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Orthogonal designs in linear models and sequences with zero autocorrelation
ffl'&llltlICS & m,,l~l ill ELSEVIER
Statistics & Probability Letters 26 (1996) 333-338
Orthogonal designs in linear models and sequences with zero autocorrelation Christos Koukouvinos Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece Received October 1994
Abstract In factorial experiments the use of orthogonal designs provides better estimates for the main effects and interactions. In 2k factorial experiments we obtain orthogonal designs from Hadamard matrices. Sequences with zero autocorrelation can be used to construct Hadamard matrices. In this paper new sequences with zero autocorrelation which are called fsequences are employed to construct some new orthogonal designs. A M S Subject Classification." Primary 62K05, 62K15, secondary 05B15 Keywords: Factorial designs; Block-circulant matrices; Auto-correlation
I. Introduction Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect o f the factors on a response. There are several special cases o f the general factorial design that are important because they are widely used in research work and also because they form the basis o f other designs o f considerable practical value. The most important o f these special cases is that o f k factors, each at only two levels. These levels may be quantitative, such as two values of temperature, pressure, or time; or they may be qualitative, such as two machines, two operators, the "high" and "low" levels o f a factor, or perhaps the presence and absence o f a factor. A complete replicate o f such a design requires n ----2 x 2 x • • • x 2 = 2 k observations and is called a 2 k factorial design. The general linear model is Y = ~ + (AlXl + .." + A k x k ) + (Al2XlX2 + " " +Ak-l,kXk-lXk) + . . . + Al2...kXlX2...x k + e,
(1)
where the errors are assumed to be uncorrelated random variables with E(e) = O, Var(e) = G2, and # is the general mean, {Ai} the main effects o f the factors, {Aq} the interactions between two factors . . . . . {&,i, ...i, } 0167-7152/96/$12.00 (E) 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 5 ) 0 0 0 2 9 - 1
C. Koukouvinos/Statistics & Probability Letters 26 (1996) 333 338
334
the interactions between t factors, 1 ~
i2 <
""
<
it<~k and
= ~ + 1 if the factor Fi appears at the "high" level, xi
L-1
if the factor Fi appears at the "low" level.
So, in order to estimate all the main effects and interactions o f the factors we need 2 k observations. In this case, since all the degrees o f freedom are exhausted we cannot estimate the variance tr z o f the errors, unless r > 1 observations are available for each case. Then 2k(r - 1) degrees of freedom are left for the estimation o f tr 2. One way o f overcoming this difficulty is to suppress the interactions o f high order, a fact that often happens in practice. Doing so, the degrees o f freedom that are left are equal to the number o f suppressed interactions. Box and Hunter (1961) (see also Montgomery, 1991) gave the notion of the resolution o f an experimental design consisting o f n observations. Let the model be o f the form Y = l~ + AlXl + . . . + Akxk + e,
(2)
and let us also suppose that all the main effects A1 . . . . . Ak and the general mean /~ as well, are estimable. If we suppress the interactions o f two or more factors, then we have a design of resolution III. Let the model be o f the form Y = I~ ~ - m l X l ~- "'" + A k x k -}-A12XlX2 + "'" + A k - l , k X k - l X k + e,
(3)
and let us also suppose that all the main effects A~ . . . . . Ak and the general mean /~ as well, are estimable. If we suppress the interactions of three or more factors, then we have a design o f resolution IV. In the experiments o f n observations, the design matrix X contains the elements + 1 (the coefficients of A i , A i j ) and the information matrix M = x T x specifies the accuracy o f the estimations. If the design X is such that (i) d e t M is maximized, then we have D-optimality. (ii) The sum of the variances o f the estimators, i.e. trM - l , is minimized, then we have A-optimality. For models o f resolution III, defined in (2), D and A-optimal designs have mostly been studied and constructed; see Kounias(1977). If n -= 0 mod 4, the orthogonal designs are D and A-optimal and they are given from Hadamard matrices. For the cases n ~- 1 mod 4, n -= 2 mod 4, n -= 3 mod 4, several optimal designs have been constructed; see Shah and Sinha (1989). If we want to make a randomization with n = 0 mod 4 observations, we need all the non-equivalent Hadamard matrices of order n and this is a difficult problem when n ~>28.
