OPTIMAL REPEATED MEASUREMENTS DESIGNS ... - Project Euclid

The Annals of Statistics 1997, Vol. 25, No. 6, 2328 ] 2344

OPTIMAL REPEATED MEASUREMENTS DESIGNS: THE LINEAR OPTIMALITY EQUATIONS1 BY H. B. KUSHNER Nathan S. Kline Institute for Psychiatric Research In approximate design theory, necessary and sufficient conditions that a repeated measurements design be universally optimal are given as linear equations whose unknowns are the proportions of subjects on the treatment sequences. Both the number of periods and the number of treatments in the designs are arbitrary, as is the covariance matrix of the normal response model. The existence of universally optimal ‘‘symmetric’’ designs is proved; the single linear equation which the proportions satisfy is given. A formula for the information matrix of a universally optimal design is derived.

1. Introduction. In the class RMDŽ t, N, p . of repeated measurements designs}where t is the number of treatments, N the number of subjects and p the number of periods}a subject is repeatedly exposed to various treatments. Many authors w Cheng and Wu Ž1980., Kunert Ž1983., Hedayat and Zhao Ž1990.x assume the response model

Ž 1.1.

yi j s tdŽ i , j. q r dŽ i , jy1. q p j q bi q « i j ,

1 F i F N, 1 F j F p,

in which the design allocates treatment dŽ i, j . to the ith subject in the jth period. tdŽ i, j. and r dŽ i, j. are the treatment and carryover effects, p j is the period effect and bi is the subject effect. The N independent vectors « i s Ž « i j ., 1 F j F p, are multivariate normal with mean 0, and p = p covariance matrix, C. Our chief interest is to determine all designs which minimize functions which depend on the proportions of subjects assigned to each treatment sequence. We use ‘‘proportions’’ in the sense customary in approximate design theory, the approach of this paper. The functions considered are the standard information functions. Previous optimality results contain restrictions on p, t, C, the competing class of designs and the class of designs proved to be optimal. A necessary and sufficient condition for universal optimality was given for the case p s 2 in Hedayat and Zhao Ž1990., who also pointed out that most other optimality

Received October 1995; revised July 1997. 1 Research supported in part by National Institute of Mental Health Grants MH 42959 and P50MH51359. AMS 1991 subject classifications. Primary 62K05; secondary 62K10. Key words and phrases. Optimal repeated measurements designs, universal optimality, treatment effect, carryover effect, treatment sequence.

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REPEATED MEASUREMENTS DESIGNS

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results relate to the situation t F p. The conditions p s t and p ' 1 Žmod t . appear in Cheng and Wu Ž1980. and p s t in Kunert Ž1985.. The case t s 2 was treated in Laska and Meisner Ž1985. and in Matthews Ž1987.. Pioneering work w Hedayat and Afsarinejad Ž1978.; Cheng and Wu Ž1980.x concentrated on the case C s Ip . However, a subject’s p responses are not likely to be independent. Moreover, the optimality of a design usually depends on C Žan exception is the case p s 2.. In almost all subsequent work w Kunert Ž1985.; Matthews Ž1987.x , C is assumed to be a first-order autoregressive matrix, but since this assumption is highly specialized, there remains the open problem of obtaining optimal designs for a general covariance matrix. Also, many authors place limitations on the class of competing designs. For example, in Cheng and Wu Ž1980., some optimality results are proved when the class of competing designs excludes those in which a treatment is preceded by itself. Even when the class of competing designs is the full class RMDŽ t, N, p ., optimality results are often only proved for interesting subclasses of designs w Cheng and Wu Ž1980., Section 3; Kunert Ž1985.x . None of the above limitations, except for p s 2, appear in Hedayat and Zhao Ž1990. Žtheir assumption that C is a correlation matrix is easy to remove.. The present paper provides a comparable treatment of the general optimal design problem in approximate theory. The main results of this paper are the following. For any p, t and C, a necessary and sufficient condition that the proportions of subjects on treatment sequences determine a universally optimal Žresp. strictly F-optimal. design is that they satisfy a system of linear equations ŽTheorems 5.3 and 5.4.. Surprisingly, ‘‘strict F-optimality’’}the optimality of a design with respect to any strictly concave information function F}is equivalent to the universal optimality of the design. A formula Ž4.13. for the information matrix of a universally optimal Žresp. strictly F-optimal. design is derived. For any F, the related formula Ž4.12. can be used both to check a design’s optimality and to compute the F-efficiency of a nonoptimal design. For any p, t or C, there exist ‘‘symmetric’’ designs which are universally optimal. In a symmetric design ŽSection 3., equal proportions of subjects are assigned to all treatment sequences within a ‘‘symmetry block’’ ŽSection 5.. These proportions are determined by just one linear equation ŽTheorem 5.5., which is also a necessary condition for any optimal design. A symmetric design consisting of at most two symmetry blocks always exists ŽCorollary 5.6.. In Kushner Ž1996, 1997., we study solutions of the optimality equations. We derive the designs of Cheng and Wu Ž1980., Kunert Ž1983. and Matthews Ž1987, 1990. and give new universally optimal designs. 2. The information matrix of treatment and carryover effects. In this section, we give a formula ŽLemma 2.4. for C d Žt , r ., the information matrix of treatment and carryover effects. The formula concretely exhibits the treatment sequence proportions whose determination is the aim of opti-

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H. B. KUSHNER

mal design theory. The matrix, C ; , which also appears in the formula, significantly affects a design’s optimality. 2.1. Notation. In denotes the n = n identity matrix, Jn the n = n matrix 1 2 of 1’s and Jm, n s Jm, n the m = n matrix of 1’s. Jm, n is the m = n matrix 1 whose first row is zero and final m y 1 rows are 1’s. 1 m s 11m s Jm, 1 and m 2 2 1 m s Jm, 1. T , 1 F m F 2, are, respectively, the Np = t design matrices of treatment and carryover effects; Ti1 and Ti2 are the p = t period-treatment and period-carryover incidence matrices for subject i, so that T m s wŽ T1m .X , ŽT2m .X , . . . , ŽTNm .X xX . The first row of Ti2 is zero; for 2 F j F p, the jth row of Ti2 is the Ž j y 1.st row of Ti1. Note that

Ž 2.1.

