数理解析研究所講究録
第 853 巻 1993 年 162-175

162

SOME NON-ALIASING RELATIONSHIP FOR SECOND-ORDER MODEL

Masahide KUWADA (田正秀)

Faculty of Integrated Arts and Sciences
Hiroshima University

ABSTRACT: We consider the second-order model based on a design which is derived from abalanced array of strength 4 and 3 symbols. In this model, when the information matrix of adesign is singular, we present some non-aliasing relationship among the factorial effects not to be

negligible.

1. Introduction
In a practical experimentation, the most interesting factorial effects are the main effects, the next

are the two-factor interactions, and so on.

Thus the experimenter wantto carry out theexperimentation such that the main effects are not confounded (or aliased) with each other, andfurthermore that if they are confounded with some effects, then these are possibly higher orderinteractions which may be negligible.

In a (fractional) factorial experiment, the aliasing (orconfounding) relationship among the factorial ($and/or$ block) effects has been studied as thedefining relationship ( $e.g.$ , Finney [2]). Tlie extended concept of resolution for$2^{m}$

factorials $(e.g.$ ,Yamamoto $and/or$ Hyodo $[5,13]$ ) and balanced $Rac0onal2^{m}$ factorial ( -BFF) designs of even$2^{m}$

resolution (e.g., Shirakura [9,10]) can be regarded as the aliasing relationship in a certain sense.
The characteristic polynomial of the information matnix for the second-order model and theeconomical second-order designs of$3^{m}$

factorials were presented by Hoke $[3,4]$ . The second-ordermodel based on 3’ factorials contains the general mean, the linear and the quadratic components of

163
the main effects and the linear by linear ones of the two-fador interactions.

Under someconditions, a balanced array (B-array) yields a balanced dcsign ( $e.g.$ , Kuwada [7]). By using thealgebraic structure of the multidimensional relationship, Kuwada [8] obtained an explicit expression
for the characteristic polynomial of the infonnation matrix of$3^{m}$-BFF designs of resolution V

derived from B-arrays of strength 4. The inversion of the infornation matrix of$3^{m}$-BFF designs ofresolution V was presented by Srivastava and Ariyaratna [11] using the another technique. Optimal$3^{m}$-BFF designs of resolution V were independently obtained by Ariyaratna [1] and Kuwada [6].

An expression for the trace of the variance-covariance matrix of a balanced $(2,0)$-symmetric designof resolution V for$3^{m}$

factorials was also obtained by Srivastava and Chopra [12].

In this paper, attention is focused on finding some non-aliasing relationship for the second-order model when the infornation matrix of a$3^{m}$-BFF design derived from a B-array of strength 4

is singular. In this situation, there are three cases: (A) the general mean and all main effects are
estimable and are not $\infty nfounded$ with the two-factor interactions, (B) all main effects are estimableand are not confounded with the general mean and the two-factor interactions, (C) the linear
components of the main effects are estimable and are not $\infty nfounded$ with the general mean, thequadratic ones of the main effects and the two-factor interactions.

2. Preliminaries
Let$\Theta_{0}$and $\Theta_{1}beann_{0}xlve\alpha orofthefa\alpha orialeffe\alpha stobe\propto timatedandann_{1}$ xl one not ofinterest and not assumed to be known, respectively, in the absence of the remaining factorial
effects. Further let $y(T)$ be a vector of$N$

observations based on a fraction Twith$m(\geqq 4)$

factors.

Then the linear model may be written as$\mathcal{E}[y(T)]=E_{0}\Theta 0+E_{1}\Theta_{1}$and$Var[j(T)]=0^{2}h$ ,(2.1)

where $E_{i}(i=0,1)$ are Nxn submatrices of the design matrix $[E_{0}; E_{1}]$ ( $=E\tau$, say). Here$i$

$Var[y]$denote the expected value and the variance-covariance matrix of a random vector

respectively, and($=\Theta’$$\epsilon[y]$$I_{p}$

is the identity matrix of order$p$. The normal equation for estimating

and
$y$

,

$(\Theta 0’;\Theta_{1’})$, say) is given byMoo$\hat{\Theta}0+M_{01}\hat{\Theta}_{1}=E_{0’}y(T)$and$M_{10}\hat{\Theta}0+M_{11}\hat{\Theta}_{1}=E_{1’}y(T)$

,

(2.2)

164
where $M_{ij}=E_{i’}E_{j}(i,j=0,1)$. Throughout this paper, we assume that Moo is nonsingular because wewant at least to estimate$e_{0}$

Then it follows ffom (2.2) that$\hat{\Theta}_{0}=M\omega^{- 1}E_{0’}y(T)_{-M\alpha\}^{- 1}}M_{01}\hat{\Theta}_{1}$andif $\det(M_{11}-M_{10}M\alpha)^{- 1}M_{01})\neq 0$ ,$\hat{\Theta}_{1}=(0\alpha’ M_{01})^{-1}(E_{1’}-M_{10}Mw^{- 1}E_{0’})y(T)$$(M_{11}$-Mio$M_{\alpha\}^{- 1}}M_{01})^{g}(\alpha)^{- 1}$$+\{I_{n_{1}}-(M_{11}-M_{1}oM\mathfrak{w}^{- 1}M_{01})^{g}(M_{11}-M_{10}Mm^{- 1}M_{01})\}z$

where $\det(A)$ and$i.e.,$ $AA^{g}A=A$

then

and

$\Theta 0$$A^{g}$

denote the determinant of a matrix, respectively, and$\Theta_{1}$

$z$

$A$

if $\det(M_{11}-M_{10}M\alpha)^{- 1}M_{01})=0$ ,and a generalized inverse of a matrix $A$

is an arbitrary vector of size$n_{1}x1$. If$\det(M_{11}-M_{10}M\infty^{- 1}M_{01})\neq 0$,

can be estimated separately. Thus in this paper, we consider the situation inwhich $\det(M_{11}-M_{10}M\infty^{- 1}M_{01})=0$ . Then we get$\hat{\Theta}0=M_{\alpha)^{- 1}}E_{0’}y(T)-M\alpha)^{- 1}M_{01}(M_{11}-M_{10}M_{(n^{- 1}M_{01})^{g}(\alpha\}^{- 1}}E_{1}’-M_{10}ME_{0’})y(T)$,$-M_{\alpha)^{- 1}}M_{01}\{I_{n_{1}}-(M_{11}-M_{10}M_{0^{- 1}}M_{01})^{g}(M_{11}-M_{10}M\alpha 1^{- 1}M_{01})\}z$(2.3a)$\hat{\Theta}_{1}=(M_{11}-M_{10}M_{0^{- 1}}M_{0I})^{g}(\alpha)^{- 1}$,$+\{I_{n_{1}}-(M_{11}-M_{1}oMm^{- 1}M_{01})^{g}(M_{11}-M_{10}M\omega^{- 1}M_{01})\}z$(2.3b)and hence$\mathcal{E}[\hat{\Theta}0]=\Theta 0+M_{\alpha)^{- 1}}M_{01}\{I_{n_{1}}-(\alpha\}^{- 1}$$\mathcal{E}[\hat{\Theta}_{1}]=(M_{11}-M_{10}Mm^{- 1}M_{01})^{g}(M_{11}-M_{10}M\infty^{- 1}M_{01})\Theta_{1}$$+\{I_{D_{1}}-(M_{11}-M_{1}oM_{tX^{-1}},M_{01})^{g}(M_{11}-M_{1}oM\alpha 1^{- 1}M_{01})\}z$.

