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The Scientific World Journal
Volume 2015, Article ID 163234, 9 pages
http://dx.doi.org/10.1155/2015/163234
Research Article

Results for Two-Level Designs with General Minimum Lower-Order Confounding

1School of Mathematical Sciences, Xinjiang University, Urumqi 830046, China
2KLAS and School of Mathematics, Northeast Normal University, Changchun 130024, China
3LPMC and School of Mathematical Sciences, Nankai University, Tianjin 300071, China

Received 2 June 2014; Accepted 19 November 2014

Academic Editor: Haijun Jiang

Copyright © 2015 Zhi Ming Li and Run Chu Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The general minimum lower-order confounding (GMC) criterion for two-level design not only reveals the confounding information of factor effects but also provides a good way to select the optimal design, which was proposed by Zhang et al. (2008). The criterion is based on the aliased effect-number pattern (AENP). Therefore, it is very important to study properties of AENP for two-level GMC design. According to the ordering of elements in the AENP, the confounding information between lower-order factor effects is more important than that of higher-order effects. For two-level GMC design, this paper mainly shows the interior principles to calculate the leading elements and in the AENP. Further, their mathematical formulations are obtained for every GMC design with according to two cases: (i) and (ii) .

1. Introduction

To find optimal designs in a more elaborate and explicit manner under effect hierarchy principle, Zhang et al. [1] first introduced the aliased effect-number pattern (AENP) and proposed a new criterion of general minimum lower-order confounding (GMC) for two-level regular design. Further, they proved that all the classification patterns conducting the existing criteria, such as maximum resolution (MR) criterion [2], minimum aberration (MA) criterion [3], clear effects (CE) criterion [4], and maximum estimation capacity (MEC) criterion [5], can be expressed as different functions of the AENP so that it can be a basis to unify these criteria.

Through the AENP, we can get a deeper understanding of properties of the above criteria and relationships among them. Zhang and Cheng [6] revealed an exact expression of the average minimum lower-order confounding property of MA design. Hu and Zhang [7] obtained an essential statistical equivalence of MEC design and MA design. From the average least confounding property between lower-order effects, MA designs are most suitable for the situation that all the factors in experiments are treated to be equally important, while GMC design has an individual least confounding property between lower-order effects and possesses the maximum numbers of clear main effects and clear two-factor interactions (2fi’s). Because of this, GMC designs can be applied to the experiments which the experimenters have some prior information to the order of the importance factors. In practice, the latter situation more often happens than the former one. Therefore, the study for GMC designs should be significantly important in both theory and application.

Now we review some definitions proposed by Zhang et al. [1]. Let be a design with factors, independent defining words, and runs. We denote the factors by . An th-order factor effect is said to be aliased with th-order factor effects at degree if it is simultaneously aliased with th-order factor effects. The 0th-order effect is the grand mean and 1st-order effect is a main effect.

Let (written by for short) be the number of th-order factor effects that are aliased with th-order factor effects. Denote ; a set is called the aliased effect-number pattern (AENP) of the design . The set reflects the overall confounding between factor effects in the design. Define and a design that sequentially maximizes the vectoris called a GMC design, where the ordering of ’s is in accordance with the rule: is before if either , or , with , or with and . In order to make main effects or 2fi’s estimable, we need to give an assumption: the interactions involving three or more factors are absent. Thus, we only study the leading terms and of AENP for two-level GMC design in this paper.

Zhang et al. [1] listed all two-level GMC designs of 16 and 32 runs, a number of 64-run GMC designs, and obtained the values of and by computer algorithm. However, the method is not suitable for designs with larger runs. Zhang and Cheng [6] and Chen and Liu [8] provided an important theory for constructing GMC designs. Cheng and Zhang [9] and Li et al. [10] finished the construction of GMC designs with . However, there are few articles that pay attention to calculating the values of elements in the AENP, especially, the confounding information between main effects and 2fi’s, or among 2fi’s of two-level GMC design.

This paper mainly reveals the interior principles for calculating the values of and for two-level GMC design. In Section 2, we introduce some notations and obtain useful lemmas to study the lower-order confounding information of two-level GMC designs. Section 3 and Section 4, respectively, obtain values of and for GMC design with resolution , for and . Concluding remarks are given in Section 5.

2. Some Notations and Lemmas

Denote and stand for independent factors. Let be the set containing all main effects and all interactions among them, formed bywhere . By Theorem  2.7.1 of Mukerjee and Wu [11], any design can be represented by an -subset of ; that is, .

Let and for . Evidently, . For , Li et al. [10] have gotten that every GMC design is constructed by the last columns of . Therefore, GMC designs with are directly formed by the last columns of . Denote with . For , there exists a number so that GMC design is formed by the last columns of . Thus, the GMC design can be written by , where consists of the last columns of . To get the lower-order confounding information of two-level GMC design, we need to study structure of last columns of for and .

Suppose consists of the last columns of , where and denotes the cardinality of a set . The following example illustrates the structure of .

Example 1. Consider ; we select the last columns of to construct . Clearly, there are 64 choices besides . For , is one of the following six forms.(i) for .(ii) for .(iii) for .(iv) for .(v) for .(vi) for
The above example provides a way to construct . Generally, for any , we consider the construction of in . Definewhere . Then, can be constructed by either of the following cases.

Case 1. One has .

Case 2. One has .

In Case 1, the number of elements in is even since . However, that of in Case 2 is odd because of .

Consider and any ; definewhich is the number of 2fi’s in aliased with . By the definition of , it can be easily obtained thatwhere . In order to get the lower-order confounding of in the above cases, we need to study for .