1 pm s Tim1 t ,

1 F i F N, 1 F m F 2.

Write Td,m i to designate the matrix Tim when it is necessary to indicate the design d. In Kunert Ž1991., T 1 s Td , T 2 s Fd , Ti1 s Td, i , Ti2 s Fd, i . Let

Ž 2.2.

M m s Ny1

N

Ý Tim ,

1 F m F 2.

is1

by

2.2. Formulas for C d Žt , r .. The matrix C ; . Define the p = p matrix C ;

Ž 2.3.

C ;s Cy1 y Cy1 Jp Cy1 r1Xp Cy1 1 p .

Note that if x is a p-dimensional vector, then

Ž 2.4.

C ; x s 0 iff

x s a 1 p , a a scalar.

LEMMA 2.1. The information matrix of treatment and carryover effects is given by

Ž 2.5.

Cd Ž t , r . s

ž

C d11 C d 21

/

C d12 , C d 22

where C d mn s ŽT m .X ST n , 1 F m , n F 2, are t = t matrices and S s Ž IN y JN rN . m C ; . See Cheng and Wu Ž1980. for the case C s Ip . An application of Lemma 2.1 is: when p s 2, optimality is independent of the covariance matrix. For any 1 y1 . C, C ; s b Ž y1 1 , b a scalar, and scalar multiples of information matrices do not affect optimality. See Hedayat and Zhao Ž1990. for the case C a correlation matrix. COROLLARY 2.2. if

Any design which is optimal when C s C0 is also optimal

Ž 2.6. C s a C0 q g 1Xp q 1 pg X ,

a a scalar , g a p-dimensional vector.

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PROOF. The C in Ž2.6. is the most general solution of the matrix equation, C ; s ay1 C0; . Hence, C d Žt , r . is the same for the covariance matrices Ž2.6. and C s a C0 . Consequently, the same situation holds for the information matrix of treatment effects. I An application of Corollary 2.2 is: designs optimal with C s Ip are also optimal with C s a Ip q g 1Xp q 1 pg X , a ‘‘type-H’’ matrix w Huynh and Feldt Ž1970.x . In fact, for any C of this type, C ;s ay1 Ž Ip y Jprp . . LEMMA 2.3. Ži. The submatrices of C d Žt , r . are given by

Ž 2.7.

N

C d mn s

Ý Ž Tim y M m . C ; Ž Tin y M n . , X

1 F m , n F 2.

is1

Žii. The row and column sums of C d mn , 1 F m , n F 2, are zero. 2.3. Treatment sequences: proportions and treatmentrcarryover incidence matrices. Let t denote a treatment sequence characterized by a p-tuple of integers, t s Ž t 1 , t 2 , . . . , t p ., 1 F t j F t, where t j is the treatment given in the jth period, 1 F j F p. The matrices Tim depend only on the treatment sequence to which a subject is assigned: Ttm s Tim, where Tt1 and Tt2 are, respectively, the common treatment and carryover incidence matrices for all subjects assigned to t . If nt is the number of such subjects, then N s Ýt nt , where t }and in subsequent sums omitting the domain of t }runs over all t p treatment sequences. With pt s ntrN the treatment sequence proportion,

Ž 2.8.

1s

Ý pt ,

pt G 0.

t

2.4. C d Žt , r . as a sum over treatment sequences and their proportions. The ˆd , Mˆdm. The matrimatrices Tt , Ttm, Tˆt , Tˆtm and their convex means, M d , M dm , M 1 ˆ ˆ w ces L d , L d , L d mn . Let Tt be the p = 2 t matrix Tt s Tt , Tt2 x . Consistent with Ž2.2., define the convex means, M dm, M d , of the matrices Ttm , Tt by M dm s Ýt pt Ttm, 1 F m F 2, and M d s Ýt pt Tt . Let L d s Ýt pt TtX C ; Tt . LEMMA 2.4. The information matrix of treatment and carryover effects is given by the sum

Ž 2.9.

C d Ž t , r . s N Ý pt Ž Tt y M d . C ; Ž Tt y M d . . X

t

From Ž2.9., C d Žt , r . may be thought of as the weighted variance of the Tt matrices occurring with multiplicity Npt . A matrix version of a familiar identity is C d Žt , r . s NL d y NM dX C ; M d . More generally, for any p = 2 t matrix, T,

Ž 2.10. C d Ž t , r . s N Ý pt Ž Tt y T . C ; Ž Tt y T . y N Ž T y M d . C ; Ž T y M d . . X

t

X

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H. B. KUSHNER

ˆd , Mˆd1, Mˆd2 and L ˆ d , the Set T s w Jp,1 t , Jp,2 t x rt and define Tˆt , Tˆt1 , Tˆt2 , M centered counterparts of the matrices without the ‘‘hats,’’ by Ž 2.11.

Tˆt s Tt y T s Tˆt1 , Tˆt2 s Tt1 y Jp1, trt , Tt2 y Jp2, trt ,

Ž 2.12.