Therefore under $\det(M_{0})\neq 0$ and ($\det(M_{11}-M_{10}M_{0^{- 1}}M_{01})=0$, a necessary and sufficient condition
for$\Theta 0$

to be estimable and not to be confounded with$\Theta_{1}$is that$M\omega^{- 1}M_{01}\{I_{n_{1}}-(0\alpha’ M_{01})^{g}(M_{I1}-M_{10}M\infty^{- 1}M_{01})\}=0_{I}b^{xn_{1}}$’and hence

Moi
where

$\{I_{n_{1}}-(M_{11}-M_{1}oMw^{- 1}M_{01})^{g}(M_{11}-M_{10}M\alpha I^{-.1}M_{01})\}=0_{I}\dagger^{xn_{1}}$$0_{T}$

xq denotes the

(2.4)

’

pxq matrix with all zero. Note that under (2.4), we have

$Var[\hat{\Theta}0]=0^{2}\{Mm^{- 1}+Mw^{- 1}M_{01}(M_{11}-M_{10}M\alpha’- 1M_{01})^{g}M_{10}M\infty^{- 1}\}$.

The following lemmas can easily be proved.

Izmma2.1. Let$(\begin{array}{ll}a bb c\end{array})$( $=A$, say) be apositive semi-definite matrix with $ac=b^{2}$. Then we have165
$A^{g}=$

if a$(\begin{array}{lll}1/ a 00 0\end{array})$,

if $c\neq 0$ ,$(\begin{array}{lll}0 0 0 1/ c\end{array})$

ifa $=c=0$ .

$(\begin{array}{ll}0 00 0\end{array})$

Ixmma2.2.

$\not\in 0$( $=A$, say) be a positive semi-definite matrix with $a>0$ and $\det(A)$Let$(\begin{array}{lll}a b cb d ec e f\end{array})$$=adf+2bce-ae^{2}-b^{2}f-c^{2}d=0$. Then we have
if $ad-b^{2}\neq 0$ ,$A^{g}=\{1/(ad-b3\}(\begin{array}{lll}d - b 0- b a 00 0 0\end{array})$

if ad-b2 $=0,$ $af-c^{2}\neq 0$ ,$\{1/(af-c^{2})\}(\begin{array}{lll}f 0 - c0 0 0-C 0 a\end{array})$

if ad-b2 $=af-c^{2}=0$ .

$(\begin{array}{llll}1/ a 0 00 0 00 0 0\end{array})$

3. TMDPB association scheme and its algebraLet($S(a_{1}a_{2})=\{(u_{1}^{a_{1}}u_{2}^{a_{z}})|1\leqq u_{1}<u_{2}\leqq m\}$$=n(a_{1}a_{2})$, say), where$|S|$

defined among the sets

, where$a_{1}a_{2}=00$, 10,01, 11. Then$|S(a_{1}a_{2})|=(_{a+}1)m_{az}$denotes the cardinality of a set S. Suppose a relation of association isin such a way that

$S(a_{I}a_{2})$and$(u_{1}^{a_{1}}u_{2}^{a_{Z}})\in S(a_{1}a_{2})$$(v_{1^{\mathfrak{h}_{1}}}v_{2}^{4})\in S(b_{1}b_{2})$are the -th$\alpha$associates if$| \{u_{1}^{K^{a_{1})}},u_{2}^{\mathfrak{W}}\}\cap\{v_{1}K^{b_{1})\alpha\omega}v_{2}\}|=\min((o(a_{1},a_{2})_{t1}\langle b_{1},b_{2}))-a$

where if$a_{\dot{r}}=0$$v_{j}^{\Phi_{j})}=v_{j})$. Especially, whenmin$(a,b)$ and(or$b_{j}=0$$\omega(a_{I},a_{2})$), then$u_{i}^{q_{a)}}$vanishes (or$a_{1}a_{2}=00$$v_{j}^{q_{b)}}$(or $b_{1}b_{2}=00$ ),,vanishes), and if$a_{i}\neq 0$$\{u_{1}^{q_{8_{1})}},u_{2}^{K9}\}=\{\phi\}$

(or

(or

$b_{j}\neq 0$), then$u_{i}^{q_{d)}}=u_{i}$$\{v_{1}^{\psi_{1})},v_{2}^{\eta))}2\}=\{\phi\}$(or). Heredenote the minimum value of integers a and , and the number of non-zero$b$elements in the vector$(a_{1},a_{2})$, respectively.

The scheme thus defined is called the triangularmultidimensional partially balanced (TMDPB) association scheme (see Yamamoto, Shirakura andKuwada $[14,15]$ ).
Let$A_{\alpha}^{(a_{1}a_{z}b_{1}4J}$

and$D_{\alpha}^{(a_{1}a_{rh^{t}9}}$be the$n(a_{1}a_{2})xn(b_{1}b_{2})$local association matrioes and the $\tau(m)x\tau(m)$ordered association matrices of the TMDPB association scheme, respectively, where $\tau(m)=1+2m$

166$+(m2)$. Further let$A_{\beta}^{*(a_{1}a_{v}b_{1}t_{b}J}$and$D_{\beta^{\#(k^{b_{1}1}b_{-})}}a_{1}$be the matrices such that

$A_{\beta^{\#(a_{1}a_{x}b_{1}43}}(=\{A_{\beta}^{t\phi_{1}4^{a_{1}\psi}}\}’)=\Sigma_{\alpha^{\oint_{(\theta^{A_{\alpha}^{(a_{1}}}}^{\alpha}b^{b_{1}1}bJ}}a_{1}+a_{2}b_{1}+b$for$a_{1}+a_{2}\leqq b_{1}+b_{b}$$D_{\beta^{\#(a_{1}a_{\not\supset}b_{1}4J}}(=\{D_{\beta}^{\varphi_{1}4^{a_{1}\psi}}\}’)=\Sigma_{\alpha^{\oint^{\alpha}}}(a_{1}+a_{2}b_{1}+\iota\theta^{D_{\alpha}^{(a_{1}a_{Z}b_{1}4J}}$

for$a_{1}+a_{2\leqq}b_{1}+b_{2}$

,

where

a

$Z^{\beta\alpha}(a_{1}+a_{2},b_{1}+b_{2})=\phi_{\beta a_{1}+a}\eta_{\alpha}^{(a_{1}+a_{f}b_{1}\star 4J}/\{(mz)(al+\alpha$$z_{\beta\alpha}^{(J}a_{1}+ab_{1}+b,$$zm_{z}- a_{1}- a_{z}b1+b- a^{1}- a^{l}+\alpha$$=\Sigma_{r}()(a+a^{z- rm- a_{1}- a_{2}-\beta+rm- a_{1}- a_{2}-\beta})()\{()(b_{1}+bz-\beta )\}^{1/2}$$/\{(b_{1}+b_{2}- a- a_{2}+r_{)\}}r^{1}$$\phi_{\beta\beta\beta}=()-()$,$xA_{\alpha}^{(a_{1}a_{z}b_{1}4)}$,$\S_{\beta\gamma}$rank($D_{\beta}^{(a_{1}*qc,J}D_{Y}^{(d_{1}tb^{b_{1}}4J}=6c_{1}d_{1}6_{C}\not\simeq b$,$A_{\beta^{\#(a_{1}*qcJ}}A_{Y}^{*(qc_{z}b_{1}19}=6_{\beta\gamma}$$\Sigma_{\beta}A_{\beta^{*\langle a_{1}}}b^{a_{1}a}\theta=I_{n(a_{1}a}p$$D_{\beta^{\#(a_{1}a_{p}b_{1}}}\eta=\phi_{\beta}$

are cited in the following:,$\Sigma_{a_{1}a\geq\beta}D_{\beta}^{\#(a_{1}}4^{a_{1}a}J=I_{\tau(m)}$$(A_{\beta^{*(a_{1}a_{t}b_{1}}}\eta=\phi_{\beta},$,)$D_{\beta}^{t(a_{1}a,b_{1}}b$$D_{a}^{(b_{1}t_{b}a_{1}aJ}=\{D_{\alpha}^{\langle a_{1}a_{p}b_{1}\ddagger 9}\}’$,$=6_{\beta\gamma}6c_{1}d_{1}6c_{2}d_{2}D_{\beta}^{*((a_{1}a_{2},b_{1}4)}$

and$A_{\beta}^{(a_{1}*c_{I}c\dot{)}}A_{Y}^{(qqb_{1}4J}=\Sigma_{\alpha}p(a_{1}a_{2},b_{1}b_{2},a;c_{1}c_{2},\beta,\gamma)$$A_{\alpha}^{(b^{a_{1}a_{2}}J}b_{1}=\{A_{\alpha}^{1^{a_{1}}t^{b_{1}}4)}\}’,$$x\Sigma_{\alpha}p(a_{1}a_{2},b_{1}b_{\underline{7}},a;c_{1}c_{2},\beta,\gamma)D_{\alpha}^{(a_{1}*b_{1}\mathfrak{h})}$

rank

’

