27 Optimal and efficient treatment-control designs

S. Ghosh and C. R. Rao, eds., Handbook of Statistics, Vol. 13 © 1996 Elsevier Science B.V. All rights reserved.

A'/

Optimal and Efficient Treatment-Control Designs

Dibyen Majumdar

1. Introduction Comparing treatments with one or more controls is an integral part of many areas of scientific experimentation. In pharmaceutical studies, for example, new drugs are the treatments, while a placebo and/or a standard treatment is the control. Most attention will be given to the use of a single control; we will consider designs for comparing treatments with a control for various experimental settings, models and inference methods. This article is expected to update, supplement and expand upon the earlier survey of the area by Hedayat, Jacroux and Majumdar (1988). The section titles are: 2. Early development. 3. Efficient block designs for estimation. 4. Efficient designs for confidence intervals. 5. Efficient row-column designs for estimation. 6. Bayes optimal designs. 7. On efficiency bounds of designs. 8. Optimal and efficient designs in various settings.

2. Early development Consider an experiment to compare v treatments (which will be called test treatments), with a control using n homogeneous experimental units. Let the control be denoted by the symbol 0 and the test treatments by the symbols 1 , . . . , v. We assume the model to be additive and homoscedastic, i.e., if treatment i is applied to unit j then the observation yij can be expressed as: y~j = # + T i + e ~ j ,

(2.1)

where # is a general mean, "ri is the effect of treatment i and e~j's are random errors that are assumed to be independently normally distributed with expectation 0 and variance cr2. A design d is characterized by the number of experimental units that are assigned to each treatment. For i = 0, 1 , . . . , v, we will denote by rdi the number of experimental units assigned to treatment i, or replication of treatment/. 1007

1008

D. Majumdar

Given n what is the best allocation of the experimental units to the control and the test treatments, i.e., what are the optimal replications? Unless the experimenter has some knowledge about the performance of the control that can be used at the designing stage, it is intuitively clear that the control should be used more often than the test treatments, since each of the v test treatments has to be compared with the same control 0. One way to determine an optimal allocation is by considering the Best Linear Unbiased Estimators (BLUE's) of the parameters of interest, which are the treatment-control contrasts, Ti -- 7"0, i = 1 , . . . , V. For a design d, let the BLUE of 7"i - 7"0 be denoted by ~di -- ~dO. For model (2.1), it is clear that ~di -- ~dO : ffdi -- ffd0, where Ydi is the average of all observations that receive treatment i. Also, (2.2)

V a r ( ~ d , - ~d0) = ~2(r~l + rd01) •

A possible allocation (design) is that which minimizes )--Jill Var(?di -- ?ao). We will presently (Definition 2.2 later in this section) call this design A-optimal for treatmentcontrol contrasts, since it minimizes the average (hence the 'A' in A-optimal) of the variances of the v treatment-control contrasts. It is easy to see that a design that is A-optimal for treatment-control contrasts is obtained by minimizing the expression, v

E (?~/1 _1_rd1 ) i=l subject to the constraint, ~i=0 v r d i = n . The following theorem is obvious. THEOREM 2.1. If v is a square and n - 0 (mod (v + v/~)), then a design do given

by rdol . . . . .

rdov = n / ( v

q- V ~ ) ,

rdoO = V ~ r c l o l

(2.3)

is A-optimal f o r treatment-control contrasts for model (2.1).

This result was noticed by Fieler (1947), and possibly even earlier (see also Finney, 1952). Thus, if v = 4 test treatments have to be compared with a control using n = 24 experimental units then the A-optimal design assigns 4 units to each test treatment and 8 to the control. Dunnett (1955) found that the same allocation performs very well for the problem of simultaneous confidence intervals for the treatment-control contrasts 7-i - TO, i = 1,..., V. More discussion of Dunnett's work and that of other researchers in the area of multiple comparisons can be found in Section 4. Next consider the situation where the experimental units are partitioned into b blocks of k homogeneous units each. Here too, we assume the model to be additive and homoscedastic, i.e., if treatment / is applied to the unit l of block j then the observation yijt can be expressed as: yijt = # + ~-i +/3j + ~ijt,

(2.4)

Optimal and efficient treatment-control designs

1009

where/33- is the effect of block j. A design d is an allocation of the v + 1 treatments to the bk experimental units. We shall use the notation 79(v + 1, b, k) to denote the set of all connected designs, i.e., those designs in which every treatment-control contrast is estimable. For a design d, let ndij denote the number of times treatment i is used in block j. Further, for treatment symbols i and i t (i ~ it), let Adii, = gS, -"v . ndi~ndi,~ -~J ~ 1 We shall use the notation BIB(v, b,r, k, ~) to denote a Balanced Incomplete Block (BIB) design based on v treatments, in b blocks of size k each, where each treatment is replicated r times and each pair of treatments appears in )~ blocks. Cox (1958), p. 238, recommended using a BIB (v, b, r, k - t, )~) design based on the test treatments with each block augmented or reinforced by t replications of the control, where t is an integer. Das (1958) called such designs reinforced BIB designs. The idea was to have all test treatments balanced, as well as to have the test treatments balanced with respect to the control, and reinforcing BIB designs is a natural way to do it. Pearce (1960) took a different approach to achieve the same goal. He proposed a general class of designs for the problem of treatment-control comparisons, of which reinforced BIB designs is a subclass. These were the designs with supplemented balance that were proposed by Hoblyn et al. (1954) in a different context. Designs with supplemented balance have v + 1 treatments, one of which is called the supplemented treatment, the control in this case. •

J

J

"

DEFINITION 2.1. A design d E :D(v + l, b, k) is called a design with supplemented balance with 0 as the supplemented treatment if there are nonnegative integers .~d0 and .kdl, such that: )~dii' :

"~dl, for i , i r = 1,... ,v (i ~ i'),

AdOi = )~dO,

(2.5)

for i = 1 , . . . , v.

For a reinforced BIB design of Das (1958), AdO = tr. Pearce provided examples of these designs, their analysis for the linear model (2.4), and computed the standard error of the BLUE for the elementary contrasts. He showed that every elementary contrast in two test treatments have the same standard error, while every treatment-control contrast have the same standard error (see also Pearce, 1963). EXAMPLE 2.1. The following design, due to Pearce, was used in an experiment involving strawberry plants at the East Mailing Research Station in 1953 (see Pearce, 1953). Four herbicides were compared with a control, which was the absence of any herbicide, in four blocks of size seven each. Denoting the herbicides by 1,2, 3, 4 and the control by 0, the design, with columns as blocks, is the following: 0000 0000 1111 2222 3333 4444 1234

D. Majumdar

1010

According to Pearce (1983), this was the very first design with supplemented balance. Note that for this design ,Xd0 = 10, )~d~ = 6. Analysis of data for the experiment can be found in Chapter 3 of Pearce (1983) as well as in Pearce (1960). Given v, b and k, the question is, which design in 79(v+ 1, b, k) should be used? The answer would depend on the particular circumstances of the experiment, on various factors and constraints that usually confront the experimenter. (In this context, we refer the reader to page 126 of Pearce (1983) for an account of the circumstances that led to the use of the design in Example 2.1.) One of the considerations, possibly the most important one, is which design results in the best inference of the treatment-control contrasts, which are the contrasts of primary interest in the experiment. Determination of optimal designs for inference and efficiencies of designs in 79(v + 1, b, k) with respect to the optimal design in this class started much later. It may be noted that usually there are several designs with supplemented balance within 79(v + 1, b, k). Often there is a design with supplemented balance that is optimal in 79(v + 1, b, k) or at least highly efficient. On the other hand, usually the class of designs with supplemented balance also contain designs that are quite inefficient. Clearly one has to make a judicious choice of a design, even when one decides to restrict oneself to the class of designs with supplemented balance. Several optimality criteria have been considered in the literature (see Hedayat et al. (1988), and the discussions in that article). For a large portion of this article, we will focus on two optimality criteria for estimation because of the natural statistical interpretation of these criteria in experiments to compare test-treatments with a control. These criteria are given in the next definition which is quite general, i.e., it is not restricted to block designs and model (2.4). DEFINITION 2.2. Given a class of designs 79, and a model, if ~di -- ~a0 denotes the

BLUE of 7a~ - 7-a0, then a design is A-optimal for treatment-control contrasts (abbreviated as A-optimal) if it minimizes ~-'~.iv 1 Var(@di -- ~d0) in 79. A design is MV-optimal for treatment-control contrasts (abbreviated as MV-optimal) if it minimizes

M a x Var(~di -- ~dO) in 79.

l<~i~v

Pesek (1974) compared a BIB design in all v + 1 treatments (which may be viewed as a design with supplemented balance) with a reinforced BIB design and noted that the latter is more efficient according to the A-optimality criterion. Optimality of the reinforced designs in a restricted class of designs was established by Constantine (19 8 3 ). Constantine's class consisted of designs that had exactly one replication of the control in each block. In this class Constantine showed that a reinforced BIB design is A-optimal. Jacroux (1984) showed that Constantine's" conclusions remain valid when the BIB design is replaced by a suitable group divisible design in the test treatments. B echhofer and Tamhane (1981) rediscovered designs with supplemented balance when they were considering the problem of constructing simultaneous confidence intervals for the treatment-control contrasts. They called their designs Balanced Treatment Incomplete Block (BTIB) designs, a terminology which has been adopted by

Optimal and efficient treatment-control designs

1011

many authors in the area since 1981, as the Bechhofer and Tamhane paper inspired much of the research on optimal and efficient designs in :D(v+ 1, b, k). More discussion on these results will appear in Sections 3 and 4. For the two-way elimination of heterogeneity model (row-column designs), Freeman (1975) considered some designs for comparing two sets of treatments. More discussion on this topic will appear in Section 5. Starting with Owen (1970), several authors incorporated prior information on the model parameters at the designing stage to determine Bayes optimal designs. Some of these results will be discussed in Section 6.