ˆd s Md y T s Mˆd1 , Mˆd2 s Md1 y Jp1, trt , Md2 y Jp2, trt , M

Ž 2.13.

ˆd s L

Ý pt TˆtX C ; Tˆt . t

From Ž2.10.,

Ž 2.14.

ˆ d y N Ž Mˆd . C ; Mˆd . C d Ž t , r . s NL X

ˆd , Finally, define the t = t submatrices of L Ž 2.15.

ˆ d mn s L

Ý pt Ž Tˆtm . C ; Tˆtn , X

1 F m , n F 2.

t

3. Permuted and symmetric designs, convex combinations of designs. In this section, we discuss permuted designs, the convex combination of designs and symmetric designs. The exploitation of designs of these types is the theme of the method of this paper. 3.1. P, the array of proportions. Permuted designs and their information matrices. A repeated measurements design d is specified by P s Ž pt ., the t p proportions on the treatment sequences. Sometimes we refer to the design d by P and indicate the relation between d and P by d l P. Let s denote a permutation of 1, 2, . . . , t 4 , that is, s is an element of the symmetric group, St . Let st be the treatment sequence st s Ž s Ž t 1 ., s Ž t 2 ., . . . , s Ž t p ... For each s g St , define a new design, the permuted design, Ps , by Ps s Ž psy1 t ., designated also by ds . Let Hs be the t = t matrix representing the permutation s . Since ds results from a relabelling of the treatments, the d and ds incidence matrices of the ith subject and their means are related by

Ž 3.1.

Tdms , i s Tdm, i Hs ,

M dms s M dm Hs ,

s g St , 1 F i F N, 1 F m F 2.

It follows from Ž2.7. and Ž3.1. that C ds mn s HsX C d mn Hs ,

Ž 3.2. Ž 3.3.

C d s Ž t , r . s Ž I2 m

HsX

Tstm s Ttm Hs ,

s g St , 1 F m , n F 2,

. C d Ž t , r . Ž I2 m Hs . ,

s g St ,

s g St , 1 F m F 2,

and, with C d Žt . the information matrix of treatment effects w see Shah and Sinha Ž1989., page 4x ,

Ž 3.4.

C ds Ž t . s HsX C d Ž t . Hs ,

s g St .

I 3.2. The convex combination of designs. For 1 s Ý is1 a i , a i G 0, 1 F i F I, i the convex combination of the I designs d i l P s Ž pti ., 1 F i F I, is da l


2333

I I I P a s Ž pta . s Ý is1 a i P i , where pta s Ý is1 a i pti . We write da s Ý is1 a i d i . The da play an important part in the sequel. However, they may not exist in exact design theory. For example, Ž1, 2. and Ž2, 1. are two sequences of integers whose convex sum, Ž1, 2.r2 q Ž2, 1.r2 s Ž3r2, 3r2., is not a sequence of integers.

3.3. Concavity of C d Žt . as a function of d. LEMMA 3.1 w Pukelsheim Ž1993., pages 74]77x . A s, the Schur complement of A, is a concave, nonincreasing function of A G 0. Let j denote all the fixed effects in Ž1.1. and let C d Ž j . denote their information matrix. THEOREM 3.2. C d Žt . is a concave function of d:

Ž 3.5.

CdaŽ t . G

I

Ý a i Cd Ž t . .

is1

i

PROOF. From C d Ž j . s N Ýt pt Bt , where Bt are fixed matrices,

Ž 3.6.

CdaŽ j . s

I

Ý a i Cd Ž j . .

is1

i

From Ž3.6., Lemma 3.1 and C d Žt . s C ds Ž j . for a suitable Schur complement, we get Ž3.5.. I 3.4. Optimality of convex combinations of designs. Let F denote a concave information function, defined on the set of symmetric nonnegative matrices, of the type standard in optimality theory: d is F-optimal if it maximizes F Ž C d Žt ... If F is strictly concave, d is strictly F-optimal. LEMMA 3.3. Ži. The set of F-optimal designs is convex: if d i , 1 F i F I, are F-optimal designs, then so is da . In particular, if d is F-optimal, then so is da s Ýs g S t as ds , 1 s Ýs g S t as , as G 0, s g St . Žii. C d Žt . s b Ž tIt y Jt ., b a scalar, for every strictly F-optimal d. PROOF. Ži. Equation Ž3.5. and the concavity of F. Žii. Equation Ž3.4. and Kiefer’s Ž1975., Proposition 1 Remark. I NOTES. Ža. In a different problem, Silvey wŽ 1980., page 17x notes the convexity of the set of optimal designs. Žb. Part Žii. also holds in exact design theory. 3.5. Symmetric optimal designs. A symmetric design generalizes the notion of a dual balanced design, introduced for t s 2 by Laska and Meisner

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H. B. KUSHNER

Ž1985.. Design d is symmetric if

Ž 3.7.

d s ds ,

s g St ,

that is, if the proportions of d satisfy

Ž 3.8.

s g St , all t .

pst s pt ,

For a model without fixed subject effects, Laska and Meisner Ž1985. proved that an optimal dual balanced design exists; for model Ž1.1., see Matthews Ž1987.. THEOREM 3.4.

A symmetric F-optimal design exists.

PROOF. Use Lemma 3.3Ži. with as s 1rt!, s g St . The proportions in the F-optimal design

Ž 3.9.

PS s

žÝ / sgS t

Ps

t!

satisfy Ž3.8.. I We call d S l PS the symmetrized design of d l P. 4. The information matrix of strictly F-optimal and universally optimal designs. Here we derive a formula for the treatment effects information matrix of a design which is either strictly F-optimal or universally optimal. Important roles in the analysis are played by the quadratic function associated to a treatment sequence, the quadratic function of the design and by symmetric and ‘‘symmetrized’’ designs. We briefly discuss the efficiency of nonoptimal designs. 4.1. The quadratic function associated to a treatment sequence. For each treatment sequence, t , define a nonnegative quadratic, qt Ž s ., by

Ž 4.1.