$A_{\alpha}^{(a_{1}a_{z}b_{1}t_{bJ}},D_{\alpha}^{(a_{1}*b_{1}\otimes},A_{\beta}^{t(a_{1}*b_{1}|bJ}$$\Sigma_{a_{1}a_{2}}D_{0}^{(a_{1}}t^{a_{1}a_{-},)}=I_{r(m)}$$xA_{\beta}^{t((a_{1}a_{2}b_{1}}\omega$$b_{1}+b_{2}- a_{1}- a_{2}$.

Some properties of$A_{0}^{(a_{1}a_{2},a_{1}a_{2})}=I_{n(a_{1}a_{2})},$$b1+b_{2-}$ al-az

$r$

$r$

where

’

,

$D_{\beta}^{\#(a_{1}a_{f}q\omega_{D_{Y}^{\theta(d}l^{(b^{b_{1}t}9}}}$

,

(3.1)

is the Kronecker‘s delta,$p(a_{1}a_{2},b_{I}b_{\underline{9}},\alpha;c_{1}c_{2},\beta,\gamma)=\Sigma_{k}^{(a,b)r}()()*(a,b)- aa_{1}+a_{2}(a,b)^{*}$$x(z)()b_{1}+b(a,b)^{*}+\alpha m- a1-az- b_{1}b_{2}+(a,b_{*})^{*}-\alpha_{+k}(b,c^{-})^{*}-\gamma- kc_{1}+Cz-(a,c)^{*}+^{-}\beta-(b,c)+\gamma$

Here (a,b)* $= \min(a_{1}+a_{2}, b_{1}+b_{2})$ , (a,c)’ $= \min(a_{1}+a_{2}, c_{1}+c_{2})$ and (b,c)* $= \min(b_{1}+b_{b}c_{1}+c_{2})$ .

Let$\Omega=$[$D_{\beta^{\#(4)}}a_{1}a_{f}b_{1}|a_{1}a_{2},b_{1}b_{2}=$CO,10,01,11,$0 \leqq\beta\leqq\min(a_{1}+a_{2},b_{1}+b_{2})$association algebra generated by the linear closure of twenty six matrices$\Omega_{\beta}=[D_{\beta^{\#(a_{1}a_{2t}b_{1}t9}}|\beta\leqq\min(a_{1}+a_{2},b_{1}+b_{2})]$

Proposition 3.1. (i) The(ii)] which is the TMDPB, and further let$D_{\beta}^{X^{a_{1}a_{Z}b_{1}}9}$

for $\beta=0,1,2$ . Then (3.1) shows the following (see [15]):$\Omega_{\beta}(\beta=0,1,2)$

are the minimal two-sided ideals of , and$\Omega$.

$\Omega_{\beta}\Omega_{Y}=6_{\beta\gamma}\Omega_{\beta}$$7Xe\Omega isdeco1I1posedintot1zedirectsumoftBreeidee1s\Omega_{\beta},$ $i.e.,$ $\Omega=\Omega 0\oplus\Omega_{1}\oplus\Omega_{2}$(iii) Each ideaI$\Omega_{\beta}$

A as$D_{\beta^{K9}}a_{1}a_{l}b_{1}$as its basis and it is isomorphic to the complete $4x4,3x3$ and lxl

matrix algebras with mvltipliciaes$\phi_{\beta}$

for $\beta=0,1,2$, respectively.

167
4. Second-order designs derivcd from B-arrays
Consider a fractional
strength 4 and size$N$$3^{m}$

factorial experiment. Let

having$m$

be a fraction derived from a B-array ofconstraints, 3 symbols and index set

which is written as $BA$( ,m,3,4;
$N$

$T$

$\{\lambda_{i_{0}i_{1}i_{2}}\}$$\{\lambda_{ii}\dot{\sigma}_{12}|i_{0}+i_{1}+i_{2}=4,\dot{k}\geq 0\}$) for brevity. In all our evaluation, we code the threesymbols of a factor as $0,1$ or 2 and employ the standard orthogonal contrasts used in the$3^{m}$

case;viz., $- 1,0,1$ and 1, $- 2,1$ for the linear and the quadratic contrasts, respectively. Then the second-order model for may be written as$T$$\mathcal{E}[y(T)]=E_{T}\Theta$

where$\Theta$and$Var\triangleright(T)$]$=0^{2}I_{N}$,$‘=(\{\Theta(\phi)\};\{\Theta(t^{1})\};\{\Theta(\mathfrak{h}\};\{\Theta(t_{1^{1}}t_{2^{1}})\})$. Here$\Theta(\phi),$$\Theta(t^{1}),$and$\Theta(r\mathfrak{h}$$\Theta(t_{1^{1}}t_{2^{1}})$denote thegeneral mean, the linear and the quadratic components of the main effects of the factor , and the$t$linear by linear components of the two-factor interactions of the factors
where ls sm and$1\leq t_{1}<t_{2}\leq m$$t$say) is isomorphic to$||\kappa_{\beta^{ij}}||$($Ko^{\alpha)}=\gamma_{40\{\nu}\kappa 0^{01}=m^{1/2}\gamma_{310},$$+(m- 1)\gamma_{220},$,$=K_{\beta}$$t_{2}$$\kappa_{0^{02}}=m^{1/2}\gamma_{301},$$\kappa 0^{03}=\{m(m- 1)/2\}^{1/2}\gamma_{220},$$\kappa_{\beta^{ij}}=K\beta^{ji}$,$\kappa 0^{11}=(2\gamma_{4(x)}+\gamma_{301})/3$$Ko^{13}=\{(m- 1)/2\}^{1/2}\{2(2\gamma_{310}+\gamma_{211})/3+(m- 2)\gamma_{130}\},$$\kappa_{0^{23}}=\{(m- 1)/2\}^{1/2}\{2\gamma_{\mathfrak{B}0}+(m- 2)\gamma_{121}\},$$\kappa_{1^{11}}=2\gamma_{4\infty}-\gamma_{301}-\gamma_{\mathfrak{B}2},$$Ko^{22}=2\gamma_{4\infty}$$Ko^{33}=(4\gamma_{4\infty}+4\gamma_{301}+\gamma_{\mathfrak{B}2})/9+2(m- 2)(2\gamma_{220}$$\kappa_{1}^{12}=(m- 2)^{1/2}(\gamma_{\mathfrak{B}0}-\gamma_{121}),$$\kappa_{1}^{02}=(m- 2)^{112}\{(2\gamma_{310}$$\kappa_{1^{22}}=(4_{Y_{4}n}+4\gamma_{301}+\gamma_{\mathfrak{B}}j/9$$+(m- 4)(2\gamma_{220}+\gamma_{121})/3-(m- 3)\gamma_{04\wp}\kappa_{2}^{\alpha}=(4\gamma_{40}+4\gamma_{301}+\gamma_{\mathfrak{U}12})/9-2(2\gamma_{\mathfrak{B}0}+\gamma_{121})/3+\gamma_{040}$