2. Orthogonal designs An orthogonal design A, o f order n, and type (Sl,S2 . . . . . Su), denoted O D ( n ; s l , s 2 . . . . ,Su) on the commuting variables (-+-xl,+x2 . . . . . ± x u , O) is a square matrix of order n with entries ±xk where each xk occurs sk times in each row and column such that the distinct rows are pairwise orthogonal. In other words, A A T = (slx~ + " "
+ suX2u)I,,
where I, is the identity matrix.
C Koukouv&oslStatistics & Probability Letters 26 (1996) 333-338
335
A weighing matrix W = W(n,k) is a square matrix with entries 0 , + 1 having k non-zero entries per row and column and inner product of distinct rows zero. Hence W satisfies WW T = kin, and W is equivalent to an orthogonal design OD(n;k). The number k is called the weight of W. If n = k, then all the elements of W are 4-1 and is called Hadamard matrix. OD(4t; t,t,t,t), otherwise called Baumert-Hall arrays, and OD(2S; a,b, 2s - a - b) have been extensively used to construct Hadamard matrices and weighing matrices; for details see Geramita and Seberry (1979). T-matrices of order t are used to construct OD(4t; t, t,t, t). Four circulant matrices T1, T2, T3, T4 of order t which have entries 0, + 1 or - 1 and which are non-zero for each of the t 2 entries for exactly one i, i.e. Ti * Tj = 0 for i ~ j, where • is the Hadamard (or element by element) product, and which satisfy 4
Ev, v?=tl, i=1
are called T-matrices of order t. Four ( 1 , - 1 ) matrices A,B, C,D of order m which satisfy (i) AA T + BB T + CC T + DD T = 4mlm, (ii) XY T = YX T, X , Y E {A,B,C,D}, are called Williamson-type matrices of order m. Theorem 1 (Cooper-Seberry-Turyn; see Geramita and Seberry (1979)). Suppose there exist Tj, T2, T3, T4 of order t (assumed to be circulant). Let a, b, c, d be commuting variables. Then
T-matrices
X =aT~ +bT2 +cT3 + d T 4 , Y = - b T l + aT2 + dT3 - cT4, Z = -cTi -- dT2 + aT3 + bT4, W = - d T l + cT2 -- bT3 + aT4, can be used in the Goethals-Seidel array I
X -YR
YR X
ZR - WTR
WR I ZTR
x _?)
GS=
-WR
-ZTR
yTR
where R = (rig) is the (0, 1) matrix o f order t defined by ri, t-i+l = 1 and rij = 0 when j ¢ t - i + 1, to form an OD(4t; t, t, t, t). Replacing the variables of Theorem 1 by Williamson-type matrices we have:
Method 1 (Cooper-Seberry-Turyn; see Geramita and Seberry(1979)). Suppose there exist T-matrices TI, T2, T3, T4 o f order t (assumed to be circulant). Let A, B, C, D be Williamson-type matrices of order m. Then X = TI x A + T2 x B + Y=T1 x-B+T2
T3 x C + T4 x D,
xA+T3
xD+T4
x-C,
336
C. Koukouvinos/ Statistics & Probability Letters 26 (1996) 333-338 Z = T1 × - C + T2 × - D +
T3 × A + T4 × B ,
W = T1 × - D + T2 × C + T3 × - B + T4 × A, can be used in the Goethals-Seidel array to f o r m an H a d a m a r d matrix o f order 4rot. L e m m a 1 (Kharaghani, (1985)). Let W be a W(n,c). say Cl, C2 .... , C,, such that
Then there are n symmetric (0, 1 , - 1 )
matrices,
(i) CiCm = 0 , if l # m, (ii) C 2 + C 2 + . . . + C 2 = c21,. Furthermore, f o r n = c, C1 m a y be taken to be the matrix o f ones. Let n = c = 4p, W = (wij), 1 ~ i , j < ~ 4 p . For k = 1,2 . . . . . 4p, let Ck = (wkiwkj). Then, we obtain 4 p symmetric (1, - 1) matrices C1, (72
.....