ž

/

ž

X

qt Ž s . s tr Tˆt1 q sTˆt2 C ; Tˆt1 q sTˆt2 s

t q11

q

t 2 q12 s

q

t q22 s2 ,

/

y` - s - `,

where

Ž 4.2. NOTE.

t qmn s tr Ž Tˆtm . C ; Tˆtn , X

1 F m , n F 2,

t t s q21 Ž q12 ..

Equation Ž4.1. defines a genuine quadratic. For, t q22 s tr Ž Tˆt2 . C ; Tˆt2 s 0 X

implies 0 s ŽTˆt2 .X C ; Tˆt2 and then, from Pukelsheim wŽ 1993., page 15x and Ž2.4., 0 s C ; Tˆt2 , and 1 p xX s Tˆt2 s Tt2 y Jp,2 trt, where x, a t-dimensional column vector, must vanish since both Tt2 and Jp,2 t have zero first rows. This t t t t .2 gives the contradictory 0 s tTt2 y Jp,2 t . Both q11 s 0 and q11 q22 y Ž q12 s0 can and do occur.


2335

LEMMA 4.1. The quadratics corresponding to treatment sequences t and st are identical: qst Ž s . s qt Ž s . ,

s g St .

PROOF. This follows from Ž3.3. and Ž2.11.. I Let Cy1 s Ž c jk ., 1 F j, k F p, so that C ;s Ž ˜ c jk . s Ž c jk y c j c krc tot . ,

Ž 4.3.

1 F j, k F p,

p p where c j are the row sums, and c tot s Ý js1 Ý ks1 c jk is the total sum, of Cy1 . Let

ž

/

dmn Ž t . s d ttjqkq1y1ymn ,

1 F j, k F p, 1 F m , n F 2

Žwhere the symbol t 0 is left undefined but 0 ' d tt 0 s dmt 0 , 1 F m F t .. A 0 t convenient computational formula for qmn is t 22 qmn s tr dmn Ž t . C ; y ˜ c 11dmn rt ,

1 F m , n F 2.

4.2. QŽ s, P ., the quadratic of a design. Information matrix of a symmetric design. For any array of proportions, P, define the quadratic of a design, QŽ s, P ., and its coefficients, qmn Ž P . w q12 Ž P . s q21Ž P .x , by Q Ž s, P . s

Ý pt qt Ž s . s q11Ž P . q 2 q12 Ž P . s q q22 Ž P . s 2 , t

so that, from Ž2.15. and Ž4.2.,

ˆ d mn , qmn Ž P . s tr L

Ž 4.4. LEMMA 4.2. QŽ s, PS ..

1 F m , n F 2.

For any design, P, and its symmetrized design, PS , QŽ s, P . s

PROOF. Use Lemma 4.1 and Ž3.9.. I LEMMA 4.3.

Ž 4.5. Ž 4.6.

Suppose that d l P is a symmetric design. Then

Cd Ž t , r . s N

q12 Ž P .

q21 Ž P .

q22 Ž P .

/

m Ž tIt y Jt . rt Ž t y 1 . ,

2 C d Ž t . s N Ž q11 Ž P . y q12 Ž P . rq22 Ž P . . Ž tIt y Jt . rt Ž t y 1 .

sN

Ž 4.7.

ž

q11 Ž P .

ˆdm s 0, M

min

y`-s-`

Q Ž s, P . Ž tIt y Jt . rt Ž t y 1 . ,

1 F m F 2.

PROOF. If d is symmetric, then Ž3.1. and Ž3.7. imply that

Ž 4.8.

M d s X m 1Xt

where X is a p = 2 matrix.

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H. B. KUSHNER

But from Ž2.1., M d 1 t s w 1 p , 12p x . Hence, X s w 1 p , 12p x rt, which, with Ž4.8., implies Ž4.7.. Similarly, Lemma 2.3Žii., Ž3.2. and Ž3.7. imply

ž

/

a

a 12 a 22 m Ž tIt y Jt . ,

Ž 4.9. C d Ž t , r . s a 11 21

d symmetric, amn scalars.

But from Ž2.14. and Ž4.7.,

ˆd , C d Ž t , r . s NL

Ž 4.10.

d symmetric.

Hence, Ž4.5. follows from Ž4.4., Ž4.9. ] Ž4.10. and

ˆ d mn , t Ž t y 1 . amn s tr C d mn s N tr L

1 F m , n F 2,

d symmetric.

Equation Ž4.6. is obtained by taking Schur complements of Ž4.5.. I 4.3. The maximum of F Ž C d Žt .., for any F. The information matrix of a strictly F-optimal Ž resp. universally optimal . design. THEOREM 4.4.

Let

Ž 4.11.

b s max P

min

y`-s-`

Q Ž s, P . .

Then Ži. for any F, max F Ž C d Ž t . . s NbF Ž tIt y Jt . rt Ž t y 1 . .

Ž 4.12.

d

Žii. d is F-optimal for F strictly concave Ž resp. universally optimal . if and only if its treatment effects information matrix is

Ž 4.13.

C d Ž t . s Nb Ž tIt y Jt . rt Ž t y 1 . .