Here$E_{T’}E_{T}(=M_{T}$, say) for $\beta=0,1,2$ , where$+\gamma_{121})/3+\{(m- 2)(m- 3)/2\}\gamma_{04\wp}\kappa_{1}^{\alpha)}=(2\gamma_{4\infty}+\gamma_{301})/3-\gamma_{\infty 1P}\kappa_{1^{01}}=\gamma_{310}-Y_{211}$$+\gamma_{211})/3-\gamma_{I30}\},$

and , respectively,. Then from Proposition 3.1, the information matrix$\kappa 0^{12}=\gamma_{310}+(m- 1)\gamma_{211},$$-\gamma_{301}+(m- 1)\gamma_{\mathfrak{U}12}$$t_{1}$

(4.1)

, and

,

$Y_{400}=\lambda_{400+\lambda_{04}\sigma\vdash\lambda\alpha\}4+4(\lambda_{310}+\lambda\Re 1+\lambda_{130}+\lambda_{031}+\lambda_{103}+\lambda_{013})+6(\lambda_{220}+\lambda\alpha)2+\lambda_{022})+12(\lambda_{211}+\lambda_{121}+\lambda_{112})}$,$Y\alpha 0=\lambda_{4}0\mathfrak{a}+\lambda\alpha)4-4(\lambda_{301}+\lambda_{103})+6\lambda_{\mathfrak{U})2}$$\gamma_{310}=-\lambda_{400+\lambda-3\lambda_{310}-2\lambda_{301}-\lambda_{130}+\lambda_{031}+2\lambda_{103}+3\lambda_{013}-3(\lambda_{22}\triangleright\lambda\circ 22+\lambda_{211}-\lambda_{112})}\alpha!4$,,$\gamma_{301}=\lambda_{40\triangleright 2\lambda_{04}\sigma\{-\lambda\alpha)4+\lambda_{31}\sigma+4\lambda_{301}-5\lambda_{13}0-5\lambda_{031}+4\lambda_{103}+\lambda_{013}-3(\lambda\infty 0-2\lambda_{\mathfrak{U})2}+\lambda_{022}-\lambda_{211}+2\lambda_{121}-\lambda_{112})}$

(4.2)

,

$\gamma_{130}=-\lambda_{4}oo+\lambda_{tX)4}-\lambda_{31}0+2(\lambda_{301}-\lambda_{103})+\lambda_{013}+3(\lambda_{211}-\lambda_{112})$$\gamma_{220}=\lambda_{W0+\lambda\alpha)4+2(\lambda_{310}+\lambda_{013})+\lambda_{22b}2\lambda_{X2}+\lambda\circ 22-2(\lambda_{211}+\lambda_{121}+\lambda_{II2})}$,$\gamma_{202}=\lambda_{40(\rangle}\vdash 4\lambda_{04}0+x_{m4}-2\lambda_{31}\sigma\{- 4(\lambda_{301}+\lambda_{130}+\lambda_{031}+\lambda_{103})-2\lambda_{013}-3(\lambda zo-2\lambda_{\mathfrak{U}12}+\lambda_{022})-6(\lambda_{211}+\lambda_{121}+\lambda_{112})$,

168$\gamma_{211}=-\lambda_{4}oo+\lambda\infty 4-2(\lambda_{301}-\lambda_{13}0+\lambda_{031}-\lambda_{103})+3(\lambda zo-\lambda_{022})$,$\gamma_{121}=\lambda_{40(\}}+\lambda\alpha)4-\lambda_{31}0-\lambda_{013}-2(\lambda_{220}+\lambda_{202}+\lambda_{022})+\lambda_{211}+4\lambda_{121}+\lambda_{112}$(see Kuwada [8]). Thus $\det(M_{T})=0$ if and only if $\det(K_{\beta})=0$ for some $\beta(\beta=0,1,2)$ . Note that thefirst, the second, the third and the last rows and columns of $4x4$ matrix $Ko$ correspond to$\{\Theta(t^{1})\},$$3x3$$\{\Theta(t\mathfrak{J}\}\cdot$one$K_{1}$

and$\{\Theta(t_{1^{1}}t_{2^{1}})\}$

correspond tocorresponds to$\{\Theta(t_{1^{1}}t_{2^{1}})\}$$\{\Theta(\phi)\}$,, respectively, the first, the second and the last rows and columns of$\{\Theta(t^{1})\},$$\{\Theta(\circ\}$and$\{\Theta(t_{1^{1}}t_{2^{1}})\}$, respectively, and the lxl one$K_{2}$.

5. Non-aliasing rclationship for second-order model$A\ddagger$

the beginning, we consider the case (A), $i.e.$ , the general mean and all main effects are

estimable and are not confounded with the two-factor interactions. In this case,$\{\Theta(t^{1})\};\{\Theta(\mathfrak{h}\})$$\{\Theta(\mathfrak{h}\}$, and$M_{1I}$

and$\Theta_{1’}=(\{\Theta(t_{1^{1}}t_{2^{1}})\})$

corresponds toin (2.1). Note that Moo corresponds to$\{\Theta(t_{1^{1}}t_{2^{1}})\}$. Let$K_{\beta}=||K_{\beta}(ij)||$and $K_{1}(00)$ are the first $3x3$ and $2x2$ submatrices of $Ko$ andare the submatrices ofTheorem 5.1.

$K_{\beta}$$\Theta 0’=(\{\Theta(\phi)\}$$\{\Theta(\phi)\},$$\{\Theta(t^{1})\}$;

andfor $\beta=0,1(i,j=0,1)$ , where $Ko(00)$$K_{1}$, respectively, and the remainings

of appropriate size. Then we have the following:

Let $Tbe$ a $BA(N,m,3,4;\{\lambda_{i_{0}i_{1}i_{2}} \})$ with $\det(M_{T})=0$ , then a necessary andsufficient condition for the generat mean and all main effects to be estimable and not to becounfounded with the two-factor interactions is thatfor$\det(K_{Y})=0$

PIoof.

$\gamma=0,1$.

if and only if$\det(K_{\beta}(w))\neq 0$$K_{\beta}(11)-K_{\beta}(10)K_{\beta}(00)^{-1}K_{\beta}(01)$$-K_{\beta}(10)K_{\beta}(00)^{-1}K_{\beta}(01))=0$morphic to. Under

for $\beta=0,1$ and, where$\det(M\infty)\neq 0,$$K_{2}$$K_{\beta}(00)$for$\beta=0,1$$M_{11}-M_{10}M_{0^{- 1}}M_{01}$, and hence

is isomorphic to, and hence $\det(M_{T})=0$ if and only if $\det(K_{\beta}(11)$for some $\beta(\beta=0,1)$ or $K_{2}=0$ . While the left hand side of (2.4) is iso-

$K_{\beta}(01)\{1-(K_{\beta}(11)-K_{\beta}(10)K_{\beta}(00)^{-1}K_{\beta}(01))^{g}(K_{\beta}(11\succ K_{\beta}(10)K_{\beta}(\alpha))^{-1}K_{\beta}(01))\}=K_{\beta}(01)$

if $\det(K_{\beta})=0$ and if $\det(K_{\beta}(00))\neq 0(\beta=0,1),$$\gamma=0,1$

for $\beta=0,1$ and that $K_{Y}(11)=0$ if

It follows from Proposition 3.1 that Mco is isomorphic to$\det(M\alpha))\neq 0$$\neq 0$$\det(K_{\beta}(00))\neq 0$$\theta_{p}=\alpha x1$$\theta_{3}$

if $\det(Ko)\neq 0,$$\theta_{2}$if$\det(K_{1})\neq 0$

and vanish if $\det(K_{2})$Therefore (2.4) implies that $K_{Y}(11)=0$ if $\det(K_{Y})=0$ and if $\det(K_{Y}(00))\neq 0$ for. This completes the proof.

