On the E-and MV-optimality of block designs having k≥ v

Ann. Inst. Statist. Math. Vol. 40, No. 2, 407418 (1988)

ON THE E- AND MV-OPTIMALITY OF BLOCK DESIGNS HAVING k>_v MIKE JACROUX 1 AND DEXTER C. WHITTINGHILL III 2 ~Departrnent of Mathematics, Washington State University, Pullman, WA 99164, U.S.A. 2Department of Statistics, Temple University, Philadelphia, PA 19122, U.S.A. (Received April 8, 1986; revised October 8, 1986)

Abstract. In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where v treatments are applied to experimental units occurring in b blocks of size k, k _ v. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having v > k fail to yield optimal designs in the case where v < k . Some sufficient conditions are derived for the E- and MV-optimality of block designs having v < k and methods for constructing designs satisfying these sufficient conditions are given. K e y words a n d phrases:

Incidence matrix, C-matrix, eigenvalue, E-

optimality, MV-optimality. I.

Introduction

Let d denote a block design having v treatments arranged in b blocks of size k. The incidence matrix of d, denoted by Nd, is a v × b matrix whose entries ndU give the number of times treatment i occurs in block j. When k = v a + t (a_>0, 0_v. In this paper, we derive several sufficient conditions for designs to be E- and MV-optimal in classes D (v, b, k) where k> v and characterize the C-matrices of certain designs which satisfy the sufficient conditions given. Examples are also given to illustrate usage of the results obtained. 2.

Main results

In this section we give our main results. We begin by giving some notation which is used throughout the sequel. For a given class of designs D (v, b, k) and using Ix] to denote the greatest integer not exceeding some real number x_>0, we shall let r

(2.1)

=

bk = vr + p , a = [r/b] = [ k / v ] , r=ba+s, k=va+t, 2 = [(rk - s(a + 1)2 - (b - s)a2)/(v- 1)] , r k - s ( a + 1)2 - ( b - s ) a 2 = ( v - 1 ) 2 + q ,

O-O is an eigenvalue ofkCd*22-rkl~+,~,12J~,~. But we also see that kCd.22-rkI~+2~2J~,~ has constant row sums v212-rk>O with all of its off-diagonal entries non-positive (since by assumption 2d.0>2~2 for all i, j = v - f f + 1,..., v, i~j). Thus each diagonal element of kCd.22-rkI~+212J~,~ is nonnegative and it is at least as large in magnitude as the sum of the absolute values of the off-diagonal elements in the corresponding row. Hence kCd.22-rkl~+2~2J~.z is positive semi-definite, and the result follows from

ON THE E- AND MV-OPTIMAL1TY OF BLOCK DESIGNS HAVING k > v

415

Corollary 2.2. COROLLARY 2.4.

LetD(v,b,k)besuchthatp 1 is an integer, and that-~22~2d,v-~+l,v-~+l . . . . . 2d, v,v and zk_>~22-222. Then d* has Zd.l=r and is E-optimal in D(v, b, k).

~22, 222 are such

PROOF. Let kCd.22=((r+z)k-~22+222)Ip-222Jp.p in Corollary 2.2 and observe that the ~ - 1 largest eigenvalues of kCd*22 are all equal to (r+z)k222+222. The condition that zk>222-222 then insures that ((r+z)k-222+ 222)/k>r, and the result follows from Corollary 2.2. In the following two examples, we give illustrations as to h o w Corollary 2.4 can be applied.

Example 2.4. Consider the class of designs D(7, 8, k) where k = 7 a + 4 and the design d* having incidence matrix

Nd.=

a+l a+l a+l a+l a a a

a+l a+l a+l a a+l a a

a+l a+l a+l a a a+l a

a+l a+l a+l a a a a+l

a a a a+l a+l a+l a+l

a a a a+l a+l a+l a+l

a a a a+l a+l a+l a+l

a a a a+l a+l a+l a+l

It is then easy to verify that for any value of a_>3, the conditions of Corollary 2.4 are satisfied and that d* is E-optimal in D(7, 8, k).

