行政院國家科學委員會專題研究計畫 成果報告
分散因子存在下估計主效應之二水準部分因子設計排序之
研究
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 96-2118-M-004-005- 執 行 期 間 ： 96 年 08 月 01 日至 97 年 10 月 31 日 執 行 單 位 ： 國立政治大學統計學系 計 畫 主 持 人 ： 丁兆平 計畫參與人員： 博士班研究生-兼任助理人員：何漢葳 處 理 方 式 ： 本計畫可公開查詢中 華 民 國 97 年 12 月 09 日
96 年度國科會計畫報告
計畫編號：NSC 96-2118-M-004-005
計畫名稱：分散因子存在下估計主效應之二水準部分
因子設計排序之研究
主持人：丁兆平
單位：國立政治大學統計系
Optimal Two-Level Fractional Factorial Designs for
Location Main Effects with Dispersion Factors
By
Chao-Ping Ting*
Department of Statistics, National Chengchi University
Abstract
In two-level fractional factorial designs, homogeneous variance is a commonly made assumption in analysis of variance. When the variance of the response variable changes when a factor changes from one level to another, we call that factor the dispersion factor. The problem of finding optimal designs when dispersion factors present is relatively unexplored, however. In this article, we focus on finding optimal designs for the estimation of all location main effects when there are one or two dispersion factors, in the class of regular unreplicated two-level fractional factorial designs of resolution III and higher. We show that by an appropriate choice of the defining contrasts, A-optimal and D-optimal designs can be identified. Efficiencies of an arbitrary design are also investigated.
Key Words: Generator; Defining relation; Dispersion effect; Location effect.
1. Introduction
Two-level fractional factorial design is one of the most commonly and widely used designs to identify important location factors in industrial, agricultural and business experiments. The assumption of constant variance is a standard one when performing the analysis. In practice, however, situations when variance of the response variable differs from one treatment combination to another do happen. Factors that are responsible for such differences are called dispersion factors.
Identification of the dispersion factors has been extensively studied recently. Box and Meyer (1986) studied the logarithm of the ratio of the residual variance and proposed an informal method to identify dispersion factors. Montgomery (1990) achieved the same goal by plotting these statistics on a normal probability plot. Wang (1989) developed a large sample test statistic to identify dispersion factors. More recently, Bergman and Hynên (1997), Liao (2000), and McGrath and Lin (2001a) developed test procedures to identify dispersion factors in unreplicated regular 2n−p fractional factorial designs. Pan (1999) and McGrath and Lin (2001b) stressed the importance of identifying the location effects before studying dispersion effects.
All of the aforementioned papers focused on identifying dispersion effects, not until Liao and Iyer (2000), the optimality property for the estimation of dispersion effects has been studied. Even though there is a growing interest in studying the optimality property for dispersion effects, the optimality property for location effects when dispersion factors present is relatively unexplored. Lin (2005) formed D-optimal designs for estimating a specific set of location effects with one dispersion factor. In this article, our interest is focused on finding A-optimal and D-optimal designs for estimating all location main effects when one or two dispersion factors present in the class of unreplicated regular 2n−p fractional factorial designs of resolution III or
higher.
In the next section, the notation used throughout this article and the information matrix for the estimation of all location main effects are stated. Section 3 gives the A-optimal and D-optimal designs for the estimation of all location main effects with one dispersion factors. A-efficiency and D-efficiency for an arbitrary design are also given. In section 4, A-optimal and D-optimal designs for estimating all location main effects with two dispersion factors are given. Catalogues of 16-run and 32-run D-efficient 2nIII−p designs are provided.
2. Preliminaries
Let F1, F2, … , Fn denote the n two-level factors and the main effects of the
corresponding factors as well. Let n
n e e e F F F 2L 2
11 denote the general effect with ei =
1 if Fi appears in the effect, and ei = 0, otherwise. Without loss of generality, we use
F1, F2, …, and Fa to denote the a factors that are responsible for the dispersion effects.
Running a full factorial design may not be desirable, especially when n is large. Instead, running a fraction of the full factorial design, which is called a fractional factorial design, is sufficed when our interest is to estimate main effects and few low-order interactions of the design. A2n−pfractional factorial design withN = 2n−p runs can be determined by appropriately selecting p independent generators, and the design can uniquely be determined by its defining relation. For example, the treatment combinations of a 26−3design are determined when the following three generators F4
= F1F2, F5 = F1F3, and F6 = F2F3 are selected. It’s corresponding defining relation is
6 5 2 1 6 4 3 1 5 4 3 2 6 5 4 6 3 2 5 3 1 4 2 1F F FFF F FF F F F F F F F FFF F FF F F F I = = = = = = = ,
main effects, and some of the low-order interactions can be estimated.
The resolution of a design depends on the alias structure. In the defining relation, an effect that is aliased with the general mean, is called a word, and the number of letters in a word is called the word length. The minimum length of all the words in the defining relation is called the resolution of the design for two-level factional factorial designs. The example above is a design of resolution III, and is denoted as26III−3.
In a regular 2n−p fractional factorial design setting, let Yv denote the response vector, and the model we employ here is
ε β+ = v v X Y ,
where βv is the (n+1)×1 vector of the overall mean and all location main effects; X =
[
xv0,xv1,L,xvn]
is the N× n( +1) model matrix, xv =0 (1,1,L,1)', and)' , , , ( 1j 2j Nj j x x x
xv = L with xij =+1 or −1 depends on whether factor j appears at
its high level or low level in the ith response; and εv is the N × 1 vector of uncorrelated random error with E(εv =) 0v and
, )
( 0I 1D1 2D2 aDa
Var εv =γ +γ +γ +L+γ
where γ0 is the dispersion mean, γj is the dispersion main effect of factor Fj by Liao
and Iyer (2000), and Dj is the N × N diagonal matrix whose diagonal elements are
j N j
j x x
x1 , 2 ,L,and .