3. Efficient b l o c k designs for estimation

Consider block designs with observations following the one-way elimination of heterogeneity model, given by (2.4). The object of the experiment is to estimate the treatment-control contrasts: (7-1-7-0, 7-2-7-0,..., 7-v-7-0)- Let Nd = (nazi), a (v+ 1) × b matrix, and

P= ( - i v Iv) where lv is a v × 1 matrix of l's and lv is the v x v identity matrix. Then the covariance matrix for the BLUE's (Fal - ~ao, ~a2 - ?ao,..., ~av - ~ao) of the treatment-control contrasts is a 2 P C ~ P p, where for a design d C D(v + 1, b, k), 1

/

Cd = Diag(rd0, rdl, . . . , rdv) -- -~NdN~t, the Fisher Information matrix for the treatments (test treatments and control). If one partitions Cd as:

Cd=(

cdOO~[dMd'YPd)

(3.1)

then it can be shown that (see Bechhofer and Tamhane, 1981),

( P C ~ P ' ) -1 = Md, i.e., Md is the information matrix for the treatment-control contrasts. Clearly (see Definition 2.2) an A-optimal design minimizes tr(Md ~) in 79(v + 1, b, k) and an MVoptimal design minimizes the maximum diagonal element of M~ -1 in 79(v + l, b, k).

3.1. Orthogonal designs It has long been known that one way to obtain an optimal block design is to construct, if possible, an orthogonal block design, such that within each block the replication of

D. Majumdar

1012

treatments are optimal for a zero-way elimination of heterogeneity model (i.e., model (2.1)). This result, and its generalization has been used by several authors, including Magda (1980) and Kunert (1983) in the context of repeated measurement designs. The idea is to establish optimality of a design for a model from the two facts: the design is optimal under a simpler model and the design is orthogonal. We state a general result, which is essentially a different version of the result given by Magda and Kunert. First some preliminaries. Let 79 be a class of designs. Consider two models for an n × 1 random vector of observations Y, when the observations are taken according to a design d E 79: Model A:

Y

=

X l d T + X2dO2 ~- c,

M o d e l B:

Y

=

X l d T ~- X2dO2 ~- X3dO3 -J-

(3.2)

where r is a vector of treatments and Oi are vectors of nuisance parameters (for example block parameters in (2.4), or row and column parameters in (5.1), and so forth). The X ' s are known matrices and ~ is a vector of random errors with E(e) = 0, V(e) = ~r2I. In Kunert's terminology, Model B is finer that Model A. THEOREM 3.1. Suppose do E 79 is A- (MV-) optimal for treatment-control contrasts under Model A, and

x;
(3.3)

then do is A- (MV-) optimal for treatment-control contrasts under Model B. Conditions (3.3) is essentially an orthogonality condition. Taking (2.1) as Model A with 02 = #, and (2.2) as Model B, we can combine Theorems 2.1 and 3.1 to get the following result: COROLLARY 3.1. Given v,b and k, suppose v is a perfect square, k =- 0 (mod(v + v~)) and do E 79(v + 1, b, k) is such that n

lj . . . . .

ndoOj = V/Vndolj,

= k/(v +

for j = 1 , . . . , b.

(3.4)

Then do is A-optimal for treatment-control contrasts in 79(v + 1, b, k). EXAMPLE 3.1. Let v = 4, k = 6. Then a design in which each block is (0, 0, 1,2, 3, 4) is A-optimal in 79(5, b, 6).

3.2. Optimal incomplete block designs Corollary 3.1 is limited in its scope since it requires that the block sizes are multiples of (v + v/-v). Consequently, k is quite large compared, to v. What is an A-optimal design in other situations? Let us first consider the incomplete block setup, i.e., 2 ~< k ~< v.

(3.5)

Optimal and efficient treatment-control designs

1013

Let us start with an arbitrary design d in D(v + 1,b, k). Using Kiefer's (1975) technique of averaging, we obtain

t r ( P C d P ' ) >~tr(PCd P'),

(3.6)

where Cd = 1 ~-~rI H C d H ' , the summation taken over all (v÷ 1) x (v÷ 1) permutation matrices H that correspond to permutations of the v test treatments only. If we partition ~dd as in (3.1), then we see that Md = (PCd p~)-i is a completely symmetric matrix. In general, there may be no design in 79(v+ 1, b, k) for which Cd is the Fisher Information matrix for the treatments. If there is such a design, then for this design, call it el, M d = M d is completely symmetric and ~/~ (see (3.1)) is a vector with all entries equal. That is, d belongs to the class of designs with supplemented balance of Pearce (1960)! Bechhofer and Tamhane (1981) postulated that the same subclass of designs was suitable for the problem of constructing simultaneous confidence intervals for the treatment-control contrasts. (More on this in Section 4.) Their terminology is given in the following definition: -

-

DEFINITION 3.1. Suppose (3.5) holds. A design d E 79(v + 1, b, k) is called a Balanced Treatment Incomplete Block (BTIB) design if condition (2.5) holds. Condition (3.5) is used in the definition because Bechhofer and Tamhane were interested exclusively in the incomplete block setup (hence the T in BTIB). Note that, for a BTIB design d, for i, i ~ = 1 , . . . , v, i ~ i ~, v_~s(?,~ - ~do) = a2k(;~eo + ad~)/(;~do(),do + V~dl)), Corr(?e~ -- ?do, ~d~, -- ?eo) = ;~dl/(~eo + ad~).

A particular type of BTIB design will be of considerable interest to us. This is defined n e x t - the notation is due to Stufken (1987). DEFINITION 3.2. A design d E D ( v + 1, b, k) is called BTIB(v, b, k; t, s) if it is a BTIB design, and, with (t, s) C {0, 1 , . . . , k - 1} x {0, 1 , . . . , b}, it holds that,

ndij c {O, 1},

for a l l ( i , j ) c { 1 , . . . , v } x { 1 , . . . , b } ,

and ndO1

. . . . .

ndOs = t + 1,

ndo(s+l)

. . . . .

ndOb = t.

For a BTIB(v, b, k; t, s), rdo = bt + s, vAdo = b t ( k - t) + s ( k - 2 t - 1), v(v ± 1))~dl = (b(k - t) - 2s)(k - t - 1). Here are a few examples.

D. Majumdar

1014

EXAMPLE 3.2. Let v = 7, b = 7, and k = 4. A BTIB(7, 7, 4; 1,0) in 79(8,7,4) is given by t h e following array, where, as elsewhere for block designs, columns are blocks: 0 0 0 0 0 0 0 1 2 3 4 5 6 7 2 3 4 5 6 7 1 4 5 6 7 1 2 3 Here AdO = 3, ,~dl = 1. EXAMPLE 3.3. Let v = 6, b = 18, and k = 5. A BTIB(6, 18,5; 1,6) in 79(7, 18,5) is given by the following array: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 2 3 3 2 2 2 2 3 4 3 3 3 3 3 4 2 _ 4 2 4 4 5 3 5 3 4 4 5 5 4 4 4 5 5 3 5 6 5 6 6 4 6 5 6 5 6 6 6 5 6 6 6 H e r e )~d0 =

14, )~dl = 6.

EXAMPLE3.4. Let v = 4, b = 12, and k = 4. The following array is a BTIB design that is not a BTIB(4, 12, 4; t, s). 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 1 1 1 2 2 3 3 3 1 2 3 4 2 3 4 3 4 4 4 4 Here "~d0

9, .Xdl = 3.