Example 2.5. Consider the class of designs D(7, 8, k) where k = 7 a + 3 , a > 1, and the design d* given by

Nd.=

a%l a+l a+l a+l a-1 a a

a+l a÷l a+l a+l

a+l a+l a+l a+l

a

a

a-1 a

a a-1

a a a

a a a

a a a

a a a

a a a

a

a

a

a

a

a+l a+l a+l

a+l a+l a+l

a+l a+l a+l

a+l a+l a+l

a+l a+l a+l

416

MIKE J A C R O U X AND DEXTER C. WH1TTINGHILL 111

It is then easy to verify that for any value of a_>4, the conditions of Corollary 2.4 are satisfied and that d* is E-optimal in D (7, 8, k).

Comment. In all previous cases known to the authors where a widely used optimality criterion such as A-, D-, or E-optimality has been used to find an optimal design in D (v, b, k) and where an optimal design has actuallybeen determined, there has always existed at least one optimal design in M(v, b, k), i.e., there has always existed at least one optimal design with its C-matrix having maximal trace. However, in Example 2.5 an E-optimal design d* e (7, 8, k) cannot be in M(7, 8, k) for any value of a>_4. This is because an E-optimal design d* must satisfy the conditions of Theorem 2.2 and a necessary condition for this to occur is that whenever treatments i a n d j have rd,i----rdv=r, ha*ix=rid,ixfor x= 1,..., b. But since v-p=4>t=3, this can never happen if d* M(7, 8, k). Thus an E-optimal design in D(7, 8, k) where k = 7 a + 3 cannot be in M(7, 8, k) for any value of a>4. The authors have found that Example 2.5 is typical of what happens in any class D(v, b, k) where k=va+t and v-p>t, i.e., it is not difficult to show that in such classes when a is sufficiently large, an E-optimal design d* ~ D(v, b, k) having Zd*l----rcannot be in M(v, b, k). Comment. The examples given here are such that the E-optimal designs given have C-matrices of the form given in Corollary 2.2, Corollary 2.3 or Corollary 2.4. The reason for this is of course that designs having C-matrices of this form are the easiest to construct. There are various numerical conditions that one can prove in order to guarantee that a design has a C-matrix of the general form given in (2.3) (though the design may not satisfy all of the conditions of Corollary 2.2). For example one set of numerical sufficient conditions for a design to have a C-matrix of the form given in (2.3) is that s(t-v)/~ and (b-s)t/~ are integers. To see this, we note that a design d* satisfying Corollary 2.2 must have an incidence matrix of the form Nd,=[ (a + l)Jv-~,s Nd,21

aJv-~.h-s] Nd.22 "

Now, if(sk-(v-~)s(a+ l))/fi is an integer and ((b-s)k-(v-~)(b-s)cO/~ is an integer, then it is easy to see that it is possible to assign treatments v - f i + 1..... v to blocks in Nd*21and Nd*22SO that the row sums in Nd*2~and Na,22 are constant. Thus the design will have 2d.ij=,;tt2 for all i= 1,..., v - ~ , j = v - ~ + 1,..., v. Now, the fact that (sk-(v-fi)s(a+ l))/fi is an integer implies that s(t-v)/~ is an integer and ((b-s)k-(v-~)(b-s)a)/~ being an integer implies that (b-s)t/~ is an integer. Whether or not a design d* satisfying the numerical conditions given above satisfies all of the conditions of Corollary 2.2 depends upon whether v{rk-(v-~)2~tJ/~>rk and just how treatments v - ~ + l , . . . , v are assigned to blocks in Nd*21and Nd*22.

ON THE E- AND

LEMMA 2.4. d ~ D (v, b, k),

MV-OPTIMALITY

OF BLOCK DESIGNS HAVING

k>v

417

Let D(v, b, k) be such t h a t p < v - 2 . Then f o r any design

max Vard(~ti- &j) ___2 / r . i~j PROOF. Since p < v - 2 , it follows that d must have at least two treatments, say treatments i and j, such that rd~+raj