Under the assumption of constant variance, that is, γ0 = σ2 and γ1 = … = γa = 0
then Var(Yv)=σ2I . The best linear unbiased estimator βˆv of βv and the corresponding covariance matrix are
and , ' ) ' ( ˆ 1 Y X X X v v= − β ly. respective , ) ' ( ) ˆ (β = X X −1σ2 Var v
In here X'X =NIn+1 since the designs we consider here are of resolution III or higher, and In is the n × n identity matrix.
Example 2.1. A 25III−2 fractional factorial design with generators F4 = F1F2, and F5 =
F1F3, then the defining relation is
5 4 3 2 5 3 1 4 2 1F F FF F F F F F F I = = = .
The design matrix X is thus determined, and is
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − − − − − − − − − − − − − − − − − = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X .
One can see that X'X =8I6, X Y
v v ' ) 8 1 ( ˆ = β , and (ˆ) ( 8) 6. 2 I Var βv = σ
For more general cases, that is, Var Y =γ I+γ D +L+γaDa =V v 1 1 0 ) ( , say, the
best linear unbiased estimator βˆv of βv, and the corresponding covariance matrix are and , ' ) ' ( ˆ 1 1 1 Y V X X V X v v= − − − β ly. respective , ) ' ( ) ˆ ( = X V−1X −1 Var βv
Let M be the information matrix for the estimation of βv, then M =X'V−1X .
3. Regular 2n−p fractional factorial design with one dispersion factor
In this section, we focus on regular unreplicated 2n−p fractional factorial
responsible for the dispersion effect, that is, γ0 > γ1 ≠ 0, γ2 = … = γa = 0, and 1 1 0 ) (Y V I D Var v = =γ N +γ . Then ) ( 1 1 1 0 2 1 2 0 1 D I V γ N γ γ γ − − = − , and ), (mij M = i,j=0,L,n, can be partitioned as , 22 ' 12 12 11 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = M M M M M where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 1 1 0 11 m m m m M
is a 2 × 2 matrix with m0 = Nγ0 (γ02−γ12) and m1 =−γ1N (γ02 −γ12); M12 is a 2 ×
(n+1) matrix of zeroes; M22 is a (n−1)×(n−1) matrix, with mii = m0, i = 2, … , n,
and for all i ≠ j = 2, … , n,
⎪⎩ ⎪ ⎨ ⎧ = otherwise. , 0 relation, defining in the word a is if , 1 1 i j ij F F F m m
For the derivation of M, see Appendix.
Example 3.1. The same 25III−2 fractional factorial design as in Example 2.1. Then D1
and M, respectively, are
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − = 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 D , ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m M .
Since F1F2F4 and F1F3F5 are words in the defining relation, m24 = m24 = m35 = m53 =
3.1. Optimal 2n−p fractional factorial design with one dispersion factor
Let θ be the number of length three words in the defining relation involving F1.
That is, θ is the number of words in the defining relation of form F1FiFj, for 2 ≤ i < j ≤
n. Then through some row and column operations, M can be transformed into MT,
where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⊗ = − − + 1 2 0 1 0 0 θ θ n T I m Q I M , , 0 1 1 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = m m m m Q
and “⊗” is the Kronecker product. The eigenvalues of M can thus easily be obtained, and they are m0 −m1 with multiplicities θ +1, m0 +m1 with multiplicities θ +1, and
0
m with multiplicities n – 2θ – 1.
Example 3.1. (Continued) For this 25III−2 design, one can see that n = 5, θ = 2, and MT
is given below. The eigenvalues of M are m0−m1 with multiplicities 3, and m0 +m1 with multiplicities 3. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m MT .
D-optimal design is the design that minimizes the determinant of M−1, or
equivalently maximizes the determinant of M. A-optimal design is the design that minimizes the trace of M−1. For most of the eigenvalues based optimality criteria, for example, D-optimality and A-optimality, one can see that the smaller the value of θ is, the “better” the corresponding design is. The smallest possible value of θ is 0, and the D-optimal and A-optimal designs are those having the following information matrix
M*, ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = −1 0 * 0 0 n I m Q M .
It is thus appropriate to define the D-efficiency, De, and A-efficiency, Ae, for an
arbitrary design as . ) ( ) ( , ) det( ) det( 1 1 * * − − = = M tr M tr A M M De e
Through some calculation, one has
, ) ) / ( 1 ( 1 0 2 θ γ γ − = e D and . 1 2 ) / )( 1 ( 2 1 2 1 0 n n Ae − + + + − = θ γ γ θ
One can observe that both De and Ae are decreasing in θ, that is, when there are more
length three words involving F1 in the defining relation, the less efficient the
corresponding design is. Also, both De and Ae are decreasing in γ1, that is, the larger
the dispersion effect is, the less efficient the corresponding design is.
Example 3.2. A 26III−2 factional factorial design with generators F5 = F2F3, and F6 =
F1F2F4. The defining relation is
6 5 4 3 1 6 4 2 1 5 3 2FF FF F F FFF F F F I = = = .
Since there is no length three word of form F1FiFj in the defining relation, hence θ = 0,
and the corresponding information matrix M is of the optimal form, that is
. 0 0 5 0 * ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = I m Q M M
The 26III−2 fractional factorial design above is thus both D-optimal and A-optimal in estimating all location main effects when F1 is responsible for the dispersion effect in
the model.
F3F4. The defining relation and the corresponding information matrix M are 6 5 4 3 2 1 6 4 3 5 2 1F F FF F FF FF F F F I = = = , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = m m m m m m m m m m m M respectively.
One can see that there is one length three word of form F1FiFj in the defining relation,
hence θ = 1, and 2 0 1/ ) ( 1− γ γ = e D , and 3 ) / ( 7 2 1 2 1 0 − − = γ γ e A .
Example 3.4. A 26III−2 factional factorial design with generators F5 = F1F2, and F6 =
F1F3. The defining relation and the corresponding information matrix M are
6 5 3 2 6 3 1 5 2 1F F FFF F FFF F I = = = , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = m m m m m m m m m m m m m M respectively.