Starting from an arbitrary design d in D(v + 1, b, k), one can use (3.6) to obtain a lower bound for the value of the A-criterion of d. This lower bound can be minimized over 79(v + 1, b, k). A design that attains this minimum would be an A-optimal design. This was done in Majumdar and Notz (1983). To state their result we need some notation. For integers v, b, k, x and z let, g(x, z) = kv(v - 1)2 [bvk(k - 1) - (bx + z)(kv - ' v + k)

+ (bx 2 + 2xz + z)]-1 + kv[k(bx + z) - (bx 2 + 2xz + z ) ] - ' , A = {0, 1 , . . . , [k/2J - 1} x { 0 , 1 , . . . , b } -

(3.7) {0,0}.

(3.8)

Optimal and efficient treatment-control designs

1015

For a design d that is BTIB(v, b, k; x, z), it holds that t r ( P C d P ' ) = g(x, z), i.e., g gives the value of the A-criterion. Note that the replication of the control for this design is rdO = bx + z. For (x, z) C A the function g and the set A can be expressed in terms of r = bx + z as: g*(r) = kv(v - 1)2[bvk(k -

l) -

r(kv - v + k) +

h(r)]-'

h(r)] -~,

(3.9)

h(r) = b(Lr/bJ) e + (2Lr/b j + 1)(r - bLr/bJ),

(3.10)

A* = { 1 , . . . , bLk/2J}.

(3.11)

+ kv[kr -

where

We are now ready to state the result of Majumdar and Notz (1983). THEOREM 3.2. Let t and s be integers defined by g(t,s) =

Min g ( x , z ) .

(x,z)EA

(3.12)

Under condition (3.5), for any design d E 79(v + 1, b, k), t r ( P C d P ' ) >~g(t, s),

(3.13)

with equality if d is BTIB(v, b, k; t, s). Hence a BTIB(v, b, k; t, s) is A-optimal for treatment-control contrasts in D(v + 1, b, k). An equivalent version of Theorem 3.2 is: THEOREM 3.2*. Let the integer r* be defined by g*(r*) = Min g*(r). tEA*

(3.14)

Under condition (3.5),for any design d E :D(v + l, b, k), t r ( P C d P ' ) >~g*(r*),

(3.15)

with equality if d is BTIB(v, b, k; t, s) where bt + s = r*. Hence a BTIB(v, b, k; $, s) with bt + s = r* is A-optimal for treatment-control contrasts in :D(v + 1, b, k ). The quantity r* may be viewed as the optimal replication of the control, since if a BTIB(v, b, k; t, s) design do is A-optimal, then rdoo = bt + s = r*. (Is the minimum of g*(r) attained at a unique point r*? The answer is yes, except in rare cases. For

1016

D. Majumdar

now, there is no loss in assuming that r* is unique. We will return to this point in Theorem 3.6.) EXAMPLE 3.5. Let v = 7, b = 7 and k = 4. Then solving the optimization problem in (3.12) we get t = 1,s = 0. Hence, r* = 7. The BTIB(7, 7, 4;1, 0) given in Example 3.2 is A-optimal. EXAMPLE 3.6. Let v = 6, b = 18 and k = 5. Here t = 1, and s = 6. Hence r* = 24. The BTIB(6, 18, 5; 1,6) given in Example 3.3 is A-optimal. The A-optimal designs given by Theorem 3.2 are BTIB(v, b, k; t, s). It can be seen that the structure of a BTIB(v, b, k;t, s) can be of two types. If s = 0, then it is called a Rectangular-type or R-type design, while if s > 0, then it is called a Steptype or S-type design. The terminology is due to Hedayat and Majumdar (1984). With columns as blocks an R-type design d may be visualized as a k × b array:

d =

Edll d2

(3.16)

'

where dl is a t x b array of controls (0), while d2 is a (k - t) x b array in the test treatments ( 1 , 2 , . . . , v) only. Clearly, d2 must be a BIB(v, b,r, k - t , )~) design. The R-type designs are thus exactly the reinforced BIB designs. An S-type design d can be visualized as the following k x b array: d=

[ d l l d~2J d21 d22 '

(3.17)

where dll is a (t + 1) x s array of controls, d12 is a t x (b - s) array of controls, d2a is a (k - t - 1) x s array of test treatments and d22 is a (k - t) x (b - s) array of test treatments. The following result of Hedayat and Majumdar (1984) gives some properties of a BTIB(v, b, k; t, s). LEMMA 3.1. (i) For the existence of a BTIB(v, b, k; t, s), the following conditions are necessary (where ro = bt + s): (b(k - t) - s ) / v = (bk - r o ) / v ( = ql, say) s ( k - t - 1 ) I v ( = q2, say)

is an integer,

(3.19)

is an integer,

[q2(k - t - 2) + (ql - q2)(k - t - 1)]/(v - 1)

(3.18)

is an integer,

(3.20)

(ii) For an R-type design it is necessary that b ~ v, while f o r an S-type design it is necessary that b >/v + 1.

Optimal and efficient treatment-controldesigns

1017

It can be shown that a BTIB(v, b, k; t, s) is equireplicate in test treatments. Condition (3.18) is necessary for this. Conditions (3.19) and (3.20) are necessary for condition (2.5). Part (ii) of the Lemma is Fisher's inequality for BTIB(v, b, k; t, s) designs. Based on Theorem 3.2 and Lemma 3.1, Hedayat and Majumdar (1984) suggested a method for obtaining optimal designs that consists of three steps: (1) Starting from v, b, k determine t, s (equivalently r* ) that minimize 9( x, z ). (2) Verify conditions of Lemma 3.1 (i), using t and s from step 1. If the conditions are not satisfied then Theorem 3.2 cannot be applied to the class D(v + 1, b, k ). If the conditions are satisfied, then go to step 3. (3) Attempt to construct a BTIB(v, b, k; t, s). (Note that even when the conditions of Lemma 3.1 are satisfied, there is no guarantee that this design exists.) Instances of design classes where this method does not PrOduce A-optimal designs is given in examples 3.9-3.11. For R-type designs, the construction problem reduces to finding BIB designs. For S-type designs the problem is clearly more involved, and unlike the case of R-type designs this case does not reduce to the construction of designs that are well studied in the literature. We return to the construction of such designs later in this section. It may be noted that Majumdar and Notz (1983) obtained optimal designs for criteria other than A-optimality also. Giovagnoli and Wynn (1985) used approximate design theory techniques to obtain results similar to Theorem 3.2. For certain values of v, b and k the minimization in (3.12) gives a nice algebraic solution. This can sometimes be exploited to obtain infinite families of A-optimal designs with elegant combinatorial properties. Here are some results. THEOREM 3.3. A BTIB(v, b, k; 1,0) is A-optimal for treatment-control contrasts in 79(v + 1, b, k) whenever ( k - 2 ) 2 + 1 ~ < v ~ < ( k - 1 ) 2.

This result is due to Hedayat and Majumdar (1985). An example of a design that is A-optimal according to Theorem 3.3 is the design in Example 3.2. It is interesting to note that when v = (k - 2) 2 + k - 1, an A-optimal design is obtained by taking the BIB design d2 in (3.16) to be a finite projective plane of order (k - 2 ) , while when v = (k - 1) 2, an A-optimal design is obtained by taking the BIB design d2 as a finite euclidean plane of order (k - 1). The next result, due to Stufken (1987), is a generalization of Theorem 3.3. THEOREM 3.4. A BTIB(v, b, k; t, 0) is A-optimal for treatment-control contrasts in D(v + 1, b, k) whenever

(k-t-

1 ) 2 + 1 ~
EXAMPLE 3.7. Let v = 9, b = 12 and k = 8. With t = 2, (k - t 1) 2 + 1 = 26, t2v = 36, and ( k - t ) 2 = 36. Hence a BTIB(9, 12, 8; 2, 0), which is a BIB(9, 12, 8, 6, 5) design with each block augmented by two replications of the control, is A-optimal. -

1018

D. Majumdar

Theorems 3.3 and 3.4 gives some sufficient conditions for optimality of the reinforced BIB designs of Das (1958). A family of A-optimal S-type designs, due to Cheng et al. (1987), is given in the next theorem. THEOREM 3.5. Let a ~ 3 be a prime or a prime power, and ~ be a positive integer. Then there exists a B T I B ( a 2 - 1 , 7 ( a + 2 ) ( a z - 1 ) , a ; 0 , 7 ( a + 1)(a 2 - 1)). This design is A-optimal for treatment-control contrasts in D ( a 2, 7 ( a + 2)(c~ 2 - 1), a). For a method of construction of the BTIB design in Theorem 3.5 the reader is referred to Cheng et al. (1987). EXAMPLE 3.8. Let v = 8, b = 40 and k = 3. With a = 3 and 3' = 1 here is an example of a BTIB(8, 40, 3; 0, 32) that is A-optimal: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 2 3 4 5 6 7 8 3 4 5 6 7 8 4 5 6 7 8 5 6 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 3 3 4 4 4 5 5 5 6 6 7 1 2 3 7 2 4 6 5 6 4 5 5 7 8 6 7 8 7 8 8 5 4 6 8 3 7 8 8 7 8 7 6 We conclude this discussion by stating a result of Cheng et al. (1988) which characterizes the point r* and the shape of the function 9*. THEOREM 3.6. Given v, b and k, r* E {r~ , r* }, where 0 <. (r* - r? ) <~ 1, (3.21)

and fort? >tEA*, 9*(r) > g * ( r + l ) ; forr*
Optimal and efficient treatment-control designs

1019

for details. Jacroux's optimal designs are BTIB designs as well as designs that belong to a class that is combinatorially more general than BTIB designs. These designs, proposed by Jacroux (1987a), are defined as follows. DEFINITION 3.3. A design d • 79(v + 1,b,k) is called a Group Divisible Treatment Design (GDTD) with parameters m, n, -~0, )~1 and .~2 if the treatments 1 , 2 , . . . , v can be divided into m groups I I 1 , . . . , Vm of n treatments each such that there are nonnegative constants )~0, )h and )~2 such that )kdO i :

)tO,

)~dii' :

)~1,

)kdii' = "~2 ,

for i = 1 , . . . ,v, for i, i I E Vp, i ¢ i', for i • Vp, i' • Vq,p,q • { 1 , . . . , m } , p 7£ q.