PROOF. From Theorem 3.4, Lemma 4.2 and Ž4.6., for any F,

Ž 4.14.

max F Ž C d Ž t . . s d

max

d symmetric

F Ž Cd Ž t . .

s NF Ž tIt y Jt . max P

min

y`-s-`

Q Ž s, P . rt Ž t y 1 . ,

which is Ž4.12.. Necessity of Ž4.13.: If F is strictly concave, Ž4.14. also shows that, in Lemma 3.3Žii., b s Nbrt Ž t y 1.. If d is universally optimal, then, of course, it is optimal for the strictly concave F’s, so Ž4.13. is also true in this case. Sufficiency of Ž4.13.: this follows from Ž4.12.. I 4.4. Minimax and quadratics. The study of the b in Ž4.11. is the subject of this section. We present an analysis for any quadratics, rather than just for the quadratics Ž4.1.. One can, for example, apply these results to the quadratics obtained from Ž4.1. with Cy1 in place of C ; , which occur in Laska and Meisner Ž1985.. Suppose that Bi Ž x ., 1 F i F I, are nonnegative, nonconstant quadratics in x: Bi Ž x . s a i q bi x q c i x 2 G 0, y` - x - `, c i ) 0, 1 F i F I.

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Define the scalars a, b and the function B Ž x . by B Ž x . s max Bi Ž x . 4 ,

y` - x - `,

1FiFI

Ž 4.15.

bs

Ž 4.16.

B Ž a. s

min

y`-x-`

BŽ x. , max Bi Ž x . 4 s b.

min

y`-x-` 1FiFI

The quantity a is uniquely defined by Ž4.16., since Bi Ž x . is a convex function and B Ž x ., the maximum of convex functions, is convex and is not constant on any interval. w For simplicity, we use the same symbol ‘‘b’’ in Ž4.11. and Ž4.15.. When the quadratics are Ž4.1., Theorem 4.5 shows that the two b’s are identical.x Let N ; 1, 2, . . . , I 4 be the set of indices for which

Ž 4.17.

b s B Ž a . s Bi Ž a . ,

i g N.

In a neighborhood of x s a, B Ž x . depends only on the quadratics whose indices are in N : for d ) 0 sufficiently small, B Ž x . s max i g N Bi Ž x .4 , a y d x - a q d . With SI the simplex in Section 3.2, define BŽ x, a . s

I

Ý a i Bi Ž x . ,

y` - x - `,

a s Ž a 1 , a 2 , . . . , a I . g SI .

is1

THEOREM 4.5. Ži. B Ž x, a . has the minimax property

Ž 4.18.

max

min

a gS I y`-x-`

BŽ x, a . s

min

max B Ž x , a . s b.

y`-x-` a gS I

Žii. The two equations

Ž 4.19a.

BŽ x, a . s b

and

Ž 4.19b.

BX Ž x , a . s

I

Ý a i BiX Ž x . s 0

is1

are simultaneously satisfied if and only if

Ž 4.20. Ž 4.21a.

x s a,

a i s 0,

Ž 4.21b.

1s

Ž 4.22.

BX Ž a, a . s

Ý

ig N

i f N,

ai ,

a i G 0,

i g N,

Ý a i BiX Ž a. s 0.

ig N

Moreover, a solution of Ž4.21. ] Ž4.22. exists with either one or two nonzero a i ’s. PROOF. Ži. It is obvious that max

min

a gS I y`-x-`

BŽ x, a . F s

min

max B Ž x , a .

min

max Bi Ž x . s

y`-x-` a gS I

y`-x-` 1FiFI

min

y`-x-`

B Ž x . s b.

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H. B. KUSHNER

To prove Ž4.18., we show that

Ž 4.23.

min

y`-x-`

B Ž x , a . s b for some a g SI .

Case 1. There is an iU g N such that BiXU Ž Ua. s 0. Then miny`- x -` BiU Ž x . s BiU Ž a. s b. Hence, Ž4.23. holds for a iU s d ii . Case 2. There are i 1 , i 2 g N such that BiX 1Ž a. ) 0 and BiX 2Ž a. - 0. Then for

a iU s yBiX 2Ž a . r Ž BiX 1Ž a . y BiX 2Ž a . . , if i s i 1 ,

Ž 4.24.

s BiX 1Ž a . r Ž BiX 1Ž a . y BiX 2Ž a . . , if i s i 2 ,

s 0, otherwise, we get BX Ž a, a U . s 0 and therefore Ž4.23. holds for the a U in Ž4.24.. The solutions of Ž4.22. given in cases 1 and 2 have at most two components of a nonzero. Case 3. The derivatives BiX Ž a., i g N , have the same sign. This case cannot occur because B Ž x . would be strictly increasing or decreasing in a neighborhood of x s a and could not attain the minimum Ž4.23. there. Žii. For an x and a satisfying Ž4.19.,

Ž 4.25. b s

I

I

is1

is1

Ý a i Bi Ž x . F Ý a i Bi Ž a . s Ý a i B i Ž a . q Ý a i Bi Ž a . . ig N

if N

But Ž 4.26.

b ) Bi Ž a . , i f N, Ž . Ž . Ž . and if 4.21a were not true, 4.17 , 4.25 and Ž4.26. would give a contradiction. If Ž4.20. were not true, then Ž4.17. and Ž4.25. with Ž4.21. inserted gives the contradictory bs

Ý a i Bi Ž x . - Ý a i Bi Ž a. s b.

ig N

ig N

Hence Ž4.20. is true and Ž4.22. results from Ž4.19b.. I The concluding result of this section is an algorithm for calculating a, b and the set N . Let 1 F i, j F I. An intersection point of Bi Ž x . and Bj Ž x ., that is, a point Ž x i j , yi j . such that

Ž x i j . s Bj Ž x i j . ,

y i j s Bi X Bi Ž x i j . F 0

is called admissible if and BXj Ž x i j . G 0. In this definition, we do not exclude i s j, in which case Ž x i i , yi i . s Žybir2 c i , a i y bi2r4c i . is still called an admissible ‘‘intersection’’ point. LEMMA 4.6. Ži. Bi Ž x . and Bj Ž x . have at most one admissible intersection point. Žii. b s max yi j 1FiFjFI Ž x i j , yi j . admissible

Žiii. a s x m n , where Ž m, n. is any pair such that b s ym n .