Since there are two length three words of form F1FiFj in the defining relation, hence θ
= 2, and 2 2 0 1/ ) ) ( 1 ( − γ γ = e D , and 1 ) / ( 7 4 1 2 1 0 − − = γ γ e A .
One can observed that for fixed values of γ0 and γ1, γ0 > γ1, the design in
De are larger. We say that the design in Example 3.3 is D-better and A-better than the
design in Example 3.4
Remark：For designs of resolution IV or higher, the shortest word in the defining relation is of length at least four, hence all their information matrices are of form M*. Thus, resolution IV or higher designs are “robust” against single dispersion factor when our interest is to estimate all location main effects.
4. Regular 2n−p fractional factorial design with two dispersion factors
In this section, we focus on regular unreplicated 2n−p fractional factorial designs with two dispersion factors. Without loss of generality, we assume that F1 and
F2 are responsible for the dispersion effects, that is, γ1 ≠ 0, γ2 ≠ 0, γ3 = … = γa = 0, and
. )
(Y V 0I 1D1 2D2
Var v = =γ +γ +γ Then γ0 > γ1 + γ2, and
2 1 3 2 2 1 1 0 1 D D m D m D m I m NV− = + + + , where ) ( ) ( ) ( ) ( 2 1 2 0 2 2 2 2 2 0 2 2 2 1 2 1 2 2 2 1 2 0 2 0 2 2 2 1 2 0 0 0 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − − + − − + − − − − = N m , ) ( ) ( ) ( ) ( 2 1 2 0 2 2 2 2 2 0 2 2 2 1 2 1 2 2 2 1 2 0 2 0 2 0 2 2 2 1 1 1 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − − + − − + − − − − = N m , ) ( ) ( ) ( ) ( 2 1 2 0 2 2 2 2 2 0 2 2 2 1 2 1 2 2 2 1 2 0 2 0 2 1 2 0 2 2 2 2 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − − + − − + − − − − = N m , ) ( ) ( ) ( 2 2 1 2 0 2 2 2 2 2 0 2 2 2 1 2 1 2 2 2 1 2 0 2 0 2 1 0 3 γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ − − + − − + − − = N m .
The information matrix M =(mij), i,j=0,L,n, for the estimation of βv again can be partitioned as
, 22 ' 12 12 11 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = M M M M M where ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 2 3 0 1 2 1 0 11 m m m m m m m m m M ;
M12 is a 3 × (n−2) matrix and for i = 0, 1, 2, j = 3, … , n, m0j =m3, m1j =m2, and
1
2 m
m j = if F1F2Fj is a word in the defining relation, otherwise mij =0; M22 is a
) 2 ( ) 2
(n− × n− matrix whose diagonal elements are m0 and off-diagonal elements mij,
i ≠ j = 3, … , n, is ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = otherwise. , 0 relation, defining in the word a is if , relation, defining in the word a is if , relation, defining in the word a is if , 2 1 3 2 2 1 1 j i j i j i ij F F F F m F F F m F F F m m
Some characteristics concerning M are listed below.
1. There is at most one j in M12 such that mij ≠0. That is, M12 is either a matrix of
zeroes, or a matrix with exactly one column of form [m3,m2,m1]′ and all the other entries are zeroes.
2. In M22, the number of appearances of mi, i = 1, 2, 3, is at most one in each row and
each column. That is, it is not possible to have two m1’s, two m2’s, or two m3’s in
any row or column.
3. In M22, if mij = m1 (or m2), mik = m2 (or m1), then mjk = m3, 3 ≤ i < j < k ≤ n.
The derivation of M and its characteristics are given in the Appendix .
Example 4.1. A 26III−2 design with generators F5 = F1F3 and F6 = F2F4. The defining
relation and the corresponding information matrix M are 6 5 4 3 2 1 6 4 2 5 3 1F F F F F FF FF FF F I = = = ,
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 2 0 1 2 0 1 0 0 3 2 3 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m M , respectively.
There is no length three word of form F1F2Fj, hence, M12 is a zero matrix. As for M22,
since F1F3F5 is in the defining relation, m35 = m53 = m1, and since F2F4F6 is in the
defining relation, m46 = m64 = m2.
Example 4.2. A 26III−2 design with generators F5 = F1F3 and F6 = F2F3, the defining
relation and the corresponding information matrix M are 6 5 2 1 6 3 2 5 3 1FF F FF FF FF F I = = = , and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 2 3 0 1 0 2 1 0 0 3 2 3 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m m m M , respectively.
There is no length three word of form F1F2Fj, hence, M12 is a zero matrix. As for M22,
since F1F3F5 is in the defining relation, m35 = m53 = m1, and since F2F3F6 is in the
defining relation, m36 = m63 = m2. Now since both F1F3F5 and F2F3F6 are in the
defining relation, F1F2F5F6 is automatically in the defining relation, and hence m56 =
m65 = m3.
Example 4.3. A 6 3
2III− design with generators F4 = F1F2, F5 = F1F3, and F6 = F2F3 ,
the defining relation and the corresponding information matrix M are
6 5 2 1 6 4 3 1 5 4 3 2 6 5 4 6 3 2 5 3 1 4 2 1F F FFF F FF F F F F FF F FFF F FF F F F I = = = = = = = , and
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 2 3 0 1 0 1 2 3 2 1 0 1 0 3 2 2 3 0 1 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m m m m m m m m m M , respectively.
There is one length three word F1F2F4 of form F1F2Fj in the defining relation, hence
there is one column of [m3,m2,m1]′ in M12. The structure of M22 in here is the same as in the previous example.
4.1. Optimal 2n−p fractional factorial designs with two dispersion factors
Same as in section 3, for most of the eigenvalues based optimality criteria, the “optimal” information matrix, M*, if exists, is of the following form
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = −2 0 11 * 0 0 n I m M M ,
that is, the defining relation of the corresponding optimal design does not contain any length three word involving F1 and F2, and length four word involving both F1 and F2.