For a GDTD all variances Var(?ai - ?d0), i = 1 , . . . , v, are equal; the variances

Var(?ai - ?de), i,i' = 1 , . . . ,v, i ¢ i', can take at most two values depending on whether i and i' belong to the same group or not. In a BTIB design the latter variances take only one value, since a BTIB design may be considered a GDTD with only one group. DEFINITION 3.4. A design d E D(v + 1, b, k) is called a GDTD(v, b, k; t, s) if it is a GDTD, and with t and s, (t,s) E {0, 1 , . . . , k - 1} x {0, 1 , . . . , b - 1}, it holds that,

ndij E {O, 1},

forall (i,j) E { 1 , . . . , v } x { 1 , . . . , b } ,

and

ndOl . . . . . halO(s+1) . . . . .

ndos = t + 1, ndOb = t~.

Jacroux originally developed the class of GDTD's to find some sufficient conditions for a design to be MV-optimal. It follows from the fact, v

Max Var(?ai - ?dO) = _1 ~

l<~i<~v

V

Var(?ai -- T'dO), for a BTIB design d,

i=l

that all BTIB designs that are A-optimal are also MV-optimal. Jacroux (1987a) established the MV-optimality of some BTIB designs in cases where this could not be concluded from their A-optimality since the designs were not known to be A-optimal; he also proved the MV-optimality of some GDTD's. Below we give a method given in Jacroux (1987a) that is often successful in finding optimal designs - for exact statement of the theorems and the proofs we refer the reader to Jacroux's paper. We shall use the notation in (3.9) and (3.10). The method consists of three-steps:

D. Majumdar

1020

(1) Over all positive integers r such that (bk - r ) / v is an positive integer, find r** such that 9" (r) is a minimum. (2) Find d* with rd*o = r** that is a BTIB(v, b, k; t, s) design or a GDTD with some special properties specified by Jacroux. (3) Verify one of several sets of sufficient conditions that guarantee that d* is MV-optimal in 73(v + 1, b, k ). The Hedayat and Majumdar method, which is based on Theorem 3.2 of Majumdar and Notz (1983), starts by minimizing the function g*(r), then verifies a set of necessary conditions, one of which is (3.18). Jacroux's method incorporates condition (3.18) in the optimization. Moreover, Jacroux goes beyond the class of BTIB designs to certain types of GDTD's in his search for optimal designs. Thus, Jacroux's method can give optimal designs for some classes 73(v + 1, b, k) where the Hedayat and Majumdar method cannot give an optimal design. This advance was possible, however, only after some difficult technical problems were overcome in Jacroux (1987a). Here are some examples of optimal designs given by Jacroux. EXAMPLE 3.9. Let v = b = 11, k = 6. Using the notation of Theorem 3.2*, it can be seen that r* = 14, but that there is no BTIB(11, 11,6; t, s) design with 1 I t ÷ s = 14, i.e., a BTIB(11, 11,6; 1,3) does not exist. Hence Theorem 3.2* cannot be used to obtain an (A- or MV-) optimal design in 73(12, 11,6). Jacroux (1987a) showed that r** = 11, and that the following BTIB(11, 11,6; 1,0) design is MV-optimal. 0

0

0 0

2

3

123

4

5

4 5

0 6

0

0

0

0

0

0

4

1

1

1

2

1

7

5

2

2

3

3

8

5

5 6 6 7 8 9 8 6 3 4 4 6

7

7 8

9

10 10

9

7

10 11 8 9 10 11 11 11 10 11 9 EXAMPLE 3.10. Let v = 6, b = 14 and k = 4. Here, r* = 18, r** = 14. Theorem 3.2* cannot be used to obtain an optimal design; the following GDTD(6, 14, 4; 1,0) design with treatment groups {1,2}, {3,4} and {5, 6} is MV-optimal in 73(7, 14, 4). 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 3 4 2 2 3 3 3 4 4 3 3 3 4 4 5 5 5 6 4 5 6 5 6 4 5 6 . 5 6 6 6 EXAMPLE 3.11. This is an example of an A-optimal design from Jacroux (1989). Let v = 6, b = 20 and k = 3. Here r* =- 18, r** = 18. Theorem 3.2* cannot be used to obtain an optimal design since a BTIB(6, 20, 3; 0, 18) does not exist. Jacroux showed

Optimal and efficient treatment-control designs

1021

that the following GDTD(6, 20, 3; 0, 18) with treatment groups {1,2, 3} and {4, 5, 6} is A-optimal in 79(7, 20, 3). 0 00

00

000

0 0 0 0 0 0 0 0 0 0

1 4

1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 2 5 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 3 6

3.3. Construction of BTIB designs and GDTD's Let us now consider the problem of constructing balanced treatment incomplete block designs and group divisible treatment designs. As mentioned earlier, construction of R-type BTIB designs reduces to the construction of BIB designs. For this, one may appeal to the vast literature that is available. For the case of S-type BTIB designs, there is no such resource to fall back on. Construction of BTIB designs for the case k = 2 has been considered in Bechhofer and Tamhane (1983b), while Notz and Tamhane (1983) gave a complete solution for the case k = 3, 3 ~< v ~< 10. Hedayat and Majumdar (1984) has some discussion on the construction of S-type designs. In this paper, a complete catalog of A-optimal designs that can be obtained from Theorem 3.2 are given for the range 2 ~ k ~< 8, k ~< v ~< 30, and v ~< b ~< 50; this catalog contains several S-type designs. Cheng et al. (1988) and Ture (1982) also have some general recommendations for constructing S-type designs. It is easy to see that a GDTD(v, b, k; t, 0) is obtained by reinforcing each block of a group divisible design in blocks of size k - t, by t replications of the control. Construction of GDTD's in general is not always easy - some GDTD's were constructed by Jacroux (1987b). Stufken (1991a) gave a complete solution to the problem of constructing GDTD's for k = 3 and v = 4, 6. From the viewpoint of practical applications, designs with small k and v are most important. It may be noted that for v = 3, 5, k = 3 the problem reduces to constructing BTIB designs, which is given in Notz and Tamhane (1983).

3.4. Optimal designs in large blocks Next, let us remove the restriction (3.5). If the conditions of Corollary 3.1 are satisfied then one can obtain an A-optimal design quite easily. What happens if these conditions do not hold? Let us start with a definition of a class of designs that generalize the class of BTIB(v, b, k; t, s) designs. DEFINITION 3.5. A design d E 79(v + 1,b,k) is called a B T B ( v , b , k ; t , s ) if it is a design with supplemented balance, i.e., satisfies (2.5), with the additional properties: I n d ~ j - - n d i , j , ] < 1,

forall(i,i')E{1,...,v)×(1,...,v}, i~

ndo1 . . . . .

i',

ndOs ~- 1. ~- 1,

(j,j')e{1,...,b)×(1,...,b}, ndo(s+l ) . . . . .

ndOb ~-- t.

D. Majumdar

1022

In addition to the sets A and A* defined in (3.8) and (3.11) and h(r) defined in (3.10), let, for integers x and z, = [(bk - r ) / b v J , p = v ( k - 1) -t- k - 2vtx, c = b v k ( k - l) + bva(v - 2k + va), f ( x , z) = k v ( v - 1)2[c - p(bx ,1. z) ,1. (bx 2 .1. 2 x z ,1. z)] -1

r = bx ,1. z,

+ kv[k(bx + z) - (bx 2 + 2xz +

z)]-l,

f * ( r ) = k v ( v - 1)2[c - rp .1. h(r)]-1 + k v [ k r - h(r)]-1

The following theorem, due to Jacroux and Majumdar (1989), can be viewed as a generalization of Theorem 3.2, as condition (3.5) is no longer imposed. THEOREM 3.7. Let t and s be integers defined by

f(t,s)

=

Min

(x,z)~A

f(x,z).