C4p o f
order 4p, such that
(i) C I C m = O , If/#m, (ii) C 2 + C~ + . . . + C4Zp = (4p)214p.
3. Sequences with zero autocorrelation and applications Given the sequence A = {al,a2 . . . . . an} of length n, the non-periodic autocorrelation function NA(S) is defined as n-s
NA(s) = ~ aiai+s,
s = O, 1. . . . . n - l ,
i=1
and the periodic autocorrelation function PA(s) is defined, reducing i + s modulo n, as PA(s) = ~ aiai+~,
s = O, 1. . . . . n - 1 .
i=1
Sequences with zero autocorrelation can be used to construct Hadamard matrices: In factorial experiments the use o f orthogonal designs provides better estimates for the main effects and interactions. In 2 k factorial experiments we obtain orthogonal designs from Hadamard matrices. In this section we give new sequences with zero autocorrelation which are called f-sequences, and then we use these sequences to construct some new orthogonal designs. Let h , i , j , k be symbols such that h 2 = i 2 = j 2 = k 2 = 1, x y = O, x # y, x , y E { h , i , j , k } . Let A = { a l , a z , . . . , a , } be a sequence o f length n with elements a~ C { ± h , +i, ± j , ± k } , and n--S
NA(S) =- ~ aiai+s = nOos, i=1
where 60s is the Kronecker delta, i.e. f
1,
s=0
O,
s#0
(s=0,1 ..... n-
6Os =
t
1).
This means that the non-periodic autocorrelation function o f the sequence A is zero and is called 6-sequence o f length n.
337
C KoukouvinoslStatistics & Probability Letters 26 (1996) 333-338
Example 1. The sequence A = {h, i , - i , j , j } fh
i
-i
j
j'~
0
h
i
-i
j
0 0
h
i
-i
0 0
0
h
i
0
0
0
h
0
is a 5-sequence of length 5. To see this we form the matrix
and we observe that the inner product of any row with any other is zero. If TI, T2, T3, T4 are circulant T-matrices of order t, then the first row of X = hTt + iT2 + jT3 + kT4
is a b-sequence of length t because X X T = h2T1T T + i2T2 T] + j2T3T~ + k2T4 T] = tit
using x y = O, x ~ y, x , y c { h , i , j , k } . We prove that: Theorem 2. I f there exists an OD(80pZ; 20p 2, 20p 2, 20p 2, 20p 2 ).
orthogonal
design
OD(4p;p,p,p,p)
then
there
exists
an
Proof. Consider the b-sequence of length 5 of Example 1 and the matrices C1, C2. . . . . Cnp of Lemma 1. We form the block-circulant matrices Dh = (CI, C2,..., Cp),
Di =
Dj = (C2p+l,C2p+2,...,C3p),
(Cp+l, Cp+2 . . . . .
C2p),
Dk = (Cbp+l,C3p+2 . . . . . Cap),
and then we construct the block-circulant matrices X, Y,Z, W replacing the elements of the b-sequence as follows: matrix X
matrix Y
matrix Z
± h ---, q-Dh
± h - ~ 4- D i
+ h --* + D j
+ h --~ 3-Dk
± i --~ + D i
d: i --~ ± D j
-t- i ---~ -4- Dk
q- i ---+ ~ Dh
+j---+ + D j
± j --~ -4- Dk
+j ~ ±Dh
±j ~ ~-Oi
± k --~ -4- Dk
±k ~ ±Dh
-4-k ~ + D i
± k ~ 5:Dj
For example, the matrix X is
X=
Dh
Di
-Di
Dj
Dj
Dh
Di
-Di
Dj
Dj
Dj
Dh
Di
-Di
-Oi
Dj
Dj
Dh
D~
Di
--Di
nj
Dj
Dh
,
matrix W
338
C. Koukouvinos I Statistics & Probability Letters 26 (1996) 333- 338
where
Cl
C2 C3 ...