Let λ1, λ2, and λ3 be the eigenvalues of M11. Then
and , 2 ) ( ) det( 3 2 1 2 0 2 3 2 2 2 1 1 0 1 0 2 0 3 2 1 * m m m m m m m m m m M n n n n − − + − + + + − = ⋅ ⋅ ⋅ =λ λ λ say. , ) 2 ( 2 2 ) ( 3 ) 2 ( ) ( 1 0 0 3 2 1 0 2 3 2 2 2 1 3 0 2 3 2 2 2 1 2 0 1 0 1 3 1 2 1 1 1 * − − − − − − − + = − + + + + − − − − = − + + + = m n m n m m m m m m m m m m m m m n M tr ϕ λ λ λ .
4.2. Efficient resolution IV designs
For resolution IV designs, there is no length three word in the defining relation, and the transformed information matrix MT is thus of the following simpler form
, 0 0 ) ( 22 11 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = IV T M M M where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⊗ = − −2 2 0 ) ( 22 0 0 δ δ n IV I m T I M , ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 3 3 0 m m m m T , and
δ is the number of length four words in the defining relation involving both F1 and F2.
The eigenvalues of M are λ1, λ2, λ3, and m0 −m3 with multiplicities δ, m0 +m3 with multiplicities δ, and m with multiplicities n – 20 δ – 2. Then De and Ae for an
arbitrary design are thus determined,
, 2 1 ) ) / ( 1 ( ) det( ) det( 2 2 2 2 1 2 0 2 1 2 0 3 * δ δ γ γ γ γ γ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = − = = m m M M De and . 2 ) 1 ) / )(( 2 ( 2 1 ) ( ) ( 2 3 0 0 1 1 * δ ϕ δ + − − + − = = −− m m n m M tr M tr Ae
Both De and Ae are decreasing in δ, that is, when there are more length four words
involving both F1 and F2 in the defining relation, the less efficient the corresponding
design is. Also, De is decreasing in γ1 and γ2, that is, the larger the dispersion effects
are, the less efficient the corresponding design is.
Example 4.4. A 26IV−2 design with generators F5 = F1F2F3, and F6 = F1F3F4. The
defining relation and the corresponding information matrix M are 6 5 4 2 6 4 3 1 5 3 2 1F F F FF F F F F F F F I = = = , and
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 3 0 3 0 0 3 2 3 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m M , respectively.
There is one word F1F2F3F5 of form F1F2FiFj in the defining relation, hence, m35 =
m53 = m3, and δ = 1, then , 2 1 2 2 2 2 1 2 0 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = γ γ γ γ γ e D and . 2 ) 1 ) / )(( 4 ( 2 1 2 3 0 0 + − + − = m m m Ae ϕ
Example 4.5. A 26IV−2 design with generators F5 = F1F2F3, and F6 = F1F2F4. The
defining relation and the corresponding information matrix M are I =F1F2F3F5 =F1F2F4F6 =F3F4F5F6, and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 0 3 3 0 3 0 0 3 2 3 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m M , respectively.
There are two length four words, F1F2F3F5 and F1F2F4F6, involving both F1 and F2,
hence m35 = m53 = m46 = m64 = m3, and δ = 2, then
, 2 1 2 2 2 2 2 1 2 0 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = γ γ γ γ γ e D and
. 4 ) 1 ) / )(( 4 ( 4 1 2 3 0 0+ − + − = m m m Ae ϕ
This design has more length four words of form F1F2FiFj, it is thus less efficient than
the design in Example 4.3. As one can also see from the values of De and Ae of the
design in Example 4.4, both of them are smaller than those in Example 4.3.
Remark：Resolution V or higher designs are robust against two dispersion effects, if our interest is focused on estimating location main effects.
4.3. Efficient resolution III designs
Efficiencies of resolution III designs depend not only on the values of γ0, γ1 and
γ2, but also on the number of length three words of forms F1F2Fj, F1FiFj, F2FiFj, and
the number of length four words of form F1F2FiFj in the defining relation. Due to the
complexity in calculating the efficiencies of an arbitrary design, we focus on investigating the D-efficiency of designs when γ1 = γ2 = γ, say. Then m1 =m2 =m, say, and ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 3 3 0 m m m m T , , 0 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = m m m m Q ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 3 0 0 11 m m m m m m m m m M . Now, let ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 0 3 3 0 3 0 m m m m m m m m m m m m m m m m U ,
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⊗ ⊗ ⊗ = ⊗ 5 4 3 11 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 δ δ δ δ δ I m Q I T I I U I M M T ,
δ1 is the number of U matrices in MT, δ2 is the number of M11 matrices in MT, and so
on. δ1, δ2, δ3, and δ4 are functions of the number of words of forms F1F2Fj, F1FiFj,
F2FiFj, and F1F2FiFj in the defining relation, where δ1 + δ2 ≥ 1 and
1 2 2 3 4δ1+ δ2 + δ3+ δ4+δ5 =n+ . Then , ) ( ) ( ) ( )) (det( )) (det( )) (det( )) (det( )) (det( ) det( 5 4 3 3 1 2 1 5 4 3 2 1 0 2 2 0 3 0 3 0 11 0 11 δ δ δ δ δ δ δ δ δ δ δ δ m m m m m m m M m Q T M U M − + − = = + + where 02 2 2 3 2 0 , det( ) ) det(T =m −m Q =m −m , 3 2 2 3 2 2 0 0 11) ( 2 ) 2 det(M =m m − m −m + m m ,
det(U)=(m0 −m3)det(M11), and De can thus be determined
. ) ( ) ( ) ( )) (det( det det 2 0 2 2 0 3 0 3 0 1 11 * 5 4 3 3 1 2 1 − − + − + − + − = = δ δ nδδδ δ δ e m m m m m m m M M M D
Example 4.1. (Continued) For this 26III−2 design, δ1 = 0, δ2 = 1, δ3 = 0, δ4 = 2, and δ5
= 0. Then det(M)=(det(M11))(m02 −m2)2, and
2 2 0) ) / ( 1 ( m m De = − .
Example 4.2. (Continued) For this 26III−2 design, δ1 = 0, δ2 = 2, δ3 = δ4 = 0, and δ5 = 1.