(3.22)

For any design d ¢ 79(v + 1, b, k), t r ( P C d P ' ) >1 f ( t , s),

(3.23)

with equality if d is B T B ( v , b , k ; t , s ) . Hence a B T B ( v , b , k ; t , s ) treatment-control contrasts in 79(v + 1, b, k).

is A-optimal f o r

An equivalent version of Theorem 3.6 is: THEOREM 3.7*. Let the integer r* be defined by f*(r*)

= Min yEA

f*(r).

(3.24)

For any design d ¢ 79(v + 1, b, k), t r ( P C d P ' ) >~ f*(r*),

(3.25)

with equality if d is BTB(v, b, k; t, s) where bt + s : r*. Hence a BTB(v, b, k; t, s) with bt + s = r* is A-optimal f o r treatment-control contrasts in :D(v + 1, b, k).

EXAMPLE 3.12. Let v = 3, b : 10 and k = 4. The following BTB(3, 10, 4; 1,3) is A- and MV-optimal in 79(4, 10, 4): 0 0 1 2

0 0 1 3

0 0 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

Optimal and efficient treatment-control designs

1023

The following corollary, due to Jacroux and Majumdar (1989), gives an infinite family of A-optimal designs. It is clear that these designs are also MV-optimal. COROLLARY 3.2. For any integer 0 > 1, a BTB(02, b, (0 + 1)z; 0 + 1, O), f o r a b such that the design exists, is A-optimal f o r treatment-control contrasts in 79(02 + 1, b, (0 + 1)2).

4. Efficient designs for confidence intervals There are two methods for deriving inferences on the treatment-control contrasts. One is estimation, and the other is simultaneous confidence intervals. Bechhofer and Tamhane, in their discussion of Hedayat et al. (1988), describe situations where the simultaneous confidence interval approach is the appropriate one for choosing a subset of best treatments from the v test treatments. More examples are available in Hochberg and Tamhane (1987). In this section we discuss optimal and efficient designs for simultaneous confidence intervals for the treatment-control contrasts.

4.1. Designs f o r the zero-way elimination of heterogeneity model First consider the 0-way elimination of heterogeneity model, i.e., Yij = # + "ci + eij. Let, #i = # + ~-i, for i = 1 , . . . , v. We have to impose some more distributional assumptions on the random variables in order to obtain confidence intervals. For i = 0, 1 , . . . , v and j = 1 , . . . , rdi let Yij's be independent with yo "~ N ( # i , ai).

(4.1)

Dunnett (1955) was the first to give simultaneous confidence intervals for the treatment-control contrasts #i - # o , i = 1 , . . . , v. Suppose aO=~l .....

a~.

(4.2)

If Ydi denotes the mean of all observations that receive treatment i, and s 2 denotes the pooled variance estimate (mean squared error), then the simultaneous 100P% lower confidence limits for #i - #0 are: fldi -- 9dO -- 5iS~/rdi 1 + r'do1,

i = 1 , . . . , V,

(4.3)

where the 5i are determined from: P ( t i < 5i, i = 1 , . . . , v ) = P.

(4.4)

The joint distribution of the random variables ~1,.-., tv is the multivariate analog of Student's t defined by Dunnett and Sobel (1954). Here the subscript d represents the design. The design problem is to allocate a total of n observations to the v test

D. Majumdar

1024

treatments and one control. Thus ~ i =v 0 rdi = n. We shall denote by D(v + 1, n) the class of all designs. The simultaneous 100P% two-sided confidence limits for #i - #o are:

Y d i - - Y d o ± g i s ~ / r ~ l+r-ldO ,

i = 1, ..., V,

where,

P(Itd < ~ , i = 1 , . . . ,v) = P. For the design rdi = n/(v + 1), for all i, and 51 . . . . . 5v = 5, Dunnett (1955) gave tables of the 5's for various values of P. For the same design and/~1 . . . . . / ~ v = he gave tables for the t~'s for various values of P. He also investigated optimal designs for lower confidence limits in the following subset of D(v + 1, n): 7~*(v+ 1,n) = {d: d G D ( v +

1,n), rd, . . . . .

rdv}.

A design was called optimal if it maximized P for a fixed value of 5Vrdi / 1 -q- rd01 . Dunnett's numerical investigations revealed that the optimal value of rdo/rdl was only slightly less than x/~, which is the value given in Theorem 2.1, as long as 5 was chosen to make the coverage probability P in (4.4) of magnitude 0.95 or larger. The tables for the simultaneous two-sided confidence limits in Dunnett (1955) were obtained using an approximation. More accurate tables for the two-sided case were given in Dunnett (1964); optimal allocations were also discussed. Bechhofer (1969) considered the design problem for Model (4.1) when the cri are known, without the restriction (4.2). The lower 100P% simultaneous confidence limits are: -

-

1 +

1,

i =

(4.5)

1,...,

where -

-

i

1,...,v)

e.

(4.6)

The quantity 5' ~/crir~-i1 + cror~o1 is called a yardstick. Let,

7di = rdi/n,

for i = 0, 1 , . . . , v ,

and

ff =

¢o;(

(riT~ 1 + aOTd-J).

For a fixed value of ff (equivalently, for a fixed yardstick), Bechhofer defined a design d to be optimal if it maximizes P in (4.6) among all designs in the subclass, 79"(v + l , n ) = {d: d C V(v + 1,n),c~Z/rdl . . . . .

cr21rdv}.

Optimal and efficient treatment-control designs

1025

We shall call this design one that maximizes the coverage probability for a fixed yardstick. Note that, if d c 79* (v + 1, n) then the treatment-control contrasts are all estimated with the same variance; thus this is a natural subclass of 79(v + 1, n) for this setup. 79* (v + 1, n) = 79* (v + 1, n), for the homoscedastic case, tr0 = ~rl . . . . . (rv. Taking the approximate design theory approach, i.e., viewing the 7ai as nonnegative reals, not restricted to being rational, such that ~iv__0 7di = 1, Bechhofer (1969) gave an explicit equation to determine an optimal design. This is given in the following theorem, where f ( . ) is the standard normal density function, Fk (x I P) is the k-variate standard normal distribution function with all correlations equal to p, and 13

¢* = (1/2)v(v - 1 ) V / - ~ F v _ 2

/3=/_..,V~a2/~r2i,0,

(0 I 1/3)

i=1

co = co(')') = ~7V//3(1 - 3')/((1 - 3' + 73)(2(1 - 7) + "7/3)). THEOREM follows is For fixed (0, 1/(1 +

4.1. Given v >~ 2, n, and cri > 0, a design do with ~/aoi = 7~ given as optimal, i.e., it maximizes the coverage probability for a fixed yardstick. 0 < ~ <~ ~*, 70 = O, and for fixed ~ > ~*, 7o is the unique root in v ~ ) ) , of the equation: -

-

+

2(1_7)+3,/3

l

-

1-3'

3 (] -- 2/5-+----.~/3 . 3 (1 - ~) ~_ 7/3

~0,

Also, for i = 1 , . . . , v, 7i = or2( 1 - 7o)/(/3a2) • EXAMPLE 4.1. Suppose v = 2 and ~rl = or2 = (7o. If ff = 2, then 3'0 = 0.32, 3'1 = "/2 = 0.34; hence 32% of the observations are allocated to the control, and 34% to each of the two test treatments. If ff = 5 then 70 = 0.40, 71 = 72 = 0.30; hence 40% of the observations are allocated to the control, and 30% to each of the two test treatments. As a corollary, Bechhofer showed that when cri = ~r0 for all i, in the limit as --+ ec, the optimal 7o/7i -+ v/-~, the allocation given in Theorem 2.1. The results of B echhofer (1969) were generalized by Bechhofer and Turnbull (1971) by allowing the ~' to vary with the treatment symbol i (i = 1 , . . . , v ) . Optimal design s for the simultaneous two-sided confidence interval were given in Bechhofer and Nocturne (1972). Bechhofer and Tamhane (1983a) gave optimal designs that minimized the total sample size for one-sided intervals for given P , / 3 and a yardstick. Taking a different approach, Spurrier and Nizam (1990) minimized the expected allowance for a fixed coverage probability. For the model given by (4.1) and (4.2), the allowance, or yardstick, for the simultaneous 100P% lower confidence limits in

D. Majumdar

1026 (4.3) when 51 . . . . .