Cp
Cp C1 C2 ...
Cp-i
Dh =
C2 C3
C4
"'"
CI
We observe that p
DhD T = Ip × ~-]~CZi, i=1
DhD T = 0 and DhD~ = 0, since CaCb = 0, a ¢ b. Let T = (tij) be the circulant matrix of order 5, with elements (0, 1) such that tij
1,
if j - i =
0,
elsewhere
1
(i,j = 1 , 2 , . . . , 5 )
:
i.e. T = (0, 1,0, 0, 0). Then we obtain X X T = Is × (DhD] + 2DiDVi + 2DjDf ) + (T + r 4) × (-DiDTi + DjDf )
Similarly for the matrices y y T ZZ T and WW T, and then we have x x T + yyV + ZZ T + WW v = 515p × ~ C 2 : 80pZi2op2. i=1
Using the matrices X, Y, Z, W in OD(80p2; 2 0 p 2, 2 0 p 2, 20p 2, 20p2).
the
Goethals-Seidel
array,
we
obtain
an
orthogonal
design
References Box, G.E.P. and J.S. Hunter (1961), The 2k-p fractional factorial designs 1, Technometrics 3, 311-352. Geramita, A.V. and J. Seberry (1979), Orthogonal desiyns. Quadratic ]brms and Hadamard Matrices. (Marcel Dekker, New York, Basel). Kharaghani, H. (1985), New class of weighing matrices, Ars Combin. 19, 69-72. Kounias, S. (1977), Optimal 2k designs of odd and even resolution, in: Recent Developments in Statistics (Noah-Holland, Amsterdam) pp. 501-506. Montgomery, D.C. (1991), Design and analysis of Experiments (Wiley, New York, 3rd ed.). Shah, K.R. and B.K. Sinha (1989), Theory of Optimal Designs, Lecture Notes in Statistics, Vol. 54 (Springer, Berlin).
Statistics & Probability Letters 26 (1996) 333-338
Orthogonal designs in linear models and sequences with zero autocorrelation Christos Koukouvinos Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece Received October 1994
Abstract In factorial experiments the use of orthogonal designs provides better estimates for the main effects and interactions. In 2k factorial experiments we obtain orthogonal designs from Hadamard matrices. Sequences with zero autocorrelation can be used to construct Hadamard matrices. In this paper new sequences with zero autocorrelation which are called fsequences are employed to construct some new orthogonal designs. A M S Subject Classification." Primary 62K05, 62K15, secondary 05B15 Keywords: Factorial designs; Block-circulant matrices; Auto-correlation
I. Introduction Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect o f the factors on a response. There are several special cases o f the general factorial design that are important because they are widely used in research work and also because they form the basis o f other designs o f considerable practical value. The most important o f these special cases is that o f k factors, each at only two levels. These levels may be quantitative, such as two values of temperature, pressure, or time; or they may be qualitative, such as two machines, two operators, the "high" and "low" levels o f a factor, or perhaps the presence and absence o f a factor. A complete replicate o f such a design requires n ----2 x 2 x • • • x 2 = 2 k observations and is called a 2 k factorial design. The general linear model is Y = ~ + (AlXl + .." + A k x k ) + (Al2XlX2 + " " +Ak-l,kXk-lXk) + . . . + Al2...kXlX2...x k + e,
(1)
where the errors are assumed to be uncorrelated random variables with E(e) = O, Var(e) = G2, and # is the general mean, {Ai} the main effects o f the factors, {Aq} the interactions between two factors . . . . . {&,i, ...i, } 0167-7152/96/$12.00 (E) 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 5 ) 0 0 0 2 9 - 1
C. Koukouvinos/Statistics & Probability Letters 26 (1996) 333 338
334
the interactions between t factors, 1 ~
i2 <
""
<
it<~k and
= ~ + 1 if the factor Fi appears at the "high" level, xi
L-1
if the factor Fi appears at the "low" level.