Then 0 2 11)) (det( ) det(M = M m , and 3 0 3 2 2 3 2 2 0 0( 2 ) 2 m m m m m m m De= − − + .
Example 4.6. A 26III−2 design with generators F5 = F1F3, and F6 = F2F3F4. The
6 5 4 2 1 6 4 3 2 5 3 1FF F FF F FF F F F F I = = = , and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m M , respectively.
For this design δ1 = 0, δ2 = 1, δ3 = 0, δ4 = 1, and δ5 = 2. Then
2 0 2 2 0 11))( ) (det( ) det(M = M m −m m , and De =1−(m/m0)2.
Example 4.7. A 26III−2 design with generators F5 = F1F2, and F6 = F2F3F4. The
defining relation and the corresponding information matrix M are 6 5 4 3 1 6 4 3 2 5 2 1F F F F F F FFF F F F I = = = , and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 3 0 0 0 3 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m m m M , respectively.
For this design δ1 = 1, δ2 = δ3 = δ4 = 0, and δ5 = 3. Then
3 0 3 0 11))( ) (det( ) det(M = M m −m m , and ) 2 ) 2 ( ( ) ( 2 4 ) 2 ( ) 2 ( 3 2 2 3 2 2 0 0 0 3 0 2 3 0 2 2 0 2 2 3 2 3 2 3 2 2 0 2 0 m m m m m m m m m m m m m m m m m m m m m De + − − − − + − + + − − = .
Through some straightforward calculations, one can see that the De of the design in
Example 4.6 is the highest, the design in Example 4.1 is the second highest, and then the design in Example 4.2. The De of the design in Example 4.7 is the lowest among
The rankings, according to the values of De, of the above four
2 6
2III− designs with different generators are not hard to obtain. In the beginning, we think that values of
δ1, … , δ5 determine the “structure” of the defining relation, hence the design. If we
can find an ordering of all possible structures, then the corresponding D-better designs can be determined regardless of the values of γ0 and γ. However, after having been
extensively investigating many designs, we realize that it is generally not true. Example 4.7 above and Example 4.8 below show how values of γ0 and γ affect the
values of De of two designs with fixed defining relations.
Example 4.8. A 26III−2 design with generators F5 = F1F3, and F6 = F1F2F4. The
defining relation and the corresponding information matrix M are 6 5 4 3 2 1 6 4 2 5 3 1FF F F F FF FF F F F I = = = , and ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 3 0 3 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m m m m m m m m m m m m m m m M , respectively.
For this design δ1 = 0, δ2 = 1, δ3 = δ4 = 1, and δ5 = 0. Then
) )( ))( (det( ) det(M = M11 m02−m2 m02−m32 , and 2 2 0 3 2 0 3 2 0) ( / ) ( / ) / ( 1 m m m m mm m De = − − + .
When γ0 /γ > 2.590901, the De of this design is larger than the De of the design in
Example 4.7, that is, Example 4.8 is D-better than Example 4.7. Whereas, when γ0 /γ ≤
2.590901, the De of this design is smaller, that is, Example 4.7 is D-better.
Although the ordering of the De of p n III
−
2 fractional factorial designs varies when
value of De if δ1 is small and δ5 is large.
Based on the catalogues of the 16-run and 32-run 2n−p fractional factorial designs in Chen, Sun, and Wu (1993), we provide tables of the factors that we suggest to be named as the factors that are responsible for the dispersion effects, such that the D-efficiency of the resulting designs are higher. For example, design 6-2.2 in Table 4.1 with generators F5 = F1F2, and F6 = F1F3F4, if we assign F3 and F4, or F3 and F6,
or F4 and F6 as the factors that are responsible for the dispersion effects, the De of the
resulting designs are higher or even the highest depending on the values of γ0 and γ.
The designs with bold faced dispersion factors are D-optimal designs, regardless of the values of γ0 and γ.
There are cases when De depends on the size of the dispersion effects. For
example, design 9-4.7 in Table 4.2, when γ≥ γ0 / 6 , that is, the dispersion effects are
relatively large, the De of the designs when dispersion factors are named as (F2,F3),
(F2,F4), or (F6,F9) are higher; and when γ < γ0 / 6 , that is, the dispersion effects are
relatively moderate, the De of the designs when dispersion factors are named as
(F3,F4), (F3,F5), or (F8,F9) are higher. These factors with moderate dispersion effects
Table 4.1. Suggested dispersion factors for 16-run 2nIII−p fractional factorial designs.