5v = fi, is given by 58~rdilq

-

td 1, where 5 is determined

from equation (4.4), i.e., P(ti < 5, i = 1 , . . . ,v) = P. Spurrier and Nizam (1990) defined the Expected Average Allowance (EAA) as, EAA = (1/v)SE(s)\/r~' + r2o1, and defined a design do to be optimal in the sense of minimizing EAA for fixed

coverage probability if it minimizes 5~/rdi 1 + rdd for fixed P, over all of D(v + 1, n). In general, determination of an optimal design is a difficult problem. Spurrier and Nizam (1990) were able to show that when v = 2, for the optimal design do, ]rdol -rdo2[ ~< 1. When v > 2, based on their numerical calculations they conjectured that [rdoi -- rdoi, [ ~< 1, for i ~ iq This result is helpful in limiting the search for an optimal design. Spurrier and Nizam gave tables of optimal designs for 2 <~ v ~< 8 and several values of n, when the rdi'S are allowed to vary by no more than 1. Analogous tables for the two-sided confidence intervals were also given. Motivated by the thrust on quality improvement and emphasis on analyzing variability, Spurrier (1992) considered the problem of inference for the variances. He studied optimality of designs in 79* (v + 1, n) for simultaneous confidence limits for the cri2/a o2 (i = 1 , . . . , v) and gave extensive tables of optimal designs. Optimal designs are different than the optimal designs for means given in Spurrier and Nizam (1990). For the one-sided confidence intervals it turns out that for small n, rd01 /> rdo0, while for large n, rdol < rdo0. In the limiting case when n --+ c~, the optimal allocation is rdoo/rdol = x/v, the design of Theorem 2.1. Sometimes the object of the experiment is solely to determine which, if any, of the test treatments perform better than the control, on the basis of the mean performance. For this purpose, Naik (1975) and Miller (1966, p. 85-86), proposed an alternative to Dunnett (1955)'s procedure that leads to considerable savings in terms of the total sample size. Let us assume that the model is given by (4.1) and (4.2). For d c

V*(v + 1,n), let Ti = (ffdi -- ~ldO)/(s~/rdi 1 q- r-~), i = 1,... ,v, and let the ordered Ti's be T(U ~< .-. ~< T(v). Suppose treatment i is better than the control if #i > P0. The alternative procedure, known as the step down test procedure, is as follows. If T(v) <~t(a, v) (a 100a% critical value obtained from Bechhofer and Dunnett (1988)) then none of the test treatments are declared to be better than the control. If not, then the test treatment which corresponds to T(v) is declared to be better than the control and the procedure moves to the next step where T(v-1) is compared with the critical value t(ce, v - 1), and so on. Hayter and Tamhane (1991) considered the design problem for the step down procedure. They called a design optimal if it achieved the smallest n among all designs, for fixed values of the probabilities of errors of types I and II, for a given yardstick. Hayter and Tamhane (1991) gave tables of optimal designs and demonstrated the savings over Dunnett's procedure.

Optimal and efficient treatment-control designs

1027

4.2. B l o c k designs

For the one way elimination of heterogeneity Model (2.4), Bechhofer and Tamhane (1981) was the first to consider the problem of finding optimal designs for simultaneous confidence intervals. Consider the problem of comparing v test treatments with a control in b blocks of size k each. As in Section 3, the set of designs is denoted by 79(v + 1, b, k). The observations pijt are independent random variables that follow the model: Yifl " N ( # + ri + fly, (72).

Bechhofer and Tamhane (1981) started with a BTIB design d. Let, D*(v + 1, b, k) = {d E 7?(v + 1, b, k): d is a BTIB design}. For a BTIB design d, and i , i r = 1 , . . . , v , i ¢ i ~, let, ¢2 = cr-2Var(~di _ ~dO) = k()~dOq-/~dl)/(,~dO(/~dO q- V/kdl)), pd = Corr(?di - ?do, ;d~, - ?do) = ~ d l / ( a d o + Ad~).

If one uses a BTIB design d then the simultaneous 100P% lower confidence limits for (~-i - ~-0)'s are: ?di -- ?ao -- 5¢dS,

i = 1 , . . . , V,

(4.7)

where 8 2 is the mean square error, and 6 is the 100(1 - P)% critical value for the multivariate, equicorrelated t (Dunnett and Sobel (1954), tables in Krishnaiah and Armitage (1966)). Simultaneous 100P% two-sided confidence limits can be similarly defined. We now give Bechhofer and Tamhane's definition of an optimal design. Let ~ be the yardstick, i.e., P = lP(Tdi -- TdO >/ ~di -- ~dO -- ~, i = 1 , . . . , V).

(4.8)

Given v, k, ~/(7 and a, a BTIB design do is defined to be optimal if among all BTIB designs with P ~> 1 - a, do has the smallest b. Determination of an optimal design is a difficult problem. As a first step, Bechhofer and Tamhane (1981) introduced a concept of admissibility that is helpful in removing those designs from consideration that can be uniformly improved upon by another design. DEFINITION 4.1. For i = 1,2, if di is a BTIB design in D * ( v + 1, bi, k), and b~ ~< b2, then the design d2 is inadmissible with respect to dl (written dl ;,- d2) if dl yields a confidence coefficient P at least as large as (larger than) that yielded by d2 when bl < b2 (bl = b2) for all values of ~/m For a BTIB design d if there is no BTIB design dl such that dl ;'- d, then d is admissible.

D. Majumdar

1o28

Bechhofer and Tamhane (1981) also gave the following simple characterization of the relation >-. THEOREM 4.2. For i = 1,2, if d~ is a BTIB design in 79"(v + 1,b~,k) then dl >- d2

if and only if bl ~ b2,

~bZd~<~ ~b2d2, Pal, >~ Pal2,

with at least one inequality strict. EXAMPLE 4.2. If v = 4 and k = 3 then for the following BTIB designs dl and d2, dl ~ d2.

dl=

(oooo112) (oooo1123) 1 1 2 0 2 3 3

2443344

,

d2=

1 1 2 4 2 4 4 4

.

2 3 3 4 3 4 4 4

For two designs dl and d2 the union dl U d2 will denote a design that consists of all blocks of dl and d2. It is interesting to note that if da ~- d2, then it is not necessarily true that for a BTIB design d3, dl U d3 >- d2 tAd3; see Bechhofer and Tamhane (1981, p. 51) for a eounterexample. For a given pair (v, k) it will be convenient to have a set of designs A(v, k), such that all admissible BTIB designs, except possibly some equivalent ones are obtainable from this set by the operation of union of designs. Among all sets A(v, k), a set with minimal cardinality is called a minimal complete class of generator designs for the pair (v, k). When k = 2, it is easily seen that there are only two designs in a minimal complete class: a BTIB(v, v, 2; 1,0) and a BTIB(v, v(v - 1)/2, 2; 0, 0). Similarly when v = k = 3, there are only two designs in a minimal complete class, a BTIB(3, 3, 3; 1,0) and the design with only one block {1,2, 3}. These are the only two simple cases - in general it is difficult to identify a minimal complete class of generator designs. Notz and Tamhane (1983) gave minimal complete classes for k = 3, 3 ~ v <~ 10, and Ture (1982, 1985) for k = 4 and k = 5 Optimal designs for k = 2 and v = 2 , . . . , 6 were given by Bechhofer and Tamhane (1983b). Bechhofer and Tamhane (1985) compiled several tables of optimal, admissible and minimal complete classes of designs. Spurrier and Edwards (1986) proposed a different definition of optimality of designs for simultaneous confidence intervals. The expected average allowance (the same criterion used by Spurrier and Nizam (1990), that was described earlier in this section) for the confidence limits given in (4.7) is, EAA = ~'~baE(s). Spurrier and Edwards (1986) defined a BTIB design to be optimal for a fixed level of significance a = 1 - P if it minimizes EAA among all designs in the class D**(v + l,b*,k) = {d: d E D*(v + 1,b,k), for some b ~< b*},

Optimal and efficient treatment-controldesigns

1029

where b* is a fixed upper limit on the total number of blocks. They established that, in the limit as b* goes to c¢, an optimal BTIB design is the union of a BTIB(v, ab, k; t, O) design and a BTIB(v, (1-a)b, k; t + 1,0) design for some integer t and some a E (0, 1]. Even though this is an asymptotic result, Spurrier and Edwards showed how to use it to find highly efficient, if not actually optimal, designs based on a finite number of blocks. Kim and Stufken (1995) expanded the search of optimal designs to classes beyond BTIB designs. For fixed v, b, k and ~/a they found optimal designs within the class of GDTD's that maximized the coverage probability P in (4.8). They gave all optimal designs for v = 4, k = 3 and b ~< 25. Kim and Stufken (1995) also found that their optimal designs correspond closely with the A-optimal GDTD's that were earlier obtained by Stufken and Kim (1992). This close relationship between optimal designs for estimation and simultaneous confidence intervals was earlier observed by Spurrier in the discussion of Hedayat et al. (1988). The connection was further investigated by Majumdar (1996). In order to compare A-optimal designs for estimation with admissible designs for simultaneous confidence intervals, the definition of admissibility was adapted to designs in 7?* (v + 1, b, k) for a fixed b. A result in Majumdar (1996) states that if an A-optimal BTIB(v, b, k; t, s) design d*, that satisfies the sufficient conditions in Theorem 3.2, exists, then any BTIB(v, b, k; x, z) design d is admissible if and only if vao <~vd*O. It was also shown that, in this case the set of admissible designs is precisely the set of Bayes optimal designs for the class of priors given by (6.9). This established a connection between the three concepts: A-optimality, admissibility and Bayes optimality of designs.

