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

Authors: Zhi Ming Li; Run Chu Zhang
Journal: The Scientific World Journal
 (2015)

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## 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 elementsC1#2 and C2#2 in the AENP. Further, their mathematical formulations are obtained for every GMC 2n-m design with N=2n-m according to two cases: (i) 5N/16+1≤n<N/2 and (ii) N/2≤n≤N-1.


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## Body

## 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 D be a 2n-m design with n factors, m independent defining words, and N=2n-m runs. We denote the factors by 1,2,…,n. An ith-order factor effect is said to be aliased with jth-order factor effects at degree k if it is simultaneously aliased with kjth-order factor effects. The 0th-order effect is the grand mean and 1st-order effect is a main effect.LetCi#j(k)(D) (written by Ci#j(k) for short) be the number of ith-order factor effects that are aliased with kjth-order factor effects. Denote Kj=nj; a set {Ci#j(k),0≤k≤Kj,0≤i,j≤n} is calledthe aliased effect-number pattern (AENP) of the design D. The set reflects the overall confounding between factor effects in the design. Define Ci#j=(Ci#j(0),Ci#j(1),…,Ci#j(Kj)) and a design that sequentially maximizes the vector(1)C#=C1#2,C2#2,C1#3,C2#3,C3#2,C3#3,…is calleda GMC design, where the ordering of Ci#j’s is in accordance with the rule: Ci#jis beforeCu#vif eithermax⁡(i,j)<max⁡(u,v),ormax⁡(i,j)=max⁡(u,v),withi<u,ormax⁡(i,j)=max⁡(u,v)withi=uandj<v. 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 C1#2 and C2#2 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 C1#2 and C2#2 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 2n-m designs with N/4+1≤n≤N-1. 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 ofC1#2 and C2#2 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 C1#2 and C2#2 for GMC 2n-m design with resolution R≥III, for 5N/16+1≤n<N/2 and N/2≤n≤N-1. Concluding remarks are given in Section 5.