2339

INDICATION OF PROOF. Use max a g S I B Ž x, a . s max 1 F i F I Bi Ž x . and Ž4.18.. For the set of quadratics qt Ž s .4 , define a, b and q Ž s . by q Ž s . s max qt Ž s . 4 ,

Ž 4.27.

y` - s - `,

t

Ž 4.28.

bs

Ž 4.29.

q Ž a. s

min

qŽ s. ,

min

max qt Ž s . 4 s b.

y`-s-` y`-s-`

t

Let T be the set of treatment sequences at which minimax Ž4.29. is achieved:

Ž 4.30.

b s q Ž a . s qt Ž a . ,

t g T.

When C is known, a, b and T can be rapidly determined by Lemma 4.6. From Ž4.21a., in an optimal design,

Ž 4.31.

pt s 0,

t g T.

Hence the support of any optimal design is a subset, possibly proper, of T. 4.5. Design efficiency. The F-efficiency of design d is defined as eff F Ž d . s F Ž C d Ž t . . rF Ž C dU Ž t . . ,

Ž 4.32.

where d is any design and dU is F-optimal. The denominator in Ž4.32. is given by Ž4.12.. If d l P is symmetric, the numerator in Ž4.32. is obtained from Ž4.6., yielding

Ž 4.33.

eff Ž d . s bdrb,

d symmetric,

where bd s

min

y`-s-`

2 Q Ž s, P . s q11 Ž P . y q12 Ž P . rq22 Ž P . .

We omit F in Ž4.33. because, for symmetric d, F-efficiency is independent of F. 5. The optimality equations. The matrix equation Ž4.13. can be considered to be a very complicated system of equations for the proportions of a universally optimal Žresp. strictly F-optimal. design. However, it is possible ˆ d mn to reduce Ž4.13. to a system of linear equations featuring the matrices L ˆdm. These matrices are much simpler than C d Žt .. The entries in Cd Žt , r . and M ˆ d and Mˆd are linear are quadratic functions of the proportions and those of L functions of the proportions. In this section, if R is a 2 t = 2 t matrix, then the t = t submatrices R i j , 1 F i, j F 2, are defined as in Ž2.5.. Rq denotes the Moore]Penrose inverse of R. If w is a scalar, then wqs 1rw, w / 0 and 0qs 0.

2340

H. B. KUSHNER

ˆsd . 5.1. The matrix L If R G 0 is a 2 t = 2 t matrix, then

LEMMA 5.1.

Ž 5.1.

q

tr R s F tr R 11 y Ž tr R 12 . Ž tr R 22 . . 2

X PROOF. From Pukelsheim wŽ 1993., page 75x , for T s Ž It , X t . , X t a t = t matrix, q T X RT s R s q Ž X t q Rq 22 R 21 . R 22 Ž X t q R 22 R 21 . . X

Ž 5.2.

Set X t s xIt , x a scalar. Then R s F T X RT s R 11 q xR12 q xR 21 q x 2 R 22 and

Ž 5.3.

tr R s F tr R 11 q 2 x tr R 12 q x 2 tr R 22 , all x.

In Ž5.3., set x s yŽtr R12 .Žtr R 22 .q to obtain Ž5.1.. I LEMMA 5.2. Ži. For any d,

ˆd . C d Ž t , r . F NL

Ž 5.4. Žii. If d is optimal, then

ˆsd . C d Ž t . s NL

Ž 5.5.

PROOF. Ži. This follows from Ž2.14.. Žii. From Ž5.4.,

ˆsd . C d Ž t . s C ds Ž t , r . F NL

Ž 5.6.

From Ž4.4., Lemma 5.1 and Ž4.13., 2 ˆsd F N Ž q11 Ž P . y q12 N tr L Ž P . rq22 Ž P . . s tr C d Ž t . .

Ž 5.7.

Equation Ž5.5. follows from Ž5.6. and Ž5.7.. I 5.2. First form of the optimality equations.

ˆdm in THEOREM 5.3. With a, b and T given in Ž4.28. ] Ž4.30., Tˆtm, M ; Ž2.11. ] Ž2.12. and C in Ž2.3., d is a universally optimal Ž resp. strictly F-optimal . design in approximate design theory iff the treatment sequence proportions, pt , satisfy the linear equations Ž 5.8.

ž

X

pt Ž Tˆt2 . C ; Tˆt1 q a Ž Tˆt2 . C ; Tˆt2 s 0, C;

X

X

žÝ

tg T

Ž 5.11.

/

Ý

tg T

Ž 5.10.

X

pt Ž Tˆt1 . C ; Tˆt1 q a Ž Tˆt1 . C ; Tˆt2 s b Ž tIt y Jt . rt Ž t y 1 . ,

tg T

Ž 5.9.

ž

Ý

ž

pt Tˆt1 q aTˆt2

/

/ / s 0, pt s 0,

if t f T .

2341


ˆ d mn given in Ž2.15., we will prove Ž5.8. ] Ž5.10. in the form PROOF. With L Ž 5.8.

X

ˆ d11 q aL ˆ d12 s b Ž tIt y Jt . rt Ž t y 1 . , L

Ž 5.9.