5. Efficient row-column designs for estimation Suppose, in addition to the blocking factor there is another factor, crossed with blocks, that makes the experimental units heterogeneous. The units can be conceptually arranged in an array, with rows and columns each representing a factor. Each cell in this array has one unit, and each unit receives a treatment, either a test treatment or the control. The primary object of the experiment is to compare the test treatments with the control. Let the bk units be arranged in k rows and b columns. If the unit in row l (1 ~< l ~< k) and column j (1 ~< j ~< b) receives treatment i (0 ~< i ~< v) then we assume that the observation Yijt follows the model: yifl = # + Ti +/33 + 7t + eij~,

(5.1)

where # is a general effect, ~-i is a treatment effect, flj' is a column effect, 7t is a row effect and cijt's are uncorrelated random variables with E(eijt) = 0, V(eifl) = cr2. A design d can be represented as a k x b array with entries from {0, 1 , . . . , v}. Let 7?RC(v + 1, b, k) denote the set of all connected designs. For d E 79RC(v + 1, b, k), let wait be the number of times treatment i appears in row l, and ndij be the number of

D. Majumdar

1030

times treatment i appears in column j. Let Wd = (WdiZ) be a (v + 1) x k matrix and Nd = (ndij) be a (v + 1) x b matrix. Wd is the treatment-row incidence matrix and Nd is the treatment-column incidence matrix of the design d. For i ¢ i t, let Adie = k ~ = 1 n d i j n d e j and PdW = ~ t = l WdilWdet. As usual, rdi denotes the replication of treatment i. The Fisher Information matrix for the treatments is: 1

I

1

I

1

t

CdR° = Diag(rd0, r dl , . . . , r dv ) -- ~ Nd N~ - -~W d W ~ + - ~ r d r_d, where r_d is the (v + 1) x 1 vector of the replications. The information matrix for the treatment-control contrasts is, as in Section 3, the v x v submatrix M ~ c obtained from C ~ c by deleting the row and column that corresponds to the control, i.e., the first row and first column. An A-optimal design is one that minimizes tr(M~C) -1, while an MV-optimal design is one that minimizes the maximum diagonal element of

(M~C)-~. Notz (1985) investigated optimal designs in this setup. His result on A-optimality is given below. THEOREM 5.1. For d E 79RC(v + 1, b, k) the following holds: tr(MffC)-l>~

M i n f**(r) O
where

f * * ( r ) = v[r + r 2 / b k + v(v -

h*(r)] -1

1) -1 [(v - 1)(bk - r) - 2 r 2 / b k + h*(r)] -1,

h* ( r ) = [r + (2r - k ) [ r / k J - k [ r / k J 2 ] / b + [r + (ar - b)k~/b] -

b[rlbJ2]lk.

The following corollary of this theorem is an extended version of a corollary given by Notz (1985). COROLLARY 5.1. Suppose v is a square,

k-O

(mod(v+v/v))

and

b=-O ( m o d ( v + x / v ) ) .

For i = 1 , . . . , k / ( v + x / ~ ) and j = 1 , . . . , b / ( v + v ~ ) let F_.ij be latin squares of order (v + x/~) each, which may be the same or different, with symbols v + 1 , . . . , v + v ~ replaced by O. Let do = ( ( £ i j ) ) be a k x b array. Then do is A-optimal f o r treatmentcontrol contrasts in DRO (v + 1, b, k).

Optimal and efficient treatment-control designs

1031

EXAMPLE 5.1. For v = 4, b = k = 6 an A-optimal design in Dne(5, 6, 6) is: 001234 400123 340012 234001 123400 012340 Notz also gave sufficient conditions when a design would attain the minimum in Theorem 5.1. One of the conditions is that, for the A-optimal design do, Ma~C = MRG do , i.e., the information matrix for the treatment-control contrasts should be completely symmetric. This condition can be viewed as the row-column counterpart of (2.5). Using the notation

5dii' = Pdie/b + Adii,/k - rdirdi, /bk, the condition can be expressed as: 8d01 . . . . .

~dOv (=~dO, say),

~dl2 . . . . .

~d(v--1)v

and ( = ~ d l , say).

A design that satisfies this condition was called a Balanced Treatment Row-Column Design (BTRCD) by Ture (1994), while Majumdar and Tamhane (1996) called it a Balanced Treatment versus Control Row-Column (BTCRC) design. If d is such a design with 5dO > 0 and 5dO-~-V~dl > O, then for (i,i t) E { 1 , . . . ,v} × { 1 , . . . ,v}, i ~ i I,

Var(?di -- ?d0) = a2(6d0 + 5dl)/(6dO(6dO + V6dl)), CO1T(T'di -- ~dO, ~di' -- ~dO) = 6d1/(6dO -]- 6dl).

(5.2)

It was noted in Majumdar (1986) that Corollary 5.1 can be alternatively derived from Theorem 3.1 by taking (2.1) for Model A and (5.1) for Model B. Jacroux (1986) applied Theorem 3.1 with (2.4) as Model A and (5.1) as Model B to get the following result. THEOREM 5.2. Let the k × b array do be such that, with columns as blocks, do is A(MV-) optimal for treatment-control contrasts in 79(v + 1, b, k) under model (2.4), and

rdi = 0 (mod k),

i=O, 1,...,v.

(5.3)

D. Majumdar

1032

Then there is a row-column design d~ C with the same column contents as do and Waoit = r d J k , i = O, 1 , . . . , v. The design donC is A- (MV-) optimal for treatmentcontrol contrasts in 7) RC ( v + 1, b, k ). EXAMPLE 5 . 2 . F o r DRC(10,24,3):

v = 9, b =

24 and

k =

0 4 1 0 2 4 5 7 3 0 9 6 1 0 5 8 0 2 0 0 5 3 0 4 3 1 0 1 4 0 2 2 0 7 3 0

3,

the following design is A-optimal in

0 0 8 0 7 9 1 6 2 3 4 5 4 6 6 9 0 7 2 1 3 8 7 9 9 5 0 6 8 0 9 7 6 4 5 8

Note that if a block design satisfies condition (5.3), then it can be converted to a row-column design with each treatment distributed uniformly over rows (i.e., Wdoil = rdi/k, i = 0, 1 , . . . ,v) by permuting symbols within blocks (columns). This is guaranteed by the results of Hall (1935), and Agrawal's (1966) generalization of Hall's results. Hedayat and Majumdar (1988) noticed that the designs that are obtained in this fashion are model robust in the sense that they are simultaneously optimal under models (2.4) and (5.1). Several infinite families of model robust designs were given in Hedayat and Majumdar (1988). We give one example. Start from a Projective Plane, which is a BIB(s 2 + s + 1, s 2 + s + 1, s + 1, s + 1, 1) design, for some prime power s, that is constructed from a difference set. Write a difference set as the first column of the design using symbols 1 , . . . , 8 2 + 8 + 1. Obtain the remaining columns of the design by successively adding 1 , . . . , 8 2 + 8, t o the first column, with the convention, s 2 + s + 2 - 1. Now delete any one column, and in the rest of the design replace all symbols that appear in the deleted column by the symbol 0. The resulting design is A- and MV-optimal for treatment-control contrasts in 79RC(s2 + 1, s 2 + 8, s + 1). Designs generated by this method form the Euclidean Family of designs. EXAMPLE 5.3. Starting from the Fano Plane, i.e., BIB(7, 7, 3, 3, 1), we get the following member of the Euclidean Family that is optimal in D n c ( 5 , 6, 3): 103420 034201 420013 All of the above methods for obtaining optimal designs utilize orthogonality in some form or the other. Without this, in general, it is very difficult to establish theoretical results that produce optimal row-column designs. An approach similar to Kiefer (1975) for the case of a set of orthonormal contrasts, has been attempted by Ting and Notz (1987). Ture (1994) started with the inequality

if--2 ~ Var(~di _ ~dO) ~ V(~dOq- ~dl)/(~d0(~d0 + V~al)) i=1

Optimal and efficient treatment-controldesigns

1033

- : ~. 1 ~ r t ffCaH', (see (5.2)), where d is the symmetrized version of d, i.e., C d = -Ca where the sum is over all permutation matrices H that represent permutations of the test treatments only. (Note that while there may be no design d with Fisher Information matrix for the treatments Cd, the matrix is well defined for any design d.) Using theoretical and computational results Ture (1994) gave two tables, one of A-optimal designs in the range 2 ~< v ~< 10, 2 ~< k ~< 10, k ~ b ~< 30 and one of designs, in the same range, that he conjectures are optimal. For the setup of Example 5.1, suppose there are only 3, not 4, test treatments to be compared with the control. What is an optimal design? Ture (1994) provides the following answer.