## 2. Some Notations and Lemmas

Denoteq=n-m and 1,2,…,q stand for q independent factors. Let Hq be the set containing all main effects 1,2,…,q and all interactions among them, formed by(2)H1=1,Hq=Hq-1,q,qHq-1,where qHq-1={qd:d∈Hq-1}. By Theorem  2.7.1 of Mukerjee and Wu [11], any 2n-m design D can be represented by an n-subset of Hq; that is, D⊂Hq.LetT1=H1 and Tr={r,rHr-1} for 1<r≤q. Evidently, Hq=∪r=1qTr. For 5N/16+1≤n≤N-1, Li et al. [10] have gotten that every GMC 2n-m design is constructed by the last n columns of Hq. Therefore, GMC 2n-m designs with 5N/16+1≤n<N/2 are directly formed by the last n columns of Tq. Denote Sqr=Hq∖Hr with 1≤r<q. For N/2≤n≤N-1, there exists a number r(<q) so that GMC 2n-m design is formed by the last n columns of Tr∪Sqr. Thus, the GMC design can be written by D0∪Sqr, where D0 consists of the last n-(N-2r) columns of Tr. To get the lower-order confounding information of two-level GMC design, we need to study structure of last n0 columns of Tr for r≤q and n0≤n.SupposeD0 consists of the last n0 columns of Tr(r≤q), where n0=#{D0} and #A denotes the cardinality of a set A. The following example illustrates the structure of D0.Example 1.
Considerr=7; we select the last n0 columns of T7 to construct D0. Clearly, there are 64 choices besides D0≡Tr. For 1≤n0≤63, D0 is one of the following six forms.(i)
(u+1)⋯7Hu for 1≤u<7.(ii)
(u+1)⋯7(Hu∖Hv) for 1≤v<u<7.(iii)
(u+1)⋯7((s+1)⋯vHs∪(Hu∖Hv)) for 1≤s<v<u<7.(iv)
(u+1)⋯7((s+1)⋯v(Hs∖Ht)∪(Hu∖Hv)) for 1≤t<s<v<u<7.(v)
(u+1)⋯7((s+1)⋯v((w+1)⋯tHw∪(Hs∖Ht))∪(Hu∖Hv)) for 1≤w<t<s<v<u<7.(vi)
(u+1)⋯7((s+1)⋯v((w+1)⋯t(Hw∖Hz)∪(Hs∖Ht))∪(Hu∖Hv)) for 1≤z<w<t<s<v<u<7.
The above example provides a way to constructD0. Generally, for any r(r≤q), we consider the construction of D0 in Tr. Define(3)Dt=Hit∖Hjt,ait=it+1it+2⋯jt-1,1≤t≤m,where 1≤jm<im<jm-1<im-1<⋯<jt<it<⋯<j1<i1<j0=r. Then, D0 can be constructed by either of the following cases.Case 1.
One hasD0=ai1ai2⋯aim-1aimDm∪Dm-1⋯∪D2∪D1.Case 2.
One hasD0=ai1(ai2(⋯(aim-1(aimHim∪Dm-1)⋯)∪D2)∪D1).In Case1, the number of elements in D0 is even since #{D0}=∑t=1m‍(2ik-2jk). However, that of D0 in Case 2 is odd because of #{D0}=∑t=1m-1‍(2ik-2jk)+2im-1.ConsiderD⊂Hq and any γ∈Hq; define(4)B2D,γ=#d1,d2:d1,d2∈D,d1d2=γ,which is the number of 2fi’s in D aliased with γ. By the definition of Ci#j(k)(D), it can be easily obtained that(5)C1#2kD=#γ:γ∈D,B2D,γ=k,(6)C2#2kD=k+1#γ:γ∈Hq,B2D,γ=k+1,where k=0,1,…,K2. In order to get the lower-order confounding of D0 in the above cases, we need to study B2(Dt,γ) for t≥1.Lemma 2.