169
Note from (4.1) and (4.2) that $K_{2}=0$ if and only if

Remark5.1. The $(2.3a,b)$ show that

$A_{\beta^{\#(11.11)}}e_{1}$.

$\lambda_{22(\vdash}-\lambda_{202}=\lambda_{022}=\lambda_{211}=\lambda_{121}=\lambda_{112}=0$

are estimable if $\det(K_{\beta})\neq 0(\beta=0,1,2)$ .

Example5.1. (I) Let be a $B\mu 12,4,2,4;\{0,0,0,0,1,1,1,0,0,0,0,0,0,0,0\}$ ), where the index set$T$$\{\hat{\text{\^{A}}}_{1}0^{1}1^{i}2\}=\{\lambda_{4\}0},\lambda\alpha 0,\lambda\alpha)4,\lambda_{31}0,\lambda_{\mathfrak{B}1},\lambda_{130},\lambda oe1,\lambda_{103},\lambda_{013},\lambda zo,\lambda_{Z)2},\lambda_{\mathfrak{N}2},\lambda_{211},\lambda_{121},\lambda_{1I2}\}$.

Then from (4.1)

and (4.2),$K_{0}=(12$$- 4$$- 120_{1}$$K_{1}=( 6 0 - 4\sqrt{2}),$$K_{2}=0$.

$(- 4^{0}\sqrt{2}$$(\begin{array}{llll}- 4 6 - 8 \theta- 12 - 8 66 00 0 0 0\end{array})$Thus $\det(K_{0})=0,$ $\det(K_{1})\neq 0,$ $K_{2}=0,$ $\det(K_{\beta}(00))\neq 0$ for $\beta=0,1$ and $Ko(11)=0$ . Therefore$(\Theta(\phi),\Theta(1^{1}),\Theta(2^{1}),\Theta(3^{1}),\Theta(4^{1}),\Theta(1^{2}),\Theta(2^{2}),\Theta(3^{2}),\Theta(4^{2}))$$=(\Theta(1^{1}2^{1}),\Theta(1^{1}3^{1}),\Theta(1^{1}4^{I}),\Theta(2^{1}3^{1}),\Theta(2^{1}4^{1}),\Theta(3^{1}4^{1}))$(II) Let$T$$\Theta 0’=$is estimable and is not confounded with

. Furthermore$A_{1}^{t(11.11)}\Theta_{1}$$\Theta_{1’}$is estimable.

be a $BA(12,4,2,4;\{1,0,1,0,0,0,1,0,0,0,1,0,0,0,0\})$, then we get$K_{0=}^{(12}$26$K_{2}=16$$0\backslash ,$,$K_{1}=(\begin{array}{lll}9 3 03 9 00 0 0\end{array}),$$(\begin{array}{llll}2 9 - 5 06 - 5 57 00 0 0 16\end{array})$

and hence$\det(K_{0})\neq 0,$$\det(K_{1})=0,$$K_{2}\neq 0,$

estimable and is not confounded with$\Theta 0$

and$e_{1}$$e_{1}$$\det(K_{\beta}(00))\neq 0$, and also

for $\beta=0,1$ and $K_{1}(11)=0$ . Therefore$A_{0}^{K^{11,11)}}e_{1}$and$A_{2}^{K11,11)}e_{1}$$\Theta 0$

isare estimable, where

are the same vectors as in (I).

Next we consider the case (B), $i.e.$ , all main effects are estimable and are not confounded withthe general mean and the two-factor interactions. Then$\{\Theta(t_{1^{1}}t_{2^{1}})\})$$K_{2}$$\Theta 0’=(\{\Theta(t^{1})\};\{\Theta(\mathfrak{h}\})$

and$e_{1^{*}}’=(\{\Theta(\phi)\}$in (2.1). Let $K_{0}=P’KoP$ ( $=||Ko(ij)||$ , say), $K_{1}=K_{1}$ ( $=||K_{1}(1j)||$ , say), and;$K_{2^{*}}=$. where$P=(0 0 1 0)$$(\begin{array}{llll}1 0 0 00 1 0 00 0 0 1\end{array})$Here$K_{\beta^{*}}(00)$

are the first $2x2$ submatrices of $K_{\beta}coI\tau espoelding$ toremainings are the submatrices ofTheorem 5.2.

Let

$T$

$K_{\beta}$$\{\Theta(t^{1})\}$

and$\{\Theta(t^{2})\}$, and theof appropriate size for $\beta=0,1$ . Then the following yields:

be a $BA$($N$ ,m,3,4;

cient condition for all main effects to be$\{\lambda_{i_{0}i_{1}i_{2}}\}$$estir\epsilon ble$) with $\det(M_{T})=0$ . $\Pi en$ a necessary an suffi$d$and not to be confounded with the general mean

170
and the two-factor interactions is that$propora_{oI1}d$ to the $thi\tau d$$=0$one ( $i.e.$ , the last column of $Ko$ is proportional to the first one) ifdet$(K_{0})$, and $K_{1^{*}}(11)=0if\det(K_{1}\gamma=0$ .

Proof. From Proposition 3.1,
$xM_{01}$

is isomorphic to$\det(M\alpha\})\neq 0$

if

for $\beta=0,1$ and that the last column of $Ko$ is$\det(K_{\beta(}\alpha)))\neq 0$$M\alpha$)

is isomorphic to$K_{\beta}(11)-K_{\beta}(10)K_{\beta}(00)^{-1}K_{\beta}(01)$

if and only if$\det(K_{\beta}(00))\neq 0$

and$\det(K_{0}(\alpha)))\neq 0$

is positive semi-definite and$A^{*g}=(\begin{array}{ll}0 00 d^{*}\end{array})$

. Let

and

$K_{2}$

for $\beta=0,1$ , and under$\det(K_{\beta^{*}}(11\succ K_{\beta^{*}}(10)K_{\beta}(00)^{-1}K_{\beta^{*}}(01))=0$$\det(K_{0})=0$

for$K_{\beta}(\alpha))$$\beta=0,1$, and hence$M_{11}-M_{1}oM\alpha 1^{- 1}$. Thus as shown in Theorem 5.1,$\det(M\alpha\})\neq 0,$

if and only$\det(M_{T})=0$

for some $\beta(\beta=0,1)$ or $K_{2}=0$ . We consider the case($-A$ , say) which$Ko(11)-Ko^{*}(10)Ko(00)^{-1}Ko(01)\approx(\begin{array}{ll}a^{*} b^{*}b^{*} c^{*}\end{array})$. Now we assume$a^{*}c^{*}=b^{*2}$, where $d^{*}=0$ if $c^{*}=0$ and $d^{*}=1/c^{*}$ if c’$a^{*}=0$, then from Lemma 2.1, it holds that‘lhus from (2.4), $Ko^{*}(01)\{I_{2}-(Ko(11)$$\neq 0.$.

