Some new orthogonal designs in linear regression models

STATISTICS& PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 29 (1996) 125-129

Some new orthogonal designs in linear regression models Christos Koukouvinos Department of Mathematics, National Technical University of Athens, Zoorafou 15773, Athens, Greece Received 1 April 1995

Abstract In two or three level factorial experiments and in other linear regression models the coefficients of the unknown parameters can take one out of two or three values. Orthogonal designs are obtained from Hadamard matrices or weighing matrices. In this paper we construct some new orthogonal designs using sequences with zero autocorrelation function.

A M S Subject Classification: Primary 62K05, 62K15; Secondary 05B15 Keywords: Factorial experiments; Linear model; Autocorrelation; Construction

1. Introduction Consider an experimental situation in which a response y depends on k factors X l , . . . , x k with the first order relationship o f the form y -- X f l + e, where y is an n × 1 vector o f observations, the design matrix X is n × (k + 1) whose j t h row is o f the form (1,xjl,xj2 . . . . . xjk), j = 1,2 . . . . . n, fl is the (k + 1) × l vector o f coefficients to be estimated, and e is the n × 1 vector o f errors. W e assume that e is a random vector distributed with mean 0 and covariance matrix a2I. In a two-level factorial design, each xij can be coded as -4-1. The design is then determined by the n × (k + 1) matrix o f elements -4-1. The ith column gives the sequence o f factor levels for factor xi, each row constitutes a run. When k - n - 1, the design is called a saturated design and the design matrix is an n × n square matrix. Note that n = k + 1 is the minimal number o f points (rows) required to estimate all coefficients o f interest (the fli's). 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 two or three levels. Factorial designs o f resolution III are such that all main effects are estimable, ignoring two-factor interactions and all higher order interactions. Several criteria have been advanced for the purpose o f comparing designs and for constructing optimal designs. One o f the most popular is the D-optimality criterion, which seeks to maximize det(XTX), the 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0167-7152(95)00165-4

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determinant of the X T X matrix. Recall that for any n × n matrix H consisting entirely of elements -4-1, the maximum determinant possible is det(Ha'H) = n n. In a three-level factorial design, each xij can be coded as 0, -4-1. Suppose we are given k objects to be weighed in n weighings with a chemical balance (two-pan balance) having no bias. Let 1 if t h e j t h object is placed in the left pan in the ith weighing

xij =

-1

if the jth object is placed in the right pan in the ith weighing

0 if the jth object is not weighed in the ith weighing. Then the n × (k + 1) matrix X = (xij) completely characterizes the weighing experiment. We assume X to be a non-singular matrix. Then the best linear unbiased estimator of fl is /~ = ( X r X ) - I x T y with covariance of fl, Cov(/~) = a2(XTX) -1. It has been shown by Hotelling (1944) that for any weighing design the variance of fli cannot be less than a2/n. Therefore, we shall call a weighing design X-optimal if it estimates each of the weights with this minimum variance, a2/n. Kiefer (1975) proved that an optimal weighing design in our sense is actually optimal with respect to a very general class of criteria. It can be shown that X is optimal if and only if X X X = nI. This means that a chemical balance weighing design X is optimal if it is an n × (k + 1) matrix of -4-1 whose columns are orthogonal. For a more detailed study of optimal weighing designs, the reader should consult Mood (1946), Raghavarao (1971) and Banerjee (1975).

2. Orthogonal designs An orthogonal design of order n and type (sl,s2 . . . . . Su) (si > 0), denoted OD(n;si,s2,...,su), on the commuting variables Xl,X2,... ,Xu is an n × n matrix A with entries from {0, -4-xl, +x2 .... , ±Xu} such that AA r =

six \i=1

In. /

Alternatively, the rows of A are formally orthogonal and each row has precisely si entries of the type +xi. In Geramita et al. (1976) where this was first defined, it was mentioned that

ArA =

six \

i=1

In /

and so our alternative description of A applies equally well to the columns of A. It was also shown in Geramita et al. (1976) that u<~p(n), where p(n) (Radon's function) is defined by p(n) = 8c + 2 a, when n = 2ab, b odd, a = 4 c + d , 0 ~ < d < 4 . 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 WWT --- k/n, and W is equivalent to an orthogonal design OD(n; k). The number k is called the weight of W. If k = n, that is, all the entries of W are -4-1 and WWT = nln, then W is called an Hadamard matrix of order n. Thus we have: any k (~
NA(S)=Zaiai+s, i=1

s=0,1,...,n--

1.