LetDt be defined in (3) for t≥1. Then (7)B2Dt,γ=2it-1-2jt-1,γ∈Hjt,2it-1-2jt,γ∈Hit∖Hjt,0,γ∈Hq∖Hit.Proof.
Forγ∈Hq∖Hit, we have B2(Dt,γ)=0. If γ∈Hjt, then (8)B2D,γ=#Hit∖Hjt2=2it-1-2jt-1.For γ∈Hit∖Hjt, there are 2it-1-1 pairs of factors in Hit so that their interactions are aliased with γ. Among these pairs, there are 2jt-1 pairs with one factor from Hjt and another from Hit∖Hjt. Thus,(9)B2Dt,γ=2it-1-1-2jt-1=2it-1-2jt.This completes the proof.Next we analyze Case1 of D0. For convenience, by (3), denote(10)Dt=ait⋯aim-1aimDm∪Dm-1⋯∪Dtfor 1<t≤m. Evidently, D(t)⊂Hjt-1 and D(1)=D0 in Case 1. When d1∈D(t) and d2∈Dt-1, we have d1d2∈Dt-1. Thus, (11)#d1,d2:d1∈Dt,d2∈Dt-1,d1d2=γ=∑l=tm#Dlfor γ∈Dt-1. Otherwise, the value is zero. Then(12)#d1,d2:d1∈Dt,d2∈Dt-1,d1d2=γ=∑l=tm2il-2jl,γ∈Dt-1,0,γ∉Dt-1.Based on Lemma 2 and (12), we can get the following result for Case 1.Lemma 3.
LetD0=ai1(ai2(⋯(aim-1(aimDm∪Dm-1)⋯)∪D2)∪D1). Then (13)
B
2
D
0
,
γ
=
c
m
+
1
2
,
γ
∈
H
j
m
,
c
m
+
1
-
c
t
+
2
i
t
2
,
γ
∈
D
t
,
t
=
1
,
…
,
m
,
c
t
2
,
γ
∈
H
j
t
-
1
∖
H
i
t
,
t
=
2
,
…
,
m
,
0
,
γ
∈
H
q
∖
H
i
1
,where(14)c1=0,ct=∑l=1t-1‍2il-2jl,t>1.Proof.
For1<l≤m, by (10), we have(15)B2ailDl∪Dl-1,γ=B2Dl∪Dl-1,γ=B2Dl,γ+B2Dl-1,γ+#d1,d2:d1∈Dl,d2∈Dl-1,d1d2=γ.Hence,(16)B2D0,γ=∑l=1m-2‍B2Dl,γ+B2aimDm∪Dm-1,γ+∑l=2m-1‍#d1,d2:d1∈Dl,d2∈Dl-1,d1d2=γ=∑l=1m‍B2Dl,γ+∑l=2m‍#d1,d2:d1∈Dl,d2∈Dl-1,d1d2=γ.
PutHr into 2m+1 incompatible parts: Hjm, Dl+1, and Hjl∖Hil+1 for l=0,1,…,m-1. Clearly, if γ∈Hq∖Hi1, then B2(D0,γ)=0. By Lemma 2 and (12), we, respectively, discuss the following cases.(i)
Ifγ∈Hjm, then (17)#d1,d2:d1∈Dl,d2∈Dl-1,d1d2=γ=0for1<l≤m-1. Thus, (18)B2D0,γ=∑l=1m‍B2Dt,γ=∑l=1m‍2il-1-2jl-1.(ii)
Ifγ∈Dt with 1<t≤m, one has(19)B2D0,γ=∑l=1t-1‍B2Dl,γ+B2Dt,γ+#d1,d2:d1∈Dt+1,d2∈Dt,d1d2=γ=∑l=1t-1‍2il-1-2jl-1+2it-1-2jt+∑l=t+1m‍2il-2jl=∑l=1m‍2il-2jl-∑l=1t-1‍2il-1-2jl-1-2it-1.(iii)
