Statistics & Probability Letters 54 (2001) 189–192

A characterization for orthogonal arrays of strength twovia a regression model

Ling-Yau Chana ; ∗, Kai-Tai Fangb, Rahul Mukerjeec

aDepartment of Industrial and Manufacturing Systems Engineering, The University of Hong Kong,Hong Kong, People’s Republic of China

bDepartment of Mathematics, Hong Kong Baptist University, Hong Kong, People’s Republic of ChinacIndian Institute of Management, Post Box No. 16757, Calcutta 700 027, India

Received August 2000

Abstract

We give a characterization for orthogonal arrays of strength two in terms of D-optimality under a multiple regressionmodel with continuous factor levels. c© 2001 Elsevier Science B.V. All rights reserved

MSC: 62K15; 62K05

Keywords: Continuous factor level; D-optimality

1. Introduction and preliminaries

The universal optimality of a fractional factorial plan given by an orthogonal array (OA) is well known.For pioneering work in this area, we refer to Cheng (1980) who considered OAs of strength two; see Deyand Mukerjee (1999) for more details. While these results relate to factors with qualitative or discrete levels,here we characterize an OA of strength two in terms of D-optimality under a regression model where thefactor levels are allowed to vary continuously in the design space. As a consequence, the entries of the arrayare quantitative decision variables, as opposed to being mere symbols or preassigned discrete values, and thequestion of their optimal determination also becomes relevant. These issues are addressed in Theorem 1 belowwhich is our main result. Since here the factor levels are continuous rather than qualitative or discrete, thecharacterization given in Theorem 1 may have the potential of facilitating the study of the existence of OAsof strength two by reducing combinatorial complexity.For ease in reference, recall that an OA involving n rows, s columns, q1; : : : ; qs symbols and of strength

two is an n × s array, with entries in the jth column from a set of qj distinct symbols (1 6 j 6 s) such

∗ Corresponding author.E-mail address: plychan@hku.hk (L.-Y. Chan).

0167-7152/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reservedPII: S0167 -7152(01)00051 -7

190 L.-Y. Chan et al. / Statistics & Probability Letters 54 (2001) 189–192

that in every n× 2 subarray all possible pairs of symbols occur equally often as rows. Such an OA will bedenoted by Ln(q1×· · ·×qs). Some more notation and a lemma will be helpful. For any positive integer j, let

Pj(u) =1j! 2 j

dj

duj(u2 − 1) j;

be the Legendre polynomial of degree j. Consider any positive integer q(¿ 2). DeDne �q1 =−1 and �qq =1.Also, for q ¿ 3, let �qt (2 6 t 6 q − 1) be the zeros of the Drst derivative of Pq−1(u). These zeros aredistinct and contained in (−1; 1). Let �(q) =∏

(�qi − �qj)2, where the product extends over 16 i¡ j 6 q.Then the following lemma, well known in optimal regression designs, holds (Silvey, 1980).

Lemma 1. Let u1; : : : ; un ∈ [ − 1; 1] and U be an n × q matrix such that for 1 6 i 6 n; the ith row of Uis given by (1; ui; : : : ; u

q−1i ). Then

(a) det(U ′U )6 (n=q)q�(q);(b) equality holds in (a) if and only if n=q is an integer; and for 16 t 6 q; exactly n=q of the quantitiesu1; : : : ; un equal �qt .

2. Main result

Consider now a multiple regression model

yi = �0 +s∑

j=1

qj−1∑

t=1

�jtxtij + ei (16 i 6 n); (1)

where yi is the observation corresponding to the design point x(i) = (xi1; : : : ; xis)′, and �0 and �jt (1 6 t 6qj − 1; 1 6 j 6 s) are unknown parameters. Also, e1; : : : ; en are random errors which, as usual, have zeromeans and are homoscedastic and uncorrelated. The design points x(i); 16 i 6 n, are chosen from the designspace [− 1; 1]s.The design matrix associated with (1) is given by

X = [1n; X1; X2; : : : ; Xs]; (2)

where 1n is the n×1 vector with all elements being unity and, for 16 j6 s, Xj is an n× (qj−1) matrix withthe ith row given by (xij; : : : ; x

qj−1ij ); 16 i 6 n. A design, that is, a selection of design points x(i) (16 i 6 n)

in [− 1; 1]s is called D-optimal if it maximizes det(X′X). Let A be an n× s matrix with the (i; j)th elementxij (16 i 6 n; 16 j 6 s).

Theorem 1. Let x(i) = [− 1; 1]s; 16 i 6 n. Then

(a) det(X′X)6 n−(s−1)s∏

j=1

{(n=qj)qj�(qj)}; (3)

(b) equality in (3) holds if and only if the n × s matrix A = [xij] is an OA Ln(q1 × · · · × qs) of strengthtwo such that; for 16 j 6 s; the entries in the jth column of A are �qjt ; 16 t 6 qj.