So, in order to estimate all the main effects and interactions o f the factors we need 2 k observations. In this case, since all the degrees o f freedom are exhausted we cannot estimate the variance tr z o f the errors, unless r > 1 observations are available for each case. Then 2k(r - 1) degrees of freedom are left for the estimation o f tr 2. One way o f overcoming this difficulty is to suppress the interactions o f high order, a fact that often happens in practice. Doing so, the degrees o f freedom that are left are equal to the number o f suppressed interactions. Box and Hunter (1961) (see also Montgomery, 1991) gave the notion of the resolution o f an experimental design consisting o f n observations. Let the model be o f the form Y = l~ + AlXl + . . . + Akxk + e,
(2)
and let us also suppose that all the main effects A1 . . . . . Ak and the general mean /~ as well, are estimable. If we suppress the interactions o f two or more factors, then we have a design of resolution III. Let the model be o f the form Y = I~ ~ - m l X l ~- "'" + A k x k -}-A12XlX2 + "'" + A k - l , k X k - l X k + e,
(3)
and let us also suppose that all the main effects A~ . . . . . Ak and the general mean /~ as well, are estimable. If we suppress the interactions of three or more factors, then we have a design o f resolution IV. In the experiments o f n observations, the design matrix X contains the elements + 1 (the coefficients of A i , A i j ) and the information matrix M = x T x specifies the accuracy o f the estimations. If the design X is such that (i) d e t M is maximized, then we have D-optimality. (ii) The sum of the variances o f the estimators, i.e. trM - l , is minimized, then we have A-optimality. For models o f resolution III, defined in (2), D and A-optimal designs have mostly been studied and constructed; see Kounias(1977). If n -= 0 mod 4, the orthogonal designs are D and A-optimal and they are given from Hadamard matrices. For the cases n ~- 1 mod 4, n -= 2 mod 4, n -= 3 mod 4, several optimal designs have been constructed; see Shah and Sinha (1989). If we want to make a randomization with n = 0 mod 4 observations, we need all the non-equivalent Hadamard matrices of order n and this is a difficult problem when n ~>28.
2. Orthogonal designs An orthogonal design A, o f order n, and type (Sl,S2 . . . . . Su), denoted O D ( n ; s l , s 2 . . . . ,Su) on the commuting variables (-+-xl,+x2 . . . . . ± x u , O) is a square matrix of order n with entries ±xk where each xk occurs sk times in each row and column such that the distinct rows are pairwise orthogonal. In other words, A A T = (slx~ + " "
+ suX2u)I,,
where I, is the identity matrix.
C Koukouv&oslStatistics & Probability Letters 26 (1996) 333-338
335
A weighing matrix W = W(n,k) is a square matrix with entries 0 , + 1 having k non-zero entries per row and column and inner product of distinct rows zero. Hence W satisfies WW T = kin, and W is equivalent to an orthogonal design OD(n;k). The number k is called the weight of W. If n = k, then all the elements of W are 4-1 and is called Hadamard matrix. OD(4t; t,t,t,t), otherwise called Baumert-Hall arrays, and OD(2S; a,b, 2s - a - b) have been extensively used to construct Hadamard matrices and weighing matrices; for details see Geramita and Seberry (1979). T-matrices of order t are used to construct OD(4t; t, t,t, t). Four circulant matrices T1, T2, T3, T4 of order t which have entries 0, + 1 or - 1 and which are non-zero for each of the t 2 entries for exactly one i, i.e. Ti * Tj = 0 for i ~ j, where • is the Hadamard (or element by element) product, and which satisfy 4
Ev, v?=tl, i=1
are called T-matrices of order t. Four ( 1 , - 1 ) matrices A,B, C,D of order m which satisfy (i) AA T + BB T + CC T + DD T = 4mlm, (ii) XY T = YX T, X , Y E {A,B,C,D}, are called Williamson-type matrices of order m. Theorem 1 (Cooper-Seberry-Turyn; see Geramita and Seberry (1979)). Suppose there exist Tj, T2, T3, T4 of order t (assumed to be circulant). Let a, b, c, d be commuting variables. Then
T-matrices
X =aT~ +bT2 +cT3 + d T 4 , Y = - b T l + aT2 + dT3 - cT4, Z = -cTi -- dT2 + aT3 + bT4, W = - d T l + cT2 -- bT3 + aT4, can be used in the Goethals-Seidel array I
X -YR
YR X
ZR - WTR
WR I ZTR
x _?)