Ifγ∈Hjt-1∖Hit for t>1, then (20)B2D0,γ=∑l=1t-1‍B2Dl,γ=∑l=1t-1‍2il-1-2jl-1.This completes the proof.Lemma3 shows that the value of B2(D0,γ) in Case 1 depends on all pairs {it,jt}1≤t≤m which relate to #{D0}=∑t=1m‍(2it-2jt). For instance, take n0=#{D0}=42 that is nearer to the number 25 than 26; we have (21)n0=25+23+2=26-25+24-23+22-2.Thus i1=6, j1=5, i2=4, j2=3, i3=2, and j3=1. And take n0=54 which is closer to the number 26 than 25; one obtains n0=26-24+6=26-24+23-2. Then i1=6, j1=4, i2=3, and j2=1.Consider Case2 of D0. Denote(22)Ht=ait⋯aim-1aimHim∪Dm-1⋯∪Dtfor 1<t≤m. Clearly, H(t)⊂Hjt-1 and H(1)=D0 in Case 2. For two factors d1∈H(t) and d2∈Dt-1, one has d1d2∈Dt-1. Therefore,(23)#d1,d2:d1∈Ht,d2∈Dt-1,d1d2=γ=∑l=tm-1‍2il-2jl+2im-1,γ∈Dt-1,0,γ∉Dt-1.Specifically, if t=m, then (24)#d1,d2:d1∈aimHim,d2∈Dm-1,d1d2=γ=2im-1,γ∈Dm-1,0,γ∉Dm-1.Form≥1 and γ∈Him, there are 2im-1-1 pairs of factors in Him, which each interaction is aliased with γ. Then(25)B2aimHim,γ=B2Him,γ=2im-1-1,γ∈Him,0,γ∈Hq∖Him.Based on the above results, we can obtain the value ofB2(D0,γ) for any γ∈Hq in Case 2.Lemma 4.
LetD0=ai1(ai2(⋯(aim-1(aimHim∪Dm-1)⋯)∪D2)∪D1). Then (26)
B
2
D
0
,
γ
=
c
m
2
+
2
i
m
-
1
-
1
,
γ
∈
H
i
m
,
c
m
-
c
t
+
2
i
t
2
+
2
i
m
-
1
,
γ
∈
D
t
,
t
=
1
,
…
,
m
-
1
,
c
t
2
,
γ
∈
H
j
t
-
1
∖
H
i
t
,
t
=
2
,
…
,
m
,
0
,
γ
∈
H
q
∖
H
i
1
,where c(t) is defined in (14).Proof.
By (22), we obtain (27)B2D0,γ=∑l=1m-1‍B2Dl,γ+B2aimHim,γ+#d1,d2:d1∈aimHim,d2∈Dm-1,d1d2=γ+∑l=2m-1‍#d1,d2:d1∈Hl,d2∈Dl-1,d1d2=γ.For γ∈Hq∖Hi1, we have B2(D0,γ)=0. By Lemma 2, (25), and (23), analyze the following cases.(i)
Forγ∈Him, we obtain (28)B2D0,γ=∑l=1m-1‍B2Dl,γ+B2aimHim,γ=∑l=1m-1‍2il-1-2jl-1+2im-1-1.(ii)
Forγ∈Dt with 1≤t≤m-1, one has (29)B2D0,γ=∑l=1t-1‍B2Dl,γ+B2Dt,γ+#d1,d2:d1∈Ht+1,d2∈Dt,d1d2=γ=∑l=1t-1‍2il-1-2jl-1+∑l=tm-1‍2il-2jl-2it-1+2im-1=cm-ct+2it2+2im-1.(iii)
Forγ∈Hjt-1∖Hit with 2≤t≤m, B2(D0,γ)=∑l=1t-1‍(2il-1-2jl-1).In Lemma4, the value of B2(D0,γ) is relative to these pairs {it,jt}1<t<m and im. For example, consider n0=#{D0}=21. Since n0=(25-24)+(23-22)+2-1, it yields i1=5, j1=4, i2=3, j2=2, and i3=1. Taking n0=29, we have n0=25-3=25-22+1; thus i1=5, j1=2, and i2=1.Lemmas3 and 4, respectively, obtain the value of B2(D0,γ) that D0 consists of the last n0 columns of Tr(r≤q) for two cases. These results play a key role in calculating C1#2 and C2#2’s for all GMC 2n-m designs with 5N/16+1≤n≤N-1. Next sections will, respectively, discuss two-level GMC designs with the factor number n satisfying (i) 5N/16+1≤n<N/2 or (ii) N/2≤n≤N-1.