(1)

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127

If A(z) = al + a2z + ... + anz n-l is the associated polynomial of the sequence A, then n

n

A(z)A(z-1) = ~

Z

n--I

aiajzi-J : NA(O) + Z N A ( s ) ( z S + z-S),

i=1 j = l

z ~ O.

(2)

s=l

Given A as above of length n the periodic autocorrelation function PA(s) is defined, reducing i + s modulo n, as n

PA(S) = ~ a i a i + s ,

s = 0, 1. . . . . n -- 1.

(3)

i=1

The following theorem which uses four circulant matrices is very useful in our construction for orthogonal designs. Theorem 1 (Geramita and Seberry, 1979, Theorem 4.49). Suppose there exist four circulant matrices A, B, C, D o f order n satisfying AA r + BB T + CC T + DD r = fin. Let R be the back diagonal matrix. Then

-BR GS =

A

-CR -DTR -DR

CTR

DTR - C T R A

BTR

-BTR

A

is a W(4n, f ) when A, B, C, D are (0, 1 , - 1 ) matrices, and an orthogonal desi#n OD(4n;sl,s2 .... ,Su) on XhX2 . . . . . X~ when A, B, C, D have entries f r o m { 0 , + x , . . . . . + Xu} and f = ~.=,(sjx~).

Corollary 1. I f there are four sequences A, B, C, D o f length n with entries f r o m {0, -¢-xl, -4-x2,5:x3, +x4} with zero periodic or non-periodic autocorrelation function, then these sequences can be used as the first rows o f circulant matrices which can be used in the Goethals-Seidel array to f o r m an OD(4n;sl,s2,s3,s4). We note that if there are sequences o f length n with zero non-periodic autocorrelation function, then there are sequences o f length n + m for all m >>.O.

3. New sequences with zero autocorrelation function and some new orthogonal designs In this section we give new sequences with entries from {0, g-a, 4-b, 4-c, 4-d} on the commuting variables a, b, c,d and zero autocorrelation function. Then we use these sequences to construct some new orthogonal designs. If the variables a,b,c,d take values from the set {0, 1 , - 1 } , then we produce some new orthogonal 2k and 3k factorial designs. Theorem 2. There exist orthogonal desions OD(28; 1,4, I0, 10), OD(28; 1,9,9,9), OD(28;2,3,6,9), OD(28;2,4,4, 18), OD (28; 4, 5, 5,9), 0 D ( 2 8 ; 5 , 5 , 9 , 9 ) constructed using four circulant matrices in the Goethals-Seidel array.

C. Koukouvinos l Statistics & Probability Letters 29 (1996) 125-129

128

Table 1 Sequences of length 7 with zero periodic autocorrelation function Design OD(28; 1,4, 10, 10)

Sequences AI= A2 = A3= A4 =

OD(28; 1,9,9,9)

A3 =

(-b (-b (a

A4

(-d

AI=

( ( ( (

AI= A2 =

OD(28; 2, 3, 6, 9)

A2= A3 = A4

OD(28; 2, 4, 4, 18)

AI = A2 = A3 = A4 =

OD(28; 4, 5, 5, 9)

AI = A2 = A3 A4 =

OD(28; 5, 5, 9, 9)

( a -d -d d -d d d) (c b b -c -c c 0) (-b -c -c -d 0 -e b) ( 0 c -c -d -d e -d)

AI= A2 = A3 = A4 =

a

0 -d a 0 a b a d

(-d ( a ( c (-d

b -d

c c -e -d -c -d

d b d d

d d b

c -c

d)

d b b -c c b

b) b) c)

0 a d -b c) b 0 -c d -d) c c 0 -d 0) c -e 0 d 0) d -b d c d) a c -d -d d) d -d -d b -a) d b d -c d)

-b c -c 0 a b) 0 -a b -c -c 0) '-b d d -d -d -b -c) d -d -d 0 -d 0 -d) a

a

d -d a b -c b c) c -d a d a -b -c) a -d -a c d c c) b d d -c d -b c)

Proof. W e u s e t h e s e q u e n c e s g i v e n in T a b l e 1, w h i c h h a v e z e r o p e r i o d i c a u t o c o r r e l a t i o n f u n c t i o n , as the first r o w s o f the c o r r e s p o n d i n g c i r c u l a n t m a t r i c e s in the G o e t h a l s - S e i d e l a r r a y to o b t a i n t h e r e q u i r e d o r t h o g o n a l designs.

