem 3.1 of [29], it is obtained thatFor a MaxC2 design, it has clear 2fi’s and . Since is not a MaxC2 design, we haveorLet be an -projection of obtained by doubling , the MaxC2 design as mentioned above. With a column permutation, rewrite asin a re-changed Yates order. Write as , where , and . Applying equations (23) and (24) in the proof of Theorem 3.1 of [29], among all the -projections of , sequentially maximizes in (3), only if with or 2. In the following, we investigate with for which the following analysis and final conclusion are the same as that for with . For with , from equations (18) and (23), it is obtained thatwhere the second equality can be easily verified by noting thatTherefore,Denote , where , and . Since , we have . Therefore, from (24) and (26), we obtainComparing (27) with (19), (20) for , and (26) for , it is obtained that if is a -GMC design, should not be worse than in terms of (3). Therefore, if is a -GMC design, should be an -projection of . This completes the proof.

The lemma below is an extension of Lemma A.2, in Appendix, introduced from [34].

Lemma 4. *Suppose consists of the last , , columns of ; then, if is ahead of in .*

*Proof. *Suppose consists of the last columns of . Then, . From (16) and (18), for any , . Straightforwardly, from Lemma A.2, it is obtained that if is ahead of in . This completes the proof.

With Lemmas 3 and 4, Theorem 2 provides the construction methods of -GMC designs with .

Theorem 2. *Let be a design with for and for some . Suppose for some ; then, is a -GMC design if consists of the last columns of and*(a)* is any -projection of when .*(b)* is any -projection of when .*(c)* is any -projection of when .*

*Proof. *By Lemma 3 and its proof, if is a -GMC design, there should be and . Substituting with in (24), we obtainLetand thenwhere . Therefore, sequentially maximizes among all the possible designs, only if sequentially minimizesFor ease of presenting, let , , and .

For (a), when , there are two different ways to choose from : (i) and (ii) . Recall that if is a -GMC design, then with . Case (i) implies that , and thus . Therefore, choosing according to case (i) results in . If is chosen according to case (ii), then which results in , , or . Clearly, if is a -GMC design, should be chosen as (i). In the following, we consider only , i.e., .

For in (a), according to (v) in Lemma 1, if with , then . Denote as the last column of in Yates order, and ; then, . According to Lemma 4, for , if is ahead of in , then by (18). For any , we have . For any such that , we have . Therefore, for ,and for ,