## 3. GMC2n-m Designs with 5N/16+1≤n<N/2

Li et al. [10] showed all GMC 2n-m designs with 5N/16+1≤n<N/2, constructed by the last n columns of Tq. In Section 2, D0 is constructed by Case 1 or Case 2, which is the last n0 columns of Tr(r≤q) for n0=#{D0}. Therefore, for any GMC 2n-m design D with 5N/16+1≤n<N/2, its construction is similar to that of D0. In (3), take j0=r=q.Theorem 5.
Consider GMC2n-m design (30)D=ai1ai2⋯aim-1aimDm∪Dm-1⋯∪D2∪D1
with5N/16+1≤n<N/2. Then(a)
(31)C1#2kD=n,k=0,0,otherwise,(b)
(32)
C
2
#
2
k
D
=
n
2
j
m
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1
2
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k
=
n
2
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1
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n
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c
t
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2
i
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2
i
t
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2
j
t
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k
=
n
-
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2
i
t
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1
,
t
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1
,
…
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m
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…
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o
t
h
e
r
w
i
s
e
,
wherec(t) is defined in (14).Proof.
Evidently,n=∑l=1m(2il-2jl); we have c(m+1)=n. By Lemma 3,(33)B2D,γ=n2,γ∈Him,n-ct+2it2,γ∈Dt,t=1,…,m,ct2,γ∈Hjt-1∖Hit,t=2,…,m,0,γ∈Hq∖Hi1.(a)
SinceD⊂Hq∖Hi1, hence by (33) and (5) (34)C1#20D=#γ:γ∈D,B2D,γ=0=#D=n.
Otherwise,C1#2(k)(D)=0 for k≠0.(b)
Following (33) and (6), we obtain (35)C2#2kD=k+1×#γ:γ∈Hjm,B2D,γ=k+1+∑t=1m‍#γ:γ∈Dt,B2D,γ=k+1+∑t=2m‍#γ:γ∈Hjt-1∖Hit,B2D,γ=k+1+#γ:γ∈Hq∖Hi1,B2D,γ=k+1.If k=n/2-1, then (36)C2#2kD=n#Hjm2=n2jm-12.For k=n-(c(t)+2it)/2-1 with 1≤t≤m, one has (37)C2#2kD=n-ct+2it2#Dt=n-ct+2it22it-2jt.And if k=c(t)/2-1 with 1<t≤m, then (38)C2#2kD=ct#Hjt-1∖Hit2=ct2jt-1-2it2.Otherwise, C2#2kD=0.For GMC2n-m design with 5N/16+1≤n<N/2, Theorem 5 reveals that the value of C1#2 only depends on the factor number n. However, the value of C2#2 is related to the numbers {it,jt}1≤t≤m besides n. We illustrate them via a simple example.Example 6.
Takeq=5 and n=10; consider GMC 210-5 design D. Since n=23+2=(24-23)+(22-2), clearly, we have i1=4, j1=3, i2=2, and j2=1. Hence, 2i1-2j1=8, 2i2-2j2=2, 2j1-2i2=4, and c(1)=0,c(2)=8. By Theorem 5, we get(39)C1#2kD=10,k=0,0,otherwise,C2#2kD=16,k=1,24,k=3,5,k=4,0,otherwise.
Theorem5 applies to the case that the factor number n of GMC design is even. If n is odd, similar to the proof of Theorem 5, by Lemma 4, one can get the result below.Theorem 7.
Consider GMC2n-m design (40)D=ai1ai2⋯aim-1aimHim∪Dm-1⋯∪D2∪D1with 5N/16+1≤n<N/2. Then(a)
(41)C1#2kD=n,k=0,0,otherwise,(b)
(42)
C
2
#
2
k
D
=
n
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1
2
i
m
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1
2
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k
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n
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1
2
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1
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n
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c
t
+
2
i
t
2
2
i
t
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2
j
t
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k
=
n
-
c
t
+
2
i
t
2
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1
,
t
=
1
,
…
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m
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1
,
c
t
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j
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i
t
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k
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2
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…
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m
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0
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o
t
h
e
r
w
i
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,where c(t) is defined in (14).Proof.
Note thatn=∑l=1m-1‍(2il-2jl)+2im-1=c(m)+2im-1.Example 8.
Letq=5 and n=11; consider GMC 211-6 design D. Here n=24-23+22-1; we have i1=4, j1=3, and i2=2. Thus, c(2)=2i1-2j1=8 and 2j1-2i2=4. Following Theorem 7, it is directly obtained by (43)C1#2kD=11,k=0,0,otherwise,C2#2kD=24,k=2,16,k=3,15,k=4,0,otherwise.