X

ˆ d 21 q aL ˆ d 22 s 0, L

Ž 5.10.

X

ž

/

ˆd1 q aMˆd2 s 0. C; M

Let f be a symmetric optimal design and g s dr2 q fr2, an optimal design ˆf , B s L ˆ d and D s L ˆ g s Ar2 q Br2. From from Lemma 3.3Ži.. Set A s L Ž5.2. for R s A, B and D, Ž4.13. and Ž5.5.,

Ž X t q Aq22 A 21 . A 22 Ž X t q Aq22 A 21 . X q q Ž X t q Bq 22 B21 . B22 Ž X t q B22 B21 . X q s Ž X t q Dq for all X t . 22 D 21 . D 22 Ž X t q D 22 D 21 . , X

Ž 5.12.

X X Ž . Setting X t s yDq 22 D 21 in 5.12 , we obtain 0 s Y A 22 Y q W B22 W, where X q q q Y s yDq 22 D 21 q A 22 A 21 and W s yD 22 D 21 q B22 B21 , implying 0 s Y A 22 Y X s W B22 W, and from Pukelsheim wŽ 1993., page 15x ,

Ž 5.13.

0 s A 22 Y s B22 W .

From Ž2.13., the row and column sums of Amn , Bmn and Dmn , 1 F m , n F 2, vanish and, therefore, so do those of Y. From Ž4.10. and Lemma 4.3, Amn s qmn Ž P .Ž tIt y Jt .rt Ž t y 1., 1 F m , n F 2, where f l P, and from Section 4.1, q22 Ž P . / 0. Hence, the first equality in Ž5.13. implies 0 s Y s yDq 22 D 21 y aŽ It y Jtrt . and then the second equality in Ž5.13. implies

Ž 5.14.

W s a Ž It y Jtrt . q Bq 22 B21 .

ˆ d 21 q aL ˆ d 22 . Equation Equations Ž5.13. ] Ž5.14. give Ž5.9.X : 0 s B21 q aB22 s L Ž5.8.X follows from Ž4.13., Ž5.5. and Ž5.9.X : ˆ d11 y L ˆ d12 L ˆq ˆ b Ž tIt y Jt . rt Ž t y 1 . s Ny1 C d Ž t . s L d 22 L d 21 ˆ d11 q aL ˆ d12 L ˆq ˆ sL d 22 L d 22 ˆ d11 q aL ˆ d12 . sL Finally, from Ž4.13., Ž5.3. with R s C d Žt , r ., Ž2.14. and Ž4.4.,

Ž 5.15.

Nb s tr C d Ž t . F tr C d11 q 2 xC d12 q x 2 C d 22

ž

/

X

ž

ˆd1 q xMˆd2 C ; Mˆd1 q xMˆd2 s NQ Ž x , P . y N tr M

/

,

ˆd1 q aMˆd2 .X C ; Ž Mˆd1 q aMˆd2 .x . and setting x s a in Ž5.15., gives b F b y trwŽ M X ; 1 2 1 2 ˆd q aMˆd . C Ž Mˆd q aMˆd .x , equivalent to Ž5.10.X . I Hence, 0 s trwŽ M 5.3. Second form of the optimality equations. Define treatment-period proportion sums by Pmj s Ýt pt d tmj s Ý t js m pt , Pmjkn s Ýt pt d tmj d tnk s ÝŽ t j, t k .sŽ m, n. pt , 1 F j, k F p, 1 F m, n F t,and treatment proportion sums by p Pm s Ý js1 Pmj , 1 F m F t. Set Pmj0n s Pm0 nj s Pm00n s Pm0 s 0, 1 F j F p, 1 F m, n F t.

2342

H. B. KUSHNER

THEOREM 5.4. The optimality equations in Theorem 5.3 are equivalent to Ž5.11. and the following linear system: b Ž t dm n y 1 . rt 2 Ž t y 1 . q Ž art . wm u p

s

Ý Ž Pmj,nk q aPmj,nky1 . ˜c jk ,

1 F m, n F t ,

j, ks1 p

ywm urt q ac˜11 rt 2 s

,k , ky1 q aPmjy1 . ˜c jk , Ý Ž Pmjy1 n n

1 F m, n F t ,

j, ks1

Pmj s wm s j q 1rt ,

1 F m F t , 1 F j F p,

where a, b and T are as given in Theorem 5.3 and ˜ c jk in Ž4.3.; s j s Ž1 y j Žya. .rŽ1 q a., 1 F j F p; wm s w Ž Pm y prt ., 1 F m F t; w s Ž1 q a. 2r p Ž p q aŽ1 q p y Žya. p ..; u s Ž1 q a.y1 Ý js1 Žya. j ˜ c j1. Ž If a s y1, formulas for s j and u are obtained by taking limits.. INDICATION OF PROOF. Sum Ž5.8. ] Ž5.10. using Ttm s Ž d tmjq 1y m ., 1 F j F p, 1 F m F t for 1 F m F 2. I 5.4. Symmetry blocks, inequivalent treatment sequences, symmetric and symmetrized designs. The ‘‘symmetry block,’’ ² k :, a set of treatment sequences, is defined by ² k : s t : t s sk , s g St 4 s the orbit of k under St . card² k : s Ž t . k s t Ž t y 1. ??? Ž t y k q 1., where k s number of distinct integers in k s Ž k 1 , k 2 , . . . , k p .. Call k 1 equivalent to k 2 iff k 1 s sk 2 , s g St . Let k run over a set of inequivalent treatment sequences. Then all t p treatment sequences s

D ²k : k

is a disjoint union. In a symmetric design, P s Ž pt ., proportions are the same for every treatment sequence in a symmetry block:

Ž 5.16a.

pt s pk , if t g ² k : ,

Ž 5.16b.