, where$-K_{0^{*}}(10)K_{0^{*}}(00)^{-1}Ko^{*}(01))^{g}(K_{0}(11\succ Ko(10)Ko(\alpha))^{-1}Ko(01))\}=(x^{*}, (1-d^{*}c^{*})y^{*})$

are the $2x1$ vectors corresponding to the first and the last columns of(2.4) implies that

. The $(1, 1)$ -element of $Ko(11)$ is$x^{*}=\theta_{2}$$K_{0^{*}}(01)$$\kappa_{0^{\infty}}=N\neq 0$$x^{*}$

and$y^{*}$, respectively. Hence. On the other hand,$x^{*}=\theta_{2}$implies that the $(1, 1)$ -element of $K_{0^{*}}(11)-K_{0}(10)Ko(00)^{-1}Ko^{*}(01)$ is $a^{*}=Ko^{(n}-x^{*r}Ko(00)^{-1}x^{*}=Ko^{\infty}$ .
This is contradict. Therefore$a^{*}\neq 0$. From Lemma 2.1,

$A^{B}=(\begin{array}{ll}1/a^{*} 00 0\end{array})$, and hence $Ko^{*}(01)\{I_{2}$$-(Ko^{*}(11\succ Ko^{*}(10)Ko(00)^{-1}Ko(01))^{g}(Ko(11)-Ko(10)Ko^{*}(00)^{-1}Ko(01))\}=(o_{2}, -(b^{*}/a^{*})x^{*}+y^{*})$(2.4) implies that $a^{*}y^{*}=b^{*}x^{*}$ . From the definition of$a^{*}=Ko^{w}-x^{*\prime}Ko^{*}(00)^{-1}x^{*}$$a^{*},$

and$b^{*}$$c^{*}$. Hence, we have,,$b^{*}=Ko^{03}-x^{*r}Ko(00)^{-1}y^{*}=\kappa 0^{03}-(b^{*}/a^{*})x^{*r}Ko(00)^{-1}x^{*}$$c^{*}=\kappa 0^{33}-y^{*r}K_{0^{*}}(00)^{-1}y^{*}=\kappa 0^{33}-(b^{*}/a^{*})^{2}x^{*\prime}Ko(00)^{-1}X^{*}$

Thus since$b^{*}\kappa 0^{X1}$$a^{*}\neq 0$

and$a^{*}c^{*}=b^{*2}$.

, if $\det(K_{0})=0$ and if $\det(K_{0}(00))\neq 0$, then$\det(K_{1}^{*}(00))\neq 0$$a^{*}\kappa 0^{\mathfrak{B}}=$$A_{\beta}^{K^{11,11)}}e_{11}^{*}$$(\{\Theta(t_{1^{I}}t_{2^{1}})\})$.

$K_{1}$‘(11) $=0$ if $\det(K_{1}’)=0$ and if. The proof is complete.

Remark5.2. It follows from $(2.3a,b)$ that
, and

and

Therefore (2.4) implies that the last column of $Ko$ is proportional to the third one. By

using the argument similar to ‘Iheorem 5.1, the (2.4) implies that$\neq 0$$a^{*}\kappa_{0}^{oe}=b^{*}\kappa 0^{\alpha\}}$are estimable if$A_{0}^{K^{0,0)}}e_{10}^{*}$

and$det(K_{\beta})\neq 0(\beta=1,2)$$A_{0}^{K^{11,11)}}e_{11}$are estimable if$\det(K_{0})$, where $e_{10}"=(\{\Theta(\phi)\})$ and $e_{11^{*}}’=$

171
Bxample 5.2.

Let

$T$

be a $BA(8,4,3,4;\{0,0,0,0,0,1,1,0,0,0,0,0,0,0,0\})$ . Then Rom (4.1) and(4.2), we get$K_{0^{*}}=$$(2 0 0 0),$$(000$

Thus$- 2008$$- 205_{0}0$$\det(K_{\beta}\gamma=0$$K_{2^{*}}=0$$K_{1^{*}}=(\begin{array}{lll}2 0 00 18 00 0 0\end{array}),$.

$0\}00$

for $\beta=0,1,2,$ $\det(K_{v}(00))\neq 0$ for

the third one, and $K_{1}(11)=0$ . Therefore

estimable and is not confounded with$\gamma=0,1$, the last column of$K_{0^{*}}$

is proportional to$\Theta_{0^{*}}’=(\Theta(1^{1}),\Theta(2^{1}),\Theta(3^{1}),\Theta(4^{1}),\Theta(1^{2}),\Theta(2^{2}),\Theta(3^{2}),\Theta(4^{2}))$

is.

.

$\Theta_{1}"=(\Theta(\phi),\Theta(1^{1}2^{1}),\Theta(1^{1}3^{1}),\Theta(1^{1}4^{1}),\Theta(2^{1}3^{1}),\Theta(2^{1}4^{1}),\Theta(3^{1}4^{1}))$

However since $\det(K_{\beta})=0$ for all , no linear $\infty mbinations$ of the elements of$\beta$$e_{1}$

are estimable.

Here $\det(Ko(00))=0$ , where $K_{0}(00)$ is the submatrix of $Ko$ given in Theorem 5.1. llius$T$does notsatisfy the conditions of Theorem 5.1.
Finally consider the case (C), $i.e.$ , the linear components of the main effects are estimable andare not confounded with the general mean, the quadratic ones of the main effects and the two-factor
interactions. Thus we have
$K_{0^{**}}=O’K_{0}O$$\Theta_{0^{*}}‘$$=(\{\Theta(t^{1})\})$

and$e_{1}^{*}’=(\{\Theta(\phi)\};\{\Theta(\mathfrak{h}\};\{\Theta(t_{1^{1}}t_{2^{1}})\})$in (2.1). Let( $=||K_{0}(ij)||$ , say), $K_{1}=K_{1}$ ( $=||K_{I}^{*}(ij)||$ , say), and $K_{2}=K_{2}$ , where$O=(0 1 0 0)$$(\begin{array}{llll}1 0 0 00 0 1 00 0 0 1\end{array})$Here$K_{\beta^{t*}}(00)$$K_{\beta}^{*}(ij)$are the first lxl submatrices ofare the submatrices of(i,j)-cofactors ofelements of$K_{\beta^{**}}$$K_{\beta}$for$K_{\beta}$$\beta=0,1$

which correspond to$K_{\beta}$$\{\Theta(t^{1})\}$, and the remainingof appropriate size $(\beta=0,1)$ . Further let $K_{\beta}(-(i,j))$ be the($i,j=0,1,2,3$ if$\beta=0$; i,j $=0,1,2$ if$\beta=1$), where$\kappa_{\beta}^{*ij}$

are the $(i,j)-$. Then we get the following:

Theorem 5.3. Let $Tbe$ a $BA$( ,m,3,4;$N$$\{\lambda_{i_{0}i_{1}i_{2}}\}$) with $\det(M_{T})=0$ . $?BeI1$ a necessary and suffi-

cient condition for the linear components of the main effects to be estimable an not to be con$d$founded with the general mean, the quadratic ones of the main effects and the two-factor inter-actions is that $\det(K_{\beta}(00))\neq 0$ for $\beta=0,1$ and that $K_{0}(-(3,0))=0if\det(K_{0^{r}})=0$ and if $K_{0}(-(3,3))$$\neq 0,$

$K_{0^{*}}(-(2,0))=0$$ofK_{0}^{*}$

if $\det(K_{0^{\alpha}})=K_{0}(-(3,3))=0$ and $iIK_{0^{r}}(-(2,2))\neq 0$ , the third and the Iast columns

are proportional to the second one ( $i.e.$ , the third and the last columns of $Ko$ are propor-

tional to the first one) $iI\det(K_{0}^{*})=K_{0}(-(3,3))=K_{0^{u}}(-(2,2))=0$ , the last column of$K_{1}$

is propor-

172
tional to the second one if $det(K_{1})=0$ and if $K_{1}(-(2,2))\neq 0,$ $\kappa_{1^{r11}}=0if\det(K_{I})=K_{1}^{**}(-(2,2))=0$
and if$K_{1^{**}}(-(1,1))\neq 0$, an $d\kappa_{1^{*\cdot 11}}=\kappa_{1}22=0ifdet(K_{1}^{*})=K_{1}^{*}(-(2,2))=K_{1}(-(1,1))=0$ .