## 4. GMC2n-m Designs with N/2≤n≤N-1

In Section2, we know that any GMC 2n-m design with N/2≤n≤N-1 is constructed by D0∪Sqr, where D0 is the last n-(N-2r) columns of Tr(r<q). Lemmas 3 and 4 have shown the confounding information of D0. Next we will study a special design Sqr=Hq∖Hr(r<q), which consists of the last N-2r columns of Hq. Since r<q, the factor number of the design Sqr satisfies N-2r≥N/2. Hence, the design Sqr has GMC. By Lemma 2, we directly give the value of B2(Sqr,γ) as follows:(44)
B
2
S
q
r
,
γ
=
N
2
-
2
r
,
γ
∈
H
q
∖
H
r
,
N
2
-
2
r
-
1
,
γ
∈
H
r
.Next we discuss the values of C1#2 and C2#2 for GMC design Sqr with r<q.Theorem 9.
Consider any GMC designSqr=Hq∖Hr for r<q. Then(a)
(45)C1#2kSqr=N-2r,k=N2-2r,0,otherwise,(b)
(46)
C
2
#
2
k
S
q
r
=
N
2
-
2
r
N
-
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r
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k
=
N
2
-
2
r
-
1
,
N
2
-
2
r
-
1
2
r
-
1
,
k
=
N
2
-
2
r
-
1
-
1
,
0
,
o
t
h
e
r
w
i
s
e
.Proof.
(a) Ifk=N/2-2r, by (44), then (47)C1#2kSqr=#γ:γ∈Sqr,B2Sqr,γ=k=#Sqr=N-2r.Otherwise, C1#2kSqr=0.(b)
Fork≥0, note that (48)C2#2kSqr=k+1×#γ:γ∈Sqr,B2Sqr,γ=k+1+#γ:γ∈Hr,B2Sqr,γ=k+1.If k=N/2-2r-1, thus by (44) (49)C2#2kSqr=N2-2r#Sqr=N2-2rN-2r.Similarly, for k=N/2-2r-1-1, we have(50)C2#2kSqr=N2-2r-1#Hr=N2-2r-12r-1.For GMC designSqr(r<q), the values of C1#2 and C2#2 only rely on two numbers N and r. In particular, if r=q-1, then Sq(q-1)=Tq. By Theorem 9, one has(51)C1#2kTq=N2,k=0,0,k≠0,C2#2kTq=N4N/2-1,k=N4-1,0,k≠N4-1.The next example is used to illustrate this above result.Example 10.
Consider GMC216-11 design S54. Since r=4 and N=32, one directly gets (52)C1#2kD=16,k=0,0,k≠0,C2#2kD=120,k=7,0,k≠7.
On the other hand, every GMC2n-m design D with n≥N/2 can be constructed by the form (D∖Sqr)∪Sqr, where D∖Sqr consists of the last n-(N-2r) columns of Tr. Then, D0=D∖Sqr. Based on Lemma  3 of Li et al. [10], we obtain the relationship of D and D0 as follows:(53)B2D,γ=n-N2,γ∈Hq∖Hr,B2D0,γ+N2-2r-1,γ∈Hr.Therefore, we can get the following result.Theorem 11.
Consider GMC2n-m design D=D0∪Sqr with r<q, where D0=ai1(ai2(⋯(aim-1(aimDm∪Dm-1)⋯)∪D2)∪D1). Then(a)
(54)
B
2
D
,
γ
=
n
2
,
γ
∈
H
j
m
,
n
-
a
t
+
2
i
t
2
,
γ
∈
D
t
,
t
=
1
,
…
,
m
,
a
t
2
,
γ
∈
H
j
t
-
1
∖
H
i
t
,
t
=
2
,
…
,
m
,
N
2
-
2
r
-
1
,
γ
∈
H
r
∖
H
i
1
,
n
-
N
2
,
γ
∈
H
q
∖
H
r
,(b)
(55)C1#2kD=N-2r,k=n-N2,n-N+2r,k=N2-2r-1,0,otherwise,(c)
(56)
C
2
#
2
k
D
=
n
2
j
m
-
1
2
,
k
=
n
2
-
1
,
n
-
a
t
+
2
i
t
2
2
i
t
-
2
j
t
,
k
=
n
-
a
t
+
2
i
t
2
-
1
,
t
=
1
,
…
,
m
,
a
t
2
j
t
-
1
-
2
i
t
2
,
k
=
a
t
2
-
1
,
t
=
2
,
…
,
m
,
N
2
-
2
r
-
1
2
r
-
2
i
1
,
k
=
N
2
-
2
r
-
1
-
1
,
n
-
N
2
N
-
2
r
,
k
=
n
-
N
2
-
1
,
0
,
o
t
h
e
r
w
i
s
e