Lemma 2. Let be defined in (3) for . Then

Proof. For , we have If , then For , there are pairs of factors in so that their interactions are aliased with . Among these pairs, there are pairs with one factor from and another from . Thus,This completes the proof.

Next we analyze Case 1 of . For convenience, by (3), denotefor . Evidently, and in Case 1. When and , we have . Thus, for . Otherwise, the value is zero. ThenBased on Lemma 2 and (12), we can get the following result for Case 1.

Lemma 3. Let . Then where

Proof. For , by (10), we haveHence,Put into incompatible parts: , , and for . Clearly, if , then By Lemma 2 and (12), we, respectively, discuss the following cases.(i)If , then for . Thus, (ii)If with , one has(iii)If for , then This completes the proof.

Lemma 3 shows that the value of in Case 1 depends on all pairs which relate to . For instance, take that is nearer to the number than ; we have Thus , , , , , and . And take which is closer to the number than ; one obtains . Then , , , and .

Consider Case 2 of . Denotefor . Clearly, and in Case 2. For two factors and , one has . Therefore,Specifically, if , then

For and , there are pairs of factors in , which each interaction is aliased with . Then

Based on the above results, we can obtain the value of for any in Case 2.

Lemma 4. Let Then where is defined in (14).

Proof. By (22), we obtain For , we have . By Lemma 2, (25), and (23), analyze the following cases.(i)For , we obtain (ii)For with , one has (iii)For with , .

In Lemma 4, the value of is relative to these pairs and . For example, consider . Since , it yields , , , , and . Taking , we have ; thus , , and .

Lemmas 3 and 4, respectively, obtain the value of that consists of the last columns of for two cases. These results play a key role in calculating and ’s for all GMC designs with . Next sections will, respectively, discuss two-level GMC designs with the factor number satisfying (i) or (ii) .

3. GMC Designs with

Li et al. [10] showed all GMC designs with , constructed by the last columns of . In Section 2, is constructed by Case 1 or Case 2, which is the last columns of for . Therefore, for any GMC design with , its construction is similar to that of . In (3), take .

Theorem 5. Consider GMC design with . Then(a)(b)
where is defined in (14).

Proof. Evidently, ; we have . By Lemma 3,(a)Since , hence by (33) and (5)
Otherwise, for .(b)Following (33) and (6), we obtain If , then For with , one has And if with , then Otherwise, .

For GMC design with , Theorem 5 reveals that the value of only depends on the factor number . However, the value of is related to the numbers besides . We illustrate them via a simple example.

Example 6. Take and ; consider GMC design . Since , clearly, we have , , , and . Hence, , , , and By Theorem 5, we getTheorem 5 applies to the case that the factor number of GMC design is even. If is odd, similar to the proof of Theorem 5, by Lemma 4, one can get the result below.

Theorem 7. Consider GMC design with . Then(a)(b)where is defined in (14).

Proof. Note that .

Example 8. Let and ; consider GMC design . Here ; we have , , and . Thus, and . Following Theorem 7, it is directly obtained by

4. GMC Designs with

In Section 2, we know that any GMC design with is constructed by , where is the last columns of . Lemmas 3 and 4 have shown the confounding information of . Next we will study a special design , which consists of the last columns of . Since , the factor number of the design satisfies . Hence, the design has GMC. By Lemma 2, we directly give the value of as follows:Next we discuss the values of and for GMC design with .

Theorem 9. Consider any GMC design for . Then(a)(b)

Proof. (a) If , by (44), then Otherwise, .(b)For , note that If , thus by (44) Similarly, for , we have

For GMC design , the values of and only rely on two numbers and . In particular, if , then . By Theorem 9, one hasThe next example is used to illustrate this above result.

Example 10. Consider GMC design . Since and , one directly gets On the other hand, every GMC design with can be constructed by the form , where consists of the last columns of . Then, . Based on Lemma  3 of Li et al. [10], we obtain the relationship of and as follows:Therefore, we can get the following result.

Theorem 11. Consider GMC design with , where Then(a)(b)(c)where and is defined in (14).

Proof. (a) By (53) and Lemma 3, note that yields (a).(b)For , by (a), . If , then
Since , for , we have (c)Since
by (a), the result follows.

When the factor number of a GMC design satisfying is even, by Theorem 11, we obtain values of the corresponding and . The next example illustrates this point.

Example 12. Let , ; consider GMC design . Since and we have , , , and . Thus, and . By (b) and (c) of Theorem 11, one obtains

Theorem 13. Consider GMC design with , where Then(a)(b)(c)
where and is defined in (14).

Proof. Only prove (a). Since one has . By (53) and Lemma 4, yields (a).

The proof of (b) and (c) is similar to those of Theorem 11. The following example serves to show its application.

Example 14. Let , , and and consider GMC design . Since , we have . By (b) and (c) of Theorem 13, one gets

5. Concluding Remark

Based on construction of GMC designs with , we obtain the mathematical formulation to calculate the values of and in the AENP. These results are very useful to analyze the confounding information among lower-order factors of two-level GMC designs. For GMC designs satisfying , some further studies in this direction are in progress.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China Grants 11171165, 10871104, and 11101074; Scientific Research Program of the Higher Education Institution of XinJiang Grant XJEDU2012S01; and Research Fund for the Doctoral Program of XinJiang University Grant BS130106. The authors are very grateful to the editor, the associate editor, and the referees for their valuable comments which have led to a substantial improvement in the paper.

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