EXAMPLE 5.4. For v = 3, b = k = 6 an A-optimal design in ~DnC(4, 6, 6) is: 001231 100232 213003 322100 032310 110023 Several combinatorial techniques for constructing BTCRC designs were given in Majumdar and Tamhane (1996). The following example illustrates one method. EXAMPLE 5.5. Start from a 4 x 4 Latin square in symbols 1,2, 3 and 4, with two parallel transversals. Replace the first transversal with controls, 0. Then use the Hedayat and Seiden (1974) method of sum composition to project the other transversal. This procedure is illustrated in the following sequence of designs. The second design in the sequence is a BTCRC design in Dnc(5, 4, 4), while the last design is a BTCRC design in Dnc(5, 5, 5): 1234 3412 4321 2143

>

1204 3410 0321 2043

02041 30104 >03012 20403 14230

6. B a y e s o p t i m a l d e s i g n s

How can we utilize prior information at the designing stage? In his pioneering work, Owen (1970) studied Bayes optimal block designs for comparing treatments with a control. It is his setup that will form the basis of this section. Even though the main focus of this section is block designs, row-column designs, as well as designs for the zero-way elimination of heterogeneity model will be reviewed briefly.

D. Majumdar

1034

6.1. Bayes optimal continuous block designs Suppose the observations follow the one-way elimination of heterogeneity model (2.4). Let 8k='r~-~-o,

i=l,...,v,

7j=#+'r0+flj,

8o=0,

j=l,...,b.

The 0i's are the treatment-control contrasts that we wish to estimate, while 7d is the expected performance of the control in block j. Model (2.4) can be written as: Yijl : Oi 4- "[j 4- £ijl.

Suppose there is a total of n observations. Let Y denote the n x 1 vector of observations, £ denote the n x 1 vector of errors, 8' = (81,..., 8v), " / = ('Yl,..., %). Then the model can be expressed as:

Y = XldO + 3227 + e, where X2 is a known matrix, and Xld is a known matrix that depends on the design d. In the Bayesian approach, 0, 7 and e are all assumed random. Specifically we shall assume the following model for the conditional distribution:

Y ]8,7 ,'~ N n ( X l d 8 4 - X z % E ) ,

(6.1)

for some covariance matrix E, where Nn denotes the n-variate normal distribution. The assumed prior is:

for some vectors/z0, #7 and matrices B*, B. From (6.1) and (6.2) it follows that the posterior distribution of 0 is:

8 I Y ~ Nv(p*a, Dd),

(6.3)

where

D21 = X~a(E + X 2 B X ~ ) - I X l d +

B *-1 ,

and

Dd-l#d. = X~d(E 4- X2BX~) -I (Y - X2#7) 4- B*-I#o.

(6.4)

Optimal and efficient treatment-control designs

1035

For the loss function L(O, 0) = ( 0 - O)'W('O- 0), where W is a positive definite matrix, the Bayes estimator of 0 is 0" = #~, with expected loss tr(WDd). Owen (1970) developed a method to find a design that minimizes tr(WDa) in

D(v + 1, b, k l , . . . , kb) = {d: d is a design based on v + 1 treatments in b blocks of sizes k ~ , . . . , kb}.

(6.5)

Note that in this subsection we do not assume that the blocks are of the same size. The total number of observations is n = y'~b=l kj. In order to fix the order of observations in the vector Y" in (6.1), for d E 79(v + 1, b, k l , . . . , kb), we shall write, X2 = d i a g ( l k ~ , . . . , lkb), a block-diagonal matrix. We will focus on the special case, W=[. DEFINITION 6.1. Given the matrices E, B and B*, a design that minimizes tr(Dd) will be called a Bayes A-optimal design. Owen (1970) took the approximate theory approach, i.e., the incidences ndij's are viewed as real numbers, rather than integers. Giovagnoli and Wynn (1981) called these designs continuous block designs. The problem reduces considerably for a certain class of priors. This is stated in the following result of Owen (1970). THEOREM 6.1. Suppose the error covariance matrix is of the form:

E = Xzff~X~ + d i a g ( e l , . . . , e~), where ei > O, for i = 1 , . . . , n , and E = (eij) is a symmetric matrix. Also, suppose, /~*

=

0"2((~1

--

~2)Iv + ~2J~,),

(6.6)

where ff 2 > 0 and (v -- 1) -1 < ~2/~1 < 1, Iv is the identity matrix of order v and Jv is a v × v matrix of unities. Moreover, suppose B + ffS is a nonnegative definite matrix. For designs with fixed ndOj, j = 1 , . . . , b, the optimal continuous block design has nd~j = (kj - ndOj)/v, for all i = 1 , . . . , v, and each j = 1 , . . . , b. It may be noted that Owen established Theorem 6.1 under a condition more general than the condition, B ÷ J~ is nonnegative definite. We use the latter in Theorem 6.1 in order to avoid more complicated technical conditions, and also since this condition would be satisfied by a large class of priors. The theorem says that for the optimal continuous design,

ndlj

. . . . .

ndvj

for j = l , . . . , b .

(6.7)

In view of this it remains to determine only the ndOj'S that minimize tr(Dd). Owen gave an algorithm for this, studied some special cases and gave an example.

D. Majumdar

1036

Instead of the trace, one can minimize other functions of Dd in order to obtain Bayes optimal designs. Taking this approach, Giovagnoli and Verdinelli (1983) considered a general class of criteria for optimality, with special emphasis on D-, and E-optimality. Verdinelli (1983) gave methods for computing Bayes D- and A-optimal designs. Optimal designs under the hierarchical model of Lindley and Smith (1972) were determined by Giovagnoli and Verdinelli (1985). The model is:

Y I 0,7 N Nn(AlldO + A12% Ell), '7

A2271

( 0 1 ) ,.~Nv+b ((A3102) 71

A3272

'

(E31 '

0

0

E22

O )) E32

'

'

Giovagnoli and Verdinelli (1985) extended Owen's results to this model. It may be noted that this work followed that of Smith and Verdinelli (1980) who investigated Bayes optimal designs for a hierarchical model with no block effects, i.e., the hierarchical model corresponding to (2.1). Smith and Verdinelli's model is:

Y 10 ~ N(AldO, Ei),

O lO1 ~ N(AzO1,E2),

O1 N(Oz, E3). ~

6.2. Bayes optimal exact designs The approximate, or continuous, block design theory is wide in its scope. As the theorems in this approach specify proportions of units which are assigned to each treatment, they give an overall idea of the nature of optimal designs. Also, one rule applies to (almost) all block sizes, and requires only minor computations when the block sizes are altered. These are very desirable properties. On the other hand, application of these designs is possible only after rounding off the treatment-block incidences to nearby integers , and this could result in loss of efficiency. Moreover, it follows from the restriction (6.7) that the known optimal designs cannot be obtained unless the block sizes are somewhat large. In particular, optimal designs cannot be obtained for the incomplete block setup, i.e., k <~ v, which is encountered very often in practice. Let us return to the exact optimality theory, i.e., retain the natural domain of nonnegative integers for the incidences ndij instead of approximating these by real numbers. In Majumdar (1992), the model given by (6.1) and (6.2) was considered along with additional properties (6.6) and B = o-26((1

-

p)Ib + PJb),

Cov(e~jt,eejv) = a27r1, Cov(eijl, eej,~,) = a27rz,

Var(sijl)

for I ¢ l', for j ~ j'.

= o -2,

(6.8)

The class of designs was D(v + 1, b, k) with 2 ~< k ~< v. (As mentioned above, Owen's result cannot be applied here since (6.7) implies that k is at least v + 1.) It was shown

Optimal and efficient treatment-control designs

1037

that under the conditions, 1 > 7q /> 1r2, 7r2 + 5p >7 0, ~2/> 0 a B T I B ( v , b , k ; t , s ) is Bayes A-optimal, where (t, s) is a point where a function gB(x, z), which depends on v, b, k and the prior parameters, attains a minimum over a set similar to A in (3.8). The prior structure for the O's in (6.8) is the same as that assumed by Owen (1970), but the "y's are assumed to be exchangeable and E's are assumed exchangeable within blocks. (Actually, these variables may be more general than exchangeable random variables, since no model for the expectations is required for the design problem.) This class of priors is, therefore, somewhat less general than Owen's class of priors given in Theorem 6.1. We will not reproduce the exact statement of the theorem since