,where a(t)=N-2r+c(t) and c(t) is defined in (14).Proof.
(a) By (53) and Lemma 3, note that (57)cm+1=#D0=∑l=1m‍2il-2jl=n-N+2ryields (a).(b)
Forγ∈Sqr, by (a), B2(D,γ)=n-N/2. If k=n-N/2, then (58)C1#2kD=#Sqr=N-2r.
SinceD0⊂Hr∖Hi1, for k=N/2-2r-1, we have (59)C1#2kD=#D0=#⋃t=1m‍Dt=∑t=1m‍2it-2jt=n-N+2r.(c)
Since(60)C2#2kD=k+1×#γ:γ∈Hjm,B2D,γ=k+1+∑t=1m‍#γ:γ∈Dl,B2D,γ=k+1+∑t=2m‍#γ:γ∈Hjt-1∖Hik,B2D,γ=k+1+#γ:γ∈Hr∖Hi1,B2D,γ=k+1+#γ:γ∈Hq∖Hr,B2D,γ=k+1,
by (a), the result follows.When the factor numbern of a GMC design satisfying N/2≤n≤N-1 is even, by Theorem 11, we obtain values of the corresponding C1#2 and C1#2. The next example illustrates this point.Example 12.
Letq=8, r=7; consider GMC 2154-146 design D=D0∪S87. Since n0=#{D0}=26 and (61)n0=25-6=25-23+22-2,we have i1=5, j1=3, i2=2, and j2=1. Thus, 2i1-2j1=24,2i2-2j2=2 and a(1)=N-2r+c(1)=128,a(2)=N-2r+c(2)=152. By (b) and (c) of Theorem 11, one obtains (62)C1#2kD=128,k=26,26,k=64,0,otherwise,C2#2kD=3328,k=25,6144,k=63,1776,k=73,456,k=75,77k=76,0,otherwise.Theorem 13.
Consider GMC2n-m design D=D0∪Sqr with r<q, where D0=ai1(ai2(⋯(aim-1(aimHim∪Dm-1)⋯)∪D2)∪D1). Then(a)
(63)
B
2
D
,
γ
=
n
-
1
2
,
γ
∈
H
i
m
,
n
-
a
t
+
2
i
t
2
,
γ
∈
D
t
,
t
=
1
,
…
,
m
-
1
,
a
t
2
,
γ
∈
H
j
t
-
1
∖
H
i
t
,
t
=
2
,
…
,
m
,
N
2
-
2
r
-
1
,
γ
∈
H
r
∖
H
i
1
,
n
-
N
2
,
γ
∈
H
q
∖
H
r
,(b)
(64)C1#2kD=N-2r,k=n-N2,n-N+2r,k=N2-2r-1,0,otherwise,(c)
(65)
C
2
#
2
k
D
=
n
-
1
2
i
m
-
1
2
,
k
=
n
-
1
2
-
1
,
n
-
a
t
+
2
i
t
2
2
i
t
-
2
j
t
,
k
=
n
-
a
t
+
2
i
t
2
-
1
,
t
=
1
,
…
,
m
-
1
,
a
t
2
j
t
-
1
-
2
i
t
2
,
k
=
a
t
2
-
1
,
t
=
2
,
…
,
m
,
N
2
-
2
r
-
1
2
r
-
2
i
1
,
k
=
N
2
-
2
r
-
1
-
1
,
n
-
N
2
N
-
2
r
,
k
=
n
-
N
2
-
1
,
0
,
o
t
h
e
r
w
i
s
e
,
wherea(t)=N-2r+c(t) and c(t) is defined in (14).Proof.
Only prove (a). Since(66)#D0=∑l=1m-1‍2il-2jl+2im-1=n-N+2r,one has c(m)=n-(N-2r)-(2im-1). By (53) and Lemma 4, yields (a).The proof of (b) and (c) is similar to those of Theorem11. The following example serves to show its application.Example 14.
Letq=8, r=7, and N=256 and consider GMC 2135-127 design D=D0∪S87. Since #{D0}=23-1, we have i1=3. By (b) and (c) of Theorem 13, one gets(67)C1#2kD=128,k=7,7,k=64,0,otherwise,C2#2kD=896,k=6,7680,k=63,469,k=66,0,otherwise.


## 5. Concluding Remark

Based on construction of GMC2n-m designs with 5N/16+1≤n≤N-1, we obtain the mathematical formulation to calculate the values of C1#2 and C2#2 in the AENP. These results are very useful to analyze the confounding information among lower-order factors of two-level GMC designs. For GMC 2n-m designs satisfying n∉[5N/16+1,N-1], some further studies in this direction are in progress.



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*Source: 163234-2015-06-16.xml*