Design Generators Dispersion Factors
5-1.3 F5=F1F2 (F3,F4) 6-2.2 F5=F1F2, F6=F1F3F4 (F3,F4), (F3,F6), (F4,F6) 6-2.3 F5=F1F2, F6=F3F4 (F1,F2), (F1,F5), (F2,F5), (F3,F4), (F3,F6), (F4,F6) 6-2.4 F5=F1F2, F6=F1F3 (F2,F4), (F3,F4), (F4,F5), (F4,F6) 7-3.2 F5=F1F2, F6=F1F3, F7=F2F3F4 (F4,F7) 7-3.3 F5=F1F2, F6=F1F3, F7=F2F4 (F3,F4), (F3,F7), (F4,F6), (F6,F7) 7-3.4 F5=F1F2, F6=F1F3, F7=F1F4 (F2,F3), (F2,F4), (F2,F6), (F2,F7), (F3,F4), (F3,F5), (F3,F7), (F4,F5), (F4,F6), (F5,F6), (F5,F7), (F6,F7) 7-3.5 F5=F1F2, F6=F1F3, F7=F2F3 (F1,F4), (F2,F4), (F3,F4), (F4,F5), (F4,F6), (F4,F7) 8-4.2 F5=F1F2, F6=F1F3, F7=F1F4, F8=F2F3F4 (F2,F8), (F3,F8), (F4,F8), (F5,F8), (F6,F8), (F7,F8) 8-4.3 F5=F1F2, F6=F1F3, F7=F2F4, F8=F3F4 (F5,F8), (F6,F7) 8-4.4 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3F4 (F4,F8) 8-4.5 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4 (F4,F7), (F4,F8), (F7,F8) 8-4.6 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3 (F1,F4), (F2,F4), (F3,F4), (F4,F5), (F4,F6), (F4,F7), (F4,F8) 9-5.1 F5=F1F2, F6=F1F3, F7=F1F4, F8=F2F3F4, F9=F1F2F3F4 (F2,F3), (F2,F4), (F2,F5), (F2,F6), (F2,F7), (F2,F8), (F2,F9), (F3,F4), (F3,F5), (F3,F6), (F3,F7), (F3,F8), (F3,F9), (F4,F5), (F4,F6), (F4,F7), (F4,F8), (F4,F9), (F5,F6), (F5,F7), (F5,F8), (F5,F9), (F6,F7), (F6,F8), (F6,F9), (F7,F8), (F7,F9), (F8,F9) 9-5.2 F5=F1F2, F6=F1F3, F7=F2F4, F8=F3F4, F9=F1F2F3F4 (F1,F2), (F1,F3), (F1,F5), (F1,F6), (F2,F4), (F2,F5), (F2,F7), (F3,F4), (F3,F6), (F3,F8), (F4,F7), (F4,F8), (F5,F8), (F5,F9), (F6,F7), (F6,F9), (F8,F9) 9-5.3 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F3F4 (F8,F9) 9-5.4 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F4 (F3,F8), (F3,F9), (F4,F6), (F4,F7), (F6,F9), (F7,F8) 9-5.5 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3, F9=F1F4 (F4,F9) 10-6.1 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4 (F7,F9), (F7,F10), (F9,F10) 10-6.2 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F4, F10=F1F3F4 (F3,F4), (F3,F10), (F4,F10) 10-6.3 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F4, F10=F3F4 (F1,F7), (F1,F9), (F1,F10), (F2,F6), (F2,F8), (F2,F10), (F3,F5), (F3,F8), (F3,F9), (F4,F5), (F4,F6), (F4,F7), (F5,F10), (F6,F9), (F7,F8) 10-6.4 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3, F9=F1F4, F10=F2F4 (F4,F9), (F4,F10), (F9,F10) 11-7.1 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F4, F10=F1F3F4, F11= F2F3F4 (F1,F2), (F1,F6), (F1,F8), (F1,F11), (F2,F7), (F2,F9), (F2,F10), (F6,F7), (F6,F9), (F6,F10), (F7,F8), (F7,F11), (F8,F9), (F8,F10), (F9,F11), (F10,F11)
Table 4.1. (Continued)
Design Generators Dispersion Factors
11-7.2 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3, F9=F1F4, F10=F2F4, F11= F3F4 (F4,F8), (F4,F9), (F4,F10), (F4,F11), (F8,F9), (F8,F10), (F8,F11), (F9,F10), (F9,F11), (F10,F11) 11-7.3 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3, F9=F1F4, F10=F2F4, F11= F1F2F4 (F3,F4), (F3,F6), (F3,F7), (F3,F8), (F3,F9), (F3,F10), (F3,F11), (F4,F6), (F4,F7), (F4,F8), (F4,F9), (F4,F10), (F4,F11), (F6,F7), (F6,F8), (F6,F9), (F6,F10), (F6,F11), (F7,F8), (F7,F9), (F7,F10), (F7,F11), (F8,F9), (F8,F10), (F8,F11), (F9,F10), (F9,F11), (F10,F11) 12-8.1 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F4, F9=F2F4, F10=F1F3F4, F11= F2F3F4, F12=F1F2F3F4 (F1,F2), (F1,F3), (F1,F4), (F1,F5), (F1,F6), (F1,F7), (F1,F8), (F1,F9), (F1,F10), (F1,F11), (F1,F12), (F2,F3), (F2,F4), (F2,F5), (F2,F6), (F2,F7), (F2,F9), (F2,F10), (F2,F11), (F2,F12), (F3,F4), (F3,F5), (F3,F6), (F3,F7), (F3,F8), (F3,F9), (F3,F10), (F3,F11), (F3,F12), (F4,F5), (F4,F6), (F4,F7), (F4,F8), (F4,F9), (F4,F10), (F4,F11), (F4,F12), (F5,F6), (F5,F7), (F5,F8), (F5,F9), (F5,F10), (F5,F11), (F5,F12), (F6,F7), (F6,F9), (F6,F10), (F6,F11), (F6,F12), (F7,F8), (F7,F9), (F7,F10), (F7,F11), (F7,F12), (F8,F9), (F8,F10), (F8,F11), (F8,F12), (F9,F10), (F9,F11), (F9,F12), (F10,F11), (F10,F12), (F11,F12) 12-8.2 F5=F1F2, F6=F1F3, F7=F2F3, F8=F1F2F3, F9=F1F4, F10=F2F4, F11= F1F2F4, F12=F3F4 (F3,F4), (F3,F9), (F3,F10), (F3,F11), (F3,F12), (F4,F6), (F4,F7), (F4,F8), (F4,F12), (F6,F9), (F6,F10), (F6,F11), (F6,F12), (F7,F9), (F7,F10), (F7,F11), (F7,F12), (F8,F9), (F8,F10), (F8,F11), (F8,F12), (F9,F12), (F10,F12), (F11,F12)