Proof. As shown in Theorems 5.1 and 5.2,

$M\omega$

is isomorphic to

is isomorphic to $K_{\beta}(11)-K_{\beta}(10)K_{\beta}(00)^{-1}K_{\beta}(01)$ for$\det(K_{0^{**}})=0$

and $det(K_{0}^{*}(00))\neq 0$ . Thenand

$K_{2}$

and $M_{11}-M_{10}M\omega^{- 1}M_{01}$“. We consider the case$Ko(11)-Ko(10)Ko^{n}(00)^{-1}Ko(01)=(\begin{array}{lll}h_{11} h_{1Z} h_{13}h_{2l} h_{2} h_{2}h_{31} h_{3Z} h_{33}\end{array})$$h_{13}(=h_{31})=(\kappa^{*\cdot\infty*\cdot 13**10\cdot\cdot 03}0Ko-K0Ko)/Ko^{*\cdot\infty},$$h_{22}=\{\kappa_{0K0-(\kappa_{0})\}/Ko}^{*\infty\cdot\cdot 2\cdot\cdot 022\cdot\cdot\infty}$$h_{9B}(=h_{32})=(Ko^{*\cdot\propto 1\cdot\cdot \mathfrak{B}}K0-Ko\alpha K0)/Ko,$$h_{33}=\{\kappa_{0Ko-(Ko)\}/Ko}^{*\infty\cdot\cdot 33}032\cdot*\alpha$

Assume$h_{11}=0$,). Then since $K_{0}(11)-Ko(10)Ko(00)^{-1}Ko(01)$ is positive semi-definite,$-K_{0^{*}}(10)Ko^{*}(00)^{-1}K_{0}^{*}(01))^{g}=diag[0,$, where,$h_{12}(=h_{21})=(K0\alpha K0-KoKo)/K0$$h_{11}=\{\kappa_{0K0-(Ko^{*01})^{2}\}/K0}^{*\cdot 0\cdot\cdot 11}\infty,$$B$$\beta=0,1$$K_{\beta}(\infty)$$(K_{0^{**}}(11)$ffl, where $K_{0^{\mathfrak{n}}}(11)-K_{0}(10)Ko(00)^{-1}Ko(01)=diag[0, B]$ andis some $2x2$ matrix. The (2.4) implies that$(\kappa_{0,Ko,Ko)\{I_{3}-(K_{0}(11\succ Ko(10)K_{0}(00)^{-1}Ko(01))^{g}(Ko(11\succ Ko(10)Ko(00)^{-1}Ko(01))\}}^{**01\cdot\cdot 02\cdot\cdot 03}$$=(Ko01(K0^{*\cdot\circ 2}, Ko^{**03})\{I_{2}-\mathcal{B}B\})$$=(0,0,0)$ ,and hence we get
contradict because

If$\kappa_{0^{*01}}=0$

.

‘Ihus

$\kappa_{0}11=\kappa 0^{\infty}=N$$h_{11}=0$. Therefore

implies$h_{11}\neq 0$$\kappa_{0}11=0$

since. This is$\kappa_{0}^{-\infty}(=K_{0^{*}}(00))\neq 0$. After some calculations, we have}$h_{11}h_{22}-h_{12}h_{21}=K_{0}(-(3,3))/K0^{*\cdot\infty}$,$h_{12}h_{13}-h_{I1}h_{23}=K_{0^{**}}(-(3,2))/\kappa 0\alpha$$h_{12}h_{23}-h_{I3}h_{22}=K_{0}(-(3,1))/\kappa 0\infty$,$h_{11}h_{33}-h_{13}h_{31}=K_{0}(-(2,2))/Ko^{*\cdot\alpha)}$$h_{13}h_{23}-h_{12}h_{33}=K_{0}(-(2,1))/\kappa 0\infty$.

$K_{0^{**}}(-(3,3))\neq 0,$$i.e.,$ $h_{11}h_{2}-h_{12}h_{21}\neq 0$$=\{1/(h_{11}h_{22}-h_{12}h_{21})\}(\begin{array}{lll}h_{2} - h_{2l} 0- h_{1Z} h_{11} 00 0 0\end{array})$,, then ffom Lemma 2.2, $(K_{0}(11)-Ko^{*}(10)Ko^{s}(00)^{-1}Ko^{n}(01))^{g}$. Thus (2.4) implies that$(K00Ko, \kappa_{0})\{I_{3}-(K_{0^{*}}(11\succ K_{0}(10)K_{0}(00)^{-1}Ko(01))^{g}(K_{0^{*}}(11\succ K_{0}(10)K_{0}^{*}(00)^{-1}K_{0}(01))\}$$=(0,0, \kappa 003+\{\kappa 001K_{0}(-(3,1))+Ko02K_{0}(-(3,2))\}/K_{0}(-(3,3)))$$=(0,0,0)$ ,and hence we get $\kappa 01K(-(3,1))+KoK_{0^{r}}(-(3,2))+Ko^{*tB}K_{0}(-(3,3))=0$. While$\kappa o^{**\infty}K_{0}(-(3,0))$. Therefore we have $K_{0}(-(3,0))=0$ since$+Ko^{*\cdot 01}K_{0}(-(3,1))+Ko^{*\cdot t2}K_{0}(-(3,2))+Ko^{-oe}K_{0}(-(3,3))=0$

173$\kappa 0^{**0}\neq 0$. Similarly if $Ko^{*}(-(3,3))=0$ and if $Ko(-(2,2,))\neq 0$ , then we can get $\kappa 0^{r01}Ko(-(2,1))+Kot2$$xK_{0^{**}}(-(2,2,))+\kappa 0^{**oe}Ko^{**}(-(2,3))=0$, and hence $K_{0^{n}}(-(2,0))=0$ . If $Ko^{s*}(-(3,3))=K_{0^{*}}(-(2,2))=0$ , from

Lemma2.2, we have $(Ko(11)-K_{0}^{*}(10)Ko(00)^{-1}K_{0}^{*}(01))^{g}=diag[h_{11},0,0]$ . Thus from (2.4),$(\kappa^{**01\cdot*02}0, K0, \kappa_{0^{**03}})\{I_{3}-(Ko(11)-Ko^{*}(10)Ko(00)^{-1}Ko^{*}(01))^{g}(Ko(11\succ Ko^{*}(10)Ko(00)^{-1}K_{0}^{*}(01))\}$$=(0, \kappa 0^{**02}-Ko^{**01}h_{1}ffi_{11}, \kappa^{*\cdot 03\cdot\cdot 01}0-Koh_{13}/h_{11})$$=(0,0,0)$ .

After some calculations,$=\kappa 0^{**01}Ko^{**\llcorner 9}$

and$\kappa 0^{u\infty\cdot*01}-Koh_{\Omega}/h_{11}=0$$\kappa_{0^{**oe}}\kappa_{0^{*11}}=\kappa_{0Ko}^{n01\cdot I3}$

last columns of$\det(K_{1^{*}}(00))\neq 0$$Ko^{**}$, we get$\kappa^{**11\cdot*2}0Ko=(\kappa_{0^{*12}})^{2}$and$\kappa_{0^{*11}}\kappa 0^{r33}=(\kappa_{0^{**13}})^{2}$.