GS=
-WR
-ZTR
yTR
where R = (rig) is the (0, 1) matrix o f order t defined by ri, t-i+l = 1 and rij = 0 when j ¢ t - i + 1, to form an OD(4t; t, t, t, t). Replacing the variables of Theorem 1 by Williamson-type matrices we have:
Method 1 (Cooper-Seberry-Turyn; see Geramita and Seberry(1979)). Suppose there exist T-matrices TI, T2, T3, T4 o f order t (assumed to be circulant). Let A, B, C, D be Williamson-type matrices of order m. Then X = TI x A + T2 x B + Y=T1 x-B+T2
T3 x C + T4 x D,
xA+T3
xD+T4
x-C,
336
C. Koukouvinos/ Statistics & Probability Letters 26 (1996) 333-338 Z = T1 × - C + T2 × - D +
T3 × A + T4 × B ,
W = T1 × - D + T2 × C + T3 × - B + T4 × A, can be used in the Goethals-Seidel array to f o r m an H a d a m a r d matrix o f order 4rot. L e m m a 1 (Kharaghani, (1985)). Let W be a W(n,c). say Cl, C2 .... , C,, such that
Then there are n symmetric (0, 1 , - 1 )
matrices,
(i) CiCm = 0 , if l # m, (ii) C 2 + C 2 + . . . + C 2 = c21,. Furthermore, f o r n = c, C1 m a y be taken to be the matrix o f ones. Let n = c = 4p, W = (wij), 1 ~ i , j < ~ 4 p . For k = 1,2 . . . . . 4p, let Ck = (wkiwkj). Then, we obtain 4 p symmetric (1, - 1) matrices C1, (72
.....
C4p o f
order 4p, such that
(i) C I C m = O , If/#m, (ii) C 2 + C~ + . . . + C4Zp = (4p)214p.
3. Sequences with zero autocorrelation and applications Given the sequence A = {al,a2 . . . . . an} of length n, the non-periodic autocorrelation function NA(S) is defined as n-s
NA(s) = ~ aiai+s,
s = O, 1. . . . . n - l ,
i=1
and the periodic autocorrelation function PA(s) is defined, reducing i + s modulo n, as PA(s) = ~ aiai+~,
s = O, 1. . . . . n - 1 .
i=1
Sequences with zero autocorrelation can be used to construct Hadamard matrices: In factorial experiments the use o f orthogonal designs provides better estimates for the main effects and interactions. In 2 k factorial experiments we obtain orthogonal designs from Hadamard matrices. In this section we give new sequences with zero autocorrelation which are called f-sequences, and then we use these sequences to construct some new orthogonal designs. Let h , i , j , k be symbols such that h 2 = i 2 = j 2 = k 2 = 1, x y = O, x # y, x , y E { h , i , j , k } . Let A = { a l , a z , . . . , a , } be a sequence o f length n with elements a~ C { ± h , +i, ± j , ± k } , and n--S
NA(S) =- ~ aiai+s = nOos, i=1
where 60s is the Kronecker delta, i.e. f
1,
s=0
O,
s#0
(s=0,1 ..... n-
6Os =
t
1).
This means that the non-periodic autocorrelation function o f the sequence A is zero and is called 6-sequence o f length n.