Table 4.2. Suggested dispersion factors for 32-run 2nIII−p fractional factorial designs
Design Generators Dispersion Factors
7-2.4 F6=F1F2, F7=F1F3F4F5 (F3,F4), (F3,F5), (F3,F7), (F4,F5), (F4,F7), (F5,F7) 7-2.5 F6=F1F2, F7=F3F4F5 (F3,F4), (F3,F5), (F3,F7), (F4,F5), (F4,F7), (F5,F7) 7-2.6 F6=F1F2, F7=F1F3F4 (F3,F5), (F4,F5), (F5,F7) 7-2.7 F6=F1F2, F7=F3F4 (F1,F5), (F2,F5), (F3,F5), (F4,F5), (F5,F6), (F5,F7) 7-2.8 F6=F1F2, F7=F1F3 (F4,F5) 8-3.5 F6=F1F2, F7=F1F3F4, F8=F2F3F5 (F4,F5), (F4,F8), (F5,F7), (F7,F8) 8-3.6 F6=F1F2, F7=F1F3, F8=F2F3F4F5 (F4,F5), (F4,F8), (F5,F8) 8-3.7 F6=F1F2, F7=F1F3F4, F8=F1F3F5 (F3,F4), (F3,F5), (F3,F7), (F3,F8), (F4,F5), (F4,F8), (F5,F7), (F7,F8) 8-3.8 F6=F1F2, F7=F3F4, F8=F1F3F5 (F5,F8) 8-3.9 F6=F1F2, F7=F1F3, F8=F2F4F5 (F4,F5), (F4,F8), (F5,F8) 8-3.10 F6=F1F2, F7=F1F3, F8=F1F4F5 (F4,F5), (F4,F8), (F5,F8) 9-4.6 F6=F1F2, F7=F1F3F4, F8=F1F2F5, F9=F2F4F5 (F3,F9) 9-4.7 F6=F1F2, F7=F1F3F4, F8=F1F3F5, F9=F1F4F5 (F2,F3), (F2,F4), (F2,F5), (F2,F7), (F2,F8), (F2,F9), (F3,F6), (F4,F6), (F5,F6), (F6,F7), (F6,F8), (F6,F9) (F3,F4), (F3,F5), (F3,F7), (F3,F8), (F3,F9), (F4,F5), (F4,F7), (F4,F8), (F4,F9), (F5,F7), (F5,F8), (F5,F9), (F7,F8), (F7,F9), (F8,F9) 9-4.8 F6=F1F2, F7=F3F4, F8=F1F3F5, F9=F2F4F5 (F5,F8), (F5,F9), (F8,F9) 9-4.9 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4F5 (F5,F9) 9-4.10 F6=F1F2, F7=F1F3, F8=F2F4, F9=F3F4F5 (F5,F9) 10-5.5 F6=F1F2, F7=F1F3F4, F8=F1F3F5, F9=F1F4F5, F10=F3F4F5 (F1,F2), (F1,F6), (F2,F3), (F2,F4), (F2,F5), (F2,F6), (F2,F7), (F2,F8), (F2,F9), (F2,F10), (F3,F4), (F3,F5), (F3,F6), (F3,F7), (F3,F8), (F3,F9), (F3,F10), (F4,F5), (F4,F6), (F4,F7), (F4,F8), (F4,F9), (F4,F10), (F5,F6), (F5,F7), (F5,F8), (F5,F9), (F5,F10), (F6,F7), (F6,F8), (F6,F9), (F6,F10), (F7,F8), (F7,F9), (F7,F10), (F8,F9), (F8,F10), (F9,F10) 10-5.6 F6=F1F2, F7=F1F3F4, F8=F1F3F5, F9=F1F4F5, F10=F2F3F4F5 (F3,F10), (F4,F10), (F5,F10), (F7,F10), (F8,F10), (F9,F10) 10-5.7 F6=F1F2, F7=F1F3F4, F8=F1F3F5, F9=F2F4F5, F10=F1F2F3F4F5 (F1,F2), (F1,F6), (F2,F6), (F3,F4), (F3,F5), (F3,F7), (F4,F7) (F5,F8), (F5,F9), (F5,F10), (F8,F9), (F8,F10), (F9,F10) 10-5.8 F6=F1F2, F7=F3F4, F8=F2F3F4, F9=F2F3F5, F10=F1F4F5 (F4,F9), (F4,F10), (F5,F8), (F5,F10), (F8,F10), (F9,F10) 10-5.9 F6=F1F2, F7=F1F3, F8=F2F3F4, F9=F2F3F4F5, F10=F2F4F5 (F4,F5), (F4,F9), (F5,F8), (F8,F9) 10-5.10 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4F5 (F5,F9), (F5,F10), (F9,F10) 11-6.3 F6=F1F2, F7=F1F3, F8=F2F3F4, F9=F2F3F5, F10=F1F4F5, F11=F1F2F3F4F5 (F4,F5), (F4,F9), (F4,F10), (F4,F11), (F5,F8), (F5,F10), (F5,F11), (F8,F9), (F8,F10), (F8,F11), (F9,F10), (F9,F11)
Table 4.2. (Continued)
Design Generators Dispersion Factors
11-6.4 F6=F1F2, F7=F1F3, F8=F2F3F4, F9=F2F3F5, F10=F2F4F5, F11=F1F3F4F5 (F4,F10), (F4,F11), (F5,F10), (F5,F11), (F8,F10), (F8,F11), (F9,F10), (F9,F11), (F10,F11) 11-6.5 F6=F1F2, F7=F1F3, F8=F2F3F4, F9=F2F3F5, F10=F2F4F5, F11=F3F4F5 (F4,F6), (F4,F7), (F5,F6), (F5,F7), (F6,F8), (F6,F9), (F6,F10), (F6,F11), (F7,F8), (F7,F9), (F7,F10), (F7,F11) (F4,F5), (F4,F9), (F4,F10), (F4,F11), (F5,F8), (F5,F10), (F5,F11), (F8,F9), (F8,F10), (F8,F11), (F9,F10), (F9,F11) 11-6.6 F6=F1F2, F7=F1F3, F8=F2F4, F9=F2F3F5, F10=F1F2F4F5, F11=F3F4F5 (F5,F11), (F9,F11) 11-6.7 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F5, F10=F2F4F5, F11=F1F3F4F5 (F5,F9), (F5,F10), (F5,F11), (F9,F10), (F9,F11), (F10,F11) 11-6.8 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F5, F10=F2F4F5, F11=F3F4F5 (F2,F7), (F2,F8), (F3,F6), (F3,F8), (F4,F6), (F4,F7) (F5,F9), (F5,F10), (F5,F11), (F9,F10), (F9,F11), (F10,F11) 11-6.9 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F3F5, F11=F2F4F5 (F5,F9), (F9,F10), (F9,F11) 11-6.10 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F5, F11=F1F3F4F5 (F9,F11) 12-7.3 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F3F5, F11=F2F4F5, F12=F1F3F4F5 (F5,F9), (F5,F10), (F5,F11), (F5,F12), (F9,F10), (F9,F11), (F9,F12), (F10,F11), (F10,F12), (F11,F12) 12-7.4 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F3F5, F11=F2F4F5, F12=F3F4F5 (F5,F9), (F9,F10), (F9,F11), (F9,F12) 12-7.5 F6=F1F2, F7=F1F3, F8=F2F4, F9=F2F3F4, F10=F2F3F5, F11=F1F2F4F5, F12=F1F3F4F5 (F5,F11), (F5,F12), (F10,F11), (F10,F12) 12-7.6 F6=F1F2, F7=F1F3, F8=F2F4, F9=F2F3F4, F10=F2F3F5, F11=F1F4F5, F12=F1F2F3F4F5 (F5,F12), (F10,F11) 12-7.7 F6=F1F2, F7=F1F3, F8=F2F3, F9=F1F2F3F4, F10=F1F2F3F5, F11=F1F4F5, F12=F2F3F4F5 (F4,F5), (F4,F10), (F4,F11), (F4,F12), (F5,F9), (F5,F11), (F5,F12), (F9,F10), (F9,F11), (F9,F12), (F10,F11), (F10,F12) 12-7.8 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F5, F11=F2F3F5, F12=F2F4F5 (F9,F11), (F9,F12), (F11,F12) 12-7.9 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F2F3F5, F12=F2F4F5 (F2,F5), (F2,F11), (F2,F12), (F3,F5), (F3,F11), (F3,F12), (F4,F5), (F4,F11), (F4,F12), (F5,F6), (F5,F7), (F5,F8), (F5,F9), (F5,F10), (F6,F11), (F6,F12), (F7,F11), (F7,F12), (F9,F11), (F9,F12), (F10,F11), (F10,F12) (F5,F11), (F5,F12), (F11,F12) 12-7.10 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F5, F11=F3F5, F12=F1F2F3F4F5 (F9,F12) 13-8.2 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F5, F11=F2F3F5, F12=F2F4F5, F13=F3F4F5 (F9,F11), (F9,F12) (F9,F13), (F11,F12), (F11,F13), (F12,F13)
Table 4.2. (Continued)
Design Generators Dispersion Factors
13-8.3 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F2F3F5, F12=F2F4F5, F13=F1F3F4F5 (F5,F11), (F5,F12), (F5,F13), (F11,F12), (F11,F13), (F12,F13) 13-8.4 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F2F3F5, F12=F2F4F5, F13=F3F4F5 (F2,F5), (F2,F11), (F2,F12), (F2,F13), (F3,F5), (F3,F11), (F3,F12), (F3,F13), (F4,F5), (F4,F11), (F4,F12), (F4,F13), (F5,F6), (F5,F7), (F5,F8), (F5,F9), (F5,F10), (F6,F11), (F6,F12), (F6,F13), (F7,F11), (F7,F12), (F7,F13), (F8,F11), (F8,F12), (F8,F13), (F9,F11), (F9,F12), (F9,F13), (F10,F11), (F10,F12), (F10,F13) (F5,F11), (F5,F12), (F5,F13), (F11,F12), (F11,F13), (F12,F13) 13-8.5 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F1F5, F12=F2F4F5, F13=F3F4F5 (F12,F13) 13-8.6 F6=F1F2, F7=F1F3, F8=F2F4, F9=F3F4, F10=F1F2F3F4, F11=F2F3F5, F12=F1F2F4F5, F13=F1F3F4F5 (F5,F11), (F5,F12), (F5,F13), (F11,F12), (F11,F13), (F12,F13) 13-8.7 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F2F5, F11=F3F5, F12=F4F5, F13=F1F2F3F4F5 (F9,F13) 13-8.8 F6=F1F2, F7=F1F3, F8=F1F4, F9=F1F2F3F4, F10=F2F5, F11=F3F5, F12=F4F5, F13=F2F3F4F5 (F6,F13), (F8,F13) (F9,F13) 13-8.9 F6=F1F2, F7=F1F3, F8=F1F4, F9=F1F2F3F4, F10=F2F5, F11=F3F5, F12=F4F5, F13=F1F2F3F4F5 (F8,F13) 13-8.10 F6=F1F2, F7=F1F3, F8=F2F3, F9=F1F4, F10=F2F3F4, F11=F1F5, F12=F2F4F5, F13=F1F3F4F5 (F12,F13) 14-9.2 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F1F5, F12=F2F3F5, F13=F2F4F5, F14=F3F4F5 (F12,F13), (F12,F14), (F13,F14) 14-9.3 F6=F1F2, F7=F1F3, F8=F1F4, F9=F2F3F4, F10=F1F2F3F4, F11=F1F5, F12=F2F3F5, F13=F1F2F3F5, F14=F2F4F5 (F2,F14), (F3,F14), (F4,F14), (F5,F14), (F6,F14), (F7,F14), (F8,F14), (F9,F14), (F10,F14), (F11,F14), (F12,F14), (F13,F14) 14-9.4 F6=F1F2, F7=F1F3, F8=F1F4, F9=F1F2F3F4, F10=F2F5, F11=F3F5, F12=F4F5, F13=F2F3F4F5, F14=F1F2F3F4F5 (F6,F7), (F6,F8), (F6,F9), (F6,F10), (F6,F11), (F6,F12), (F6,F13), (F7,F8), (F7,F9), (F7,F10), (F7,F11), (F7,F12), (F7,F13), (F8,F9), (F8,F10), (F8,F11), (F8,F12), (F8,F13), (F9,F10), (F9,F11), (F9,F12), (F9,F13), (F10,F11), (F10,F12), (F10,F13), (F11,F12), (F11,F13), (F12,F13)