1s

Ý card ² k : pk ,

pk G 0,

k

where the k in Ž5.16b. }and in subsequent sums in this section}runs over a set of inequivalent treatment sequences. In any design, P, let

Ž 5.17a.

Pk s

Ý

tg ² k :

pt

denote the sum of proportions over symmetry block ² k :. Thus

Ž 5.17b.

1s

Ý Pk . k

The proportions of the symmetrized design Ž3.9. are given by

Ž 5.18.

ptS s Pkrcard ² k : , if t g ² k : .


2343

5.5. A condition necessary for an optimal design and necessary and sufficient for a symmetric optimal design. THEOREM 5.5. Ži. The equation

Ž 5.19.

0s

Ý

Pk qkX Ž a .

kg T

is a necessary condition that d be universally optimal, where Pk is given in Ž5.17a. and the k in Ž5.19. runs over inequivalent treatment sequences in T. Žii. A symmetric design is optimal iff Ž5.11. and Ž5.19. hold. PROOF. Lemma 4.2, Theorem 4.4Ži. and Theorem 4.5 imply Ž5.19.. If d is symmetric and Ž5.19. holds, then Ž4.6. and Theorem 4.5 imply that d is optimal. I NOTE. any F.

We write simply ‘‘optimal’’ in Theorem 5.5 because it is true for

COROLLARY 5.6. If T contains two or more inequivalent treatment sequences, then there exists a universally optimal symmetric design, consisting of just two symmetry blocks, ² k 1 : and ² k 2 :, with sum of proportions Pk i , 1 F i F 2, given by

Ž 5.20.

Pk 1 s yqkX 2Ž a . r Ž qkX 1Ž a . y qkX 2Ž a . . , Pk 2 s qkX 1Ž a . r Ž qkX 1Ž a . y qkX 2Ž a . .

and proportions pk i s Pk irŽ t .k i . If all treatment sequences in T are equivalent, that is, if T s ² k :, then there exists a universally optimal symmetric design with common proportion pk s 1rŽ t . k . PROOF. Equation Ž5.20. results from Ž5.19. when just two Pk ’s are nonzero. If T consists of two symmetry blocks, then the derivatives in Ž5.20. are automatically of opposite sign. Otherwise, choose k 1 and k 2 to be any treatment sequences in T such that qkX iŽ a. have opposite signs. I Acknowledgments. I am deeply grateful to Morris Meisner for many valuable discussions and to Eugene Laska, who suggested the topic of this research. In a number of places in this paper, my debt to their previous research in this field is also recorded. I am especially indebted to a referee, whose long labor resulted in a huge shortening of the paper, substantial gains in clarity and many contributions. Among these are his short proof of Theorem 4.5Ži. and the use of the matrix inequality Ž5.4.. He has also shown that Theorem 4.4 can be derived from this inequality and Kiefer’s Proposition 1.

2344

H. B. KUSHNER

REFERENCES CHENG, C. S. and WU, C. F. Ž1980.. Balanced repeated measurements designs. Ann. Statist. 8 1272]1283. ŽCorrection Ž1983.. Ann. Statist. 11 349.. HEDAYAT, A. and AFSARINEJAD, K. Ž1978.. Repeated measurements designs, II. Ann. Statist. 6 619]628. HEDAYAT, A. and ZHAO, W. Ž1990.. Optimal two-period repeated measurements designs. Ann. Statist. 18 1805]1816. HUYNH, H. and FELDT, L. S. Ž1970.. Conditions under which mean square ratios in repeated measurements designs have exact F-distributions. J. Amer. Statist. Assoc. 65 1582]1589. KIEFER, J. Ž1975.. Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models ŽJ. N. Srivistava, ed... North-Holland, Amsterdam. KUNERT, J. Ž1983.. Optimal design and refinement of the linear model with applications to repeated measurements design. Ann. Statist. 11 247]257. KUNERT, J. Ž1985.. Optimal repeated measurements designs for correlated observations and analysis by weighted least squares. Biometrika 72 375]389. KUNERT, J. Ž1991.. Cross-over designs for two treatments and correlated errors. Biometrika 78 315]324. KUSHNER, H. B. Ž1996.. Optimal and efficient repeated measurements designs for uncorrelated observations. Submitted for publication. KUSHNER, H. B. Ž1997.. Optimality and efficiency of two-treatment repeated measurements designs. Biometrika 84 455]468. LASKA, E. and MEISNER, M. Ž1985.. A variational approach to optimal two-treatment crossover designs: application to carryover-effect models. J. Amer. Statist. Assoc. 80 704]710. MATTHEWS, J. N. S. Ž1987.. Optimal crossover designs for the comparison of two treatments in the presence of carryover effects and auto-correlated errors. Biometrika 74 311]320. MATTHEWS, J. N. S. Ž1990.. Optimal dual-balanced two treatment crossover designs. Sankhya 52 332]337. PUKELSHEIM, F. Ž1993.. Optimal Design of Experiments. Wiley, New York. SHAH, K. R. and SINHA, B. K. Ž1989.. Theory of Optimal Designs. Lecture Notes in Statist. 54. Springer, New York. SILVEY, S. D. Ž1980.. Optimal Design. An Introduction to the Theory for Parameter Estimation. Chapman and Hall, London. NATHAN S. KLINE INSTITUTE FOR PSYCHIATRIC RESEARCH STATISTICAL SCIENCES AND EPIDEMIOLOGY DIVISION 140 OLD ORANGEBURG ROAD ORANGEBURG , NEW YORK 10962 E-MAIL : [email protected]