Proof. (a) First note that if G and H are matrices such that G = [1n;H], then

det(G′G) = n det(Q′Q); (4)

L.-Y. Chan et al. / Statistics & Probability Letters 54 (2001) 189–192 191

where Q= (In − n−11n1′n)H and In is the n× n identity matrix. Hence by (2),det(X′X) = n det(R′R); (5)

where

R = [R1; R2; : : : ; Rs]; with Rj = (In − n−11n1′n)Xj; 16 j6 s: (6)

Also, writing

X̃ j = [1n; Xj]; 16 j 6 s; (7)

by (4) and (6),

det(X̃′jX̃ j) = n det(R

′jRj); 16 j 6 s: (8)

By (5)–(8) and Lemma 1(a),

det(X′X) = n det(R′R)6 ns∏

j=1

det(R′jRj) (9)

= n−(s−1)s∏

j=1

det(X̃′jX̃ j)6 n

−(s−1)s∏

j=1

{(n=qj)qj�(qj)} (10)

which proves (a).(b) Equality holds in (3) if and only if equality holds in (9) and (10). Now, by (6) and (7), equality holds

in (9) if and only if R′jRk = 0 for every j �= k, that is, if and only ifX̃

′j(In − n−11n1′n)X̃ k = 0 for every j �= k: (11)

Also, by Lemma 1(b), equality holds in (10) if and only if, for every 16 j 6 s, n=qj is an integer, and theentries in the jth column of A are �qjt ; 16 t 6 qj, each repeated n=qj times.We now prove the “only if” part of the result in (b). Suppose equality holds in (3). Then, as noted

above, (11) holds and A is an OA of strength unity, the entries in the jth column of A (1 6 j 6 s) being�qjt ; 1 6 t 6 qj. We shall show that A is actually an OA of strength two. Consider the n subarray of Agiven by its Drst two columns. For 1 6 t 6 q1, 1 6 t′ 6 q2, let ftt′ be the frequency with which the pair(�q1t ; �q2t′) appears as a row of this subarray. DeDne the q1 × q2 matrix F = [ftt′ ]. Also, for j = 1; 2, let Bjbe a qj × qj matrix with rows (1; �qjt ; : : : ; �qj−1qjt ); 1 6 t 6 qj. Since qjt ; 1 6 t 6 qj, are distinct, clearlyBj is nonsingular ( j = 1; 2). As A is an OA of strength unity with entries as stated above, from (7) and thedeDnitions of B1; B2 and F,

X̃′1X̃ 2 = B

′1FB2; 1′nX̃ j = (n=qj)1

′qjBj (j = 1; 2):

Hence by (11),

B′1{F− (n=(q1q2))1q11′q2}B2 = 0:Since both B1 and B2 are nonsingular, it follows that ftt′ = n=(q1q2) for all t; t′; that is, all pairs (�q1t ; �q2t′)occur equally often as rows in the subarray of A given by its Drst two columns. Similarly, the same conclusionholds for every n× 2 subarray of A. This proves the “only if” part.

The “if” part follows by reversing the above steps.

192 L.-Y. Chan et al. / Statistics & Probability Letters 54 (2001) 189–192

Remark 1. In particular, if one considers the Drst degree regression model in (1) then qj=2 and �(qj)=4; 16j 6 s. As such, Theorem 1 reduces to the well-known result which states that det(X′X)6 ns+1, with equalityif and only if A is Ln(2s) with entries ± 1. Theorem 1 extends this result to the general case of arbitraryq1; : : : ; qs.

Remark 2. Interestingly, a counterpart of Theorem 1 may not hold under other common optimality criteria.As an illustration, let n = 24; s = 2; q1 = 2; q2 = 3, and consider the A-criterion where the objective is tominimize tr(X′X)−1. Let d0 be a design which chooses the design points (± 1;± 1) each with frequency 3and (±1; 0) each with frequency 6. Under d0, tr(X′X)−1 =3=8. On the other hand, for every design based onan OA of strength two, that is, consisting of design points of the form (ai; bj) ∈ [− 1; 1]2 (i=1; 2; j=1; 2; 3),each with frequency 4, it can be shown that tr(X′X)−1¿ 3=8. Hence, although an OA of strength two triviallyexists in this setup, it does not yield an A-optimal design. This point, which may be contrasted with whathappens with qualitative factor levels, underscores the signiDcance of Theorem 1 by showing that results forcontinuous factor levels cannot be guessed by mere analogy with the qualitative or discrete case.

Acknowledgements

This work was supported by a grant from Hong Kong Research Grant Council and Centre for Managementand Development Studies, Indian Institute of Management, Calcutta.

References

Cheng, C.S., 1980. Orthogonal arrays with variable numbers of symbols. Ann. Statist. 8, 447–453.Dey, A., Mukerjee, R., 1999. Fractional Factorial Plans. Wiley, New York.Silvey, S.D., 1980. Optimal Design. Chapman and Hall, London.