337
C KoukouvinoslStatistics & Probability Letters 26 (1996) 333-338
Example 1. The sequence A = {h, i , - i , j , j } fh
i
-i
j
j'~
0
h
i
-i
j
0 0
h
i
-i
0 0
0
h
i
0
0
0
h
0
is a 5-sequence of length 5. To see this we form the matrix
and we observe that the inner product of any row with any other is zero. If TI, T2, T3, T4 are circulant T-matrices of order t, then the first row of X = hTt + iT2 + jT3 + kT4
is a b-sequence of length t because X X T = h2T1T T + i2T2 T] + j2T3T~ + k2T4 T] = tit
using x y = O, x ~ y, x , y c { h , i , j , k } . We prove that: Theorem 2. I f there exists an OD(80pZ; 20p 2, 20p 2, 20p 2, 20p 2 ).
orthogonal
design
OD(4p;p,p,p,p)
then
there
exists
an
Proof. Consider the b-sequence of length 5 of Example 1 and the matrices C1, C2. . . . . Cnp of Lemma 1. We form the block-circulant matrices Dh = (CI, C2,..., Cp),
Di =
Dj = (C2p+l,C2p+2,...,C3p),
(Cp+l, Cp+2 . . . . .
C2p),
Dk = (Cbp+l,C3p+2 . . . . . Cap),
and then we construct the block-circulant matrices X, Y,Z, W replacing the elements of the b-sequence as follows: matrix X
matrix Y
matrix Z
± h ---, q-Dh
± h - ~ 4- D i
+ h --* + D j
+ h --~ 3-Dk
± i --~ + D i
d: i --~ ± D j
-t- i ---~ -4- Dk
q- i ---+ ~ Dh
+j---+ + D j
± j --~ -4- Dk
+j ~ ±Dh
±j ~ ~-Oi
± k --~ -4- Dk
±k ~ ±Dh
-4-k ~ + D i
± k ~ 5:Dj
For example, the matrix X is
X=
Dh
Di
-Di
Dj
Dj
Dh
Di
-Di
Dj
Dj
Dj
Dh
Di
-Di
-Oi
Dj
Dj
Dh
D~
Di
--Di
nj
Dj
Dh
,
matrix W
338
C. Koukouvinos I Statistics & Probability Letters 26 (1996) 333- 338
where
Cl
C2 C3 ...
Cp
Cp C1 C2 ...
Cp-i
Dh =
C2 C3
C4
"'"
CI
We observe that p
DhD T = Ip × ~-]~CZi, i=1
DhD T = 0 and DhD~ = 0, since CaCb = 0, a ¢ b. Let T = (tij) be the circulant matrix of order 5, with elements (0, 1) such that tij
1,
if j - i =
0,
elsewhere
1
(i,j = 1 , 2 , . . . , 5 )
:
i.e. T = (0, 1,0, 0, 0). Then we obtain X X T = Is × (DhD] + 2DiDVi + 2DjDf ) + (T + r 4) × (-DiDTi + DjDf )
Similarly for the matrices y y T ZZ T and WW T, and then we have x x T + yyV + ZZ T + WW v = 515p × ~ C 2 : 80pZi2op2. i=1
Using the matrices X, Y, Z, W in OD(80p2; 2 0 p 2, 2 0 p 2, 20p 2, 20p2).
the
Goethals-Seidel
array,
we
obtain
an
orthogonal
design
References Box, G.E.P. and J.S. Hunter (1961), The 2k-p fractional factorial designs 1, Technometrics 3, 311-352. Geramita, A.V. and J. Seberry (1979), Orthogonal desiyns. Quadratic ]brms and Hadamard Matrices. (Marcel Dekker, New York, Basel). Kharaghani, H. (1985), New class of weighing matrices, Ars Combin. 19, 69-72. Kounias, S. (1977), Optimal 2k designs of odd and even resolution, in: Recent Developments in Statistics (Noah-Holland, Amsterdam) pp. 501-506. Montgomery, D.C. (1991), Design and analysis of Experiments (Wiley, New York, 3rd ed.). Shah, K.R. and B.K. Sinha (1989), Theory of Optimal Designs, Lecture Notes in Statistics, Vol. 54 (Springer, Berlin).