[]

Theorem 3. There exist orthogonal designs

OD(4n;2,4,8,9), OD(4n;1,8,16), OD(4n; 1 , 9 , 1 6 ) , OD(4n;2,7,10), OD(4n;2,7,13), OD (4n; 2, 8, 18), OD (4n; 3, 9, 14), OD (4n; 7, 10)

for all n >~7, constructed us&g four circulant matrices in the Goethals-Seidel array. P r o o f . W e u s e the s e q u e n c e s g i v e n in T a b l e 2, w h i c h h a v e zero n o n - p e r i o d i c a u t o c o r r e l a t i o n f u n c t i o n , as t h e first r o w s o f t h e c o r r e s p o n d i n g c i r c u l a n t m a t r i c e s in t h e G o e t h a l s - S e i d e l a r r a y to o b t a i n the r e q u i r e d o r t h o g o n a l designs. S i n c e t h e s e s e q u e n c e s h a v e z e r o n o n - p e r i o d i c a u t o c o r r e l a t i o n f u n c t i o n , the s e q u e n c e s are first p a d d e d w i t h sufficient zeros a d d e d to the e n d to m a k e t h e i r l e n g t h n >~ 7. [ ]

C Koukouvinos I Statistics & Probability Letters 29 (1996) 125-129

129

Table 2 Sequences o f length 7 with zero non-periodic autocorrelation function Design

Sequences

0/)(28;2,4,8,9)

AI =

(-c

A2=(c A3 = ( - c

A4=(c OD(28; 1,8,16)

Al=

(--c

A2 =

OD(28; 1,9, 16)

OD(28; 2, 7, 10)

b

OD(28; 2, 8, 18)

-b b

c 0

b

-c -b

,41 ~

--C

-b

A2 =

c

A3 = A4 =

--c c

AI =

-c

b 0

A3 ----

--c

A4 =

--c

0 b

a a c c

A1=

b -b -c

b -b b

b c

c a -b

0 c -c c -b

0 -d a c

0 0

-b -d

0

0

b

c)

-c -c

b -b

c) -c)

b

-c

-b

-c)

0

c

c

0)

b 0

-c c

0 b

0) c)

0 0 b

-c c b

0 0 c

0) 0) c)

b

c

-c

c

-b b b

b

-c

c

0 -c

-c)

-b

-c

'--c '--c --c

b

0

a

A2 = A3 : A4 =

c) -c)

-c

a

A1 :

c -c

c) -c)

b b

c

c

c)

a

c -c

b c

b

-c

-c) c) -c)

-c -c

c

c

b

-d

b

c

-d

-d

c

-c

b

a a

-c -c

0 a

~: ---=

-c

0

b

A4=

OD(28;7, 10)

d d

--C

A2 ---A3 =

OD(28; 3, 9, 14)

-b

A4 =

A1 A2 A3 A4

0 0

-d

A3 =

A 2 -:

OD(28; 2, 7, 13)

d -b

c

c -c

b a

-c

c -c

-c) b)

c

-b)

c

-b)

c

-b)

c

a)

-b

c)

0 c

-b b

-c c

c b

a) -c) -b)

AI =

(-b

0

0

a

b

0

A2 = A3 =

(--a (-b

b 0

-a 0

0 a

a -b

0 0

0) 0)

A4 =

(-b

a

b

0

b

a

b)

References Banerjee, K.S. (1975), Weighing Designs for Chemistry, Medicine, Economics, Operations Research and Statistics (Marcel Dekker, New York). A.V., J.M. Geramita and J. Seberry Wallis (1976), Orthogonal designs, Linear and Multilinear Al#ebra 3, 281-306. A.V. and J. Seberry (1979), Orthogonal Desi#ns. Quadratic forms and Hadamard Matrices (Marcel Dekker, New York). H. (1944), Some improvements in weighing and other experimental techniques, Ann. Math. Statist. 15, 297-306. (1975), Construction and optimality of generalized Youden designs, in: J.N. Srivastava, ed., Statistical Design and Linear Models (North-Holland, Amsterdam) pp. 333-353. Mood, A. (1946), On Hotelling's weighing problem, Ann. Math. Statist. 17, 4 3 2 - 4 4 6 . Raghavarao, D. (1971), Constructions and Combinatorial Problems in Design of Experiments (Wiley, New York). Geramita, Geramita, Hotelling, Kiefer, J.