By using the argument similar to the case $\det(K_{0})=0$ and $\det(Ko(00))\neq 0$ in$\kappa_{1}^{*(n}\kappa_{1}$“$11-(\kappa_{1}01)^{2}\neq 0$, then (2.4) implies that th last column$e$is proportional to the second one. If $K_{1^{**}}(-(2,2))=0$ and $K_{1}^{*}(-(1,1))\neq 0,$ $i.e.,$$-(\kappa_{1}^{**02})^{2}\neq 0$$\kappa_{1}\kappa_{I}(n\sim z$, it follows from Lemma 2.1 that $(K_{1}(11)-K_{1}^{n}(10)K_{1^{**}}(00)^{-1}K_{1}^{**}(01))^{g}=diag[0$ ,$1/\{\kappa_{1}^{**\alpha)}\kappa_{1^{**2}}-(\kappa_{1}^{**02})^{2}\}]$. Thus we get$K_{1}^{**}(01)\{I_{2}-(K_{1}^{u}(11\succ K_{1}^{n}(10)K_{1}^{**}(00)^{-1}K_{1}^{*}(01))^{g}(K_{1^{**}}(11)$. The (2.4) implies that$-K_{1^{**}}(10)K_{1}(00)^{-1}K_{I^{*}}(01))\}=(\kappa_{1}010)$

$=0$

,

are proportional to the second. Next we consider the case $\det(K_{1}^{*})=0$ andTheorem 5.2, if $K_{1}(-(2,2))\neq 0,$ $i.e.,$““$KoK0\alpha\cdot*11$, respectively. ‘Iherefore if $Ko^{*}(-(3,3))=K_{0}(-(2,2))=0$ , then the third and the$\kappa_{0^{**11}}\kappa=$$K_{1}$

mean that$\kappa_{0^{*oe}}-\kappa 0^{ro1}h_{13}/h_{11}=0$, respectively. While from $K_{0}^{*}(-(3,3))=K_{0}(-(2,2))=0$ and$\det(Ko^{*z}(11)-Ko^{**}(10)Ko^{*}(00)^{-1}Ko^{*}(01))=0$

of

and

. Therefore$\kappa_{1}11=0$$\kappa_{1}^{-01}=0$

and hence“‘12$\kappa_{1^{*11}}(=\kappa_{1}$$)$

if $K_{1}(-(2,2))=0$ and if $K_{1}(-(1,1))\neq 0$ . Lastly consider the case $K_{1}(-(2,2))$. Then we have

, and henoe$K_{1^{*}}(01)\{I_{2}$$-(K_{1^{*s}}(11)-K_{1}^{*}(10)K_{1}(00)^{-1}K_{1}(01))^{g}(K_{1^{*}}(11)-K_{1}(10)K_{1}(00)^{-I}K_{1}(01))\}=K_{1^{u}}(01)$. The (2.4)$=K_{1}^{**}(-(1,1))=0$

implies that$=0$

$K_{1^{n}}(01)=\theta_{2’}$. From$K_{1}(11)-K_{1}^{*}(10)K_{1}(00)^{-1}K_{1}^{*}(01)=\alpha_{x2}$$K_{1^{*}}(-(2,2))=K_{1^{n}}(-(1,1))=0$

and$K_{1}^{*}(01)=\theta_{2’}$, we get$\kappa_{1^{*11}}=\kappa_{1^{rae}}$. The theorem is thus established.

Remark5.3. The $(2.3a,b)$ show that
$A_{1}^{K^{01,01)}}\Theta_{11^{*}}$$\det(K_{0^{**}})\neq 0,$

estimable if$\det(K_{2^{**}})\neq 0$

Example5.3.
Then we hav$e$(I) Letand, where$T$$A_{0}^{K^{0,0)}}e_{10^{*},A_{0}^{K^{01,0I)}}e_{11^{*}}}$$A_{1}^{*11,11)}e_{12^{\alpha}}$$e_{10^{*}}‘$

are estimable if$=(\{\Theta(\phi)\}),$

and$A_{0}^{K^{11,11)}}\Theta_{12^{*}}$$det(K_{1^{\alpha}})\neq 0$$\Theta_{11}^{s,}=(\{\Theta(t\mathfrak{h}\})$

and

, and

$\Theta_{12}",$are estimable if$Az^{K11,11)}\Theta_{12^{n}}$$=(\{\Theta(t_{1^{1}}t_{2^{1}})\})$

is.

be a $BA$( $2x+6y,4,3,4;\{x,0$ ,x,O,O,O,O,O,O,O, $y,0,0,0,0\}$ ), where $x,y\geq 1$ .

174

$K_{2}^{*}=16y$$K_{1^{**}}=(\begin{array}{lll}8y 0 00 0 00 0 0\end{array}),$$K_{0=}^{**(8_{0}x}(|^{0}0$Thus$\det(K_{\beta}^{*})=0$$K_{0^{*}}(-(3,3))=0,$

and$\det(K_{\beta}(\alpha)))\neq 0$$K_{0}^{*}(-(2,2))\neq 0,$

for $\beta=0,1$ , and$K_{0}(-(2,0))=0,$$7herefore\Theta 0^{*r,}=(\Theta(1^{1}),\Theta(2^{1}),\Theta(3^{1}),\Theta(4^{1}))$$K_{2^{r}}\neq 0$. After some calculations, we get$K_{1}(-(2,2))=K_{1}(-(1,1))=0$ and$A_{2}^{\mathscr{V}^{11,11)}}e_{12^{**}}$

is estimable, where

While since $\det(K_{1^{*}}(00))=0,$
submatrix of(II) Let$T$$K_{1^{*}}$$T$.

does not satisfy the $\infty nditions$ of Theorem 5.2, where$0$$x\geq 0$and$0$$0\backslash ,$$- 4^{x_{(x_{0^{- 4y)}}^{+8y}}}$$K_{1}^{*}(00)$.

is the$\det(K_{\beta^{**}})=0$$y\geq 1$. Then.

$16(X+- 4(x_{0^{-}}4_{2}y_{y})_{)} 00)0$

for $\beta=0,1,2$, and $\det(K_{Y}(00))\neq 0$ for$K_{I^{**}}(-(2,2))=0,$ $K_{1^{*}}(-(1,1))\neq 0$

and$\kappa_{1^{*\cdot 11}}=0$

. If

$x=0$

$\gamma=0,1$. After some calculations, we have

, then $K_{0}^{*}(-(3,3))=K_{0}(-(2,2))=0$, and the third

and the last columns of $Ko$ are propontional to the second one. On the other hand, if$e_{1}$$=$

Furthermore since$K_{1^{*}}=(\begin{array}{lll}8y 0 00 0 00 0 16y\end{array})’ K_{2}^{*}=0$$\neq 0$.

in Theorem 5.2.

$Ko^{**}=(8y$Thus$\Theta_{1^{**}}$$e_{J2^{r,}=}(\Theta(1^{1}2^{1}),\Theta(1^{1}3^{1}),\Theta(1^{1}4^{1}),\Theta(2^{1}3^{1}),\Theta(2^{1}4^{1}),\Theta(3^{1}4^{1}))$be a $BA(x+8y,4,3,4;\{0,x,0,0,y,0,0,y,0,0,0,0,0,0,0\})$ , where$(000$$\kappa_{1^{u}}"=\kappa_{1^{*\cdot 2}}=0$

is estimable and is not confounded with$(\Theta(\phi),\Theta(1^{2}),\Theta(2^{2}),\Theta(3^{2}),\Theta(4^{2}),\Theta(1^{1}2^{1}),\Theta(1^{1}3^{1}),\Theta(1^{1}4^{1}),\Theta(2^{1}3^{1}),\Theta(2^{1}4^{1}),\Theta(3^{1}4^{1}))$$K_{2^{**}}\neq 0,$.

$4^{4(x+}8(x+3y_{y^{)})}\sqrt{6}(x^{3_{-}y_{)}}0 4_{4(3x^{0}+^{-}y)}2\sqrt{6}(x- y)^{1}\sqrt{6}(xy))$$2^{2(}\sqrt{6}^{x+}4(x+3y)(x^{3_{-}y}y^{)})0$, then $Ko^{*}(-(3,3))\neq 0$ and $Ko(-(3,0))=0$ . Therefore$\Theta 0$$x\geq 1,$$i.e.,$$x$is estimable and is not confounded with““. Obviously $\det(K_{1^{*}}(00))=0$ . Thus Tdoes not satisfy th conditions of Theorem 5.2.
$e$References[1] Ariyaratna, W.M. (1979) : Optimal balanced resolution V designs of the$3^{m}$

series. Plr.D

Dissemcion, Colorado State University, Co.
[2] Finey, D.J. (1945) : The fractional replication of factorial arrangements. Ann. Eugen. 12,

291-301.
[3] Hoke, AT. (1974) : Economical second-order designs based on irregular fractions of the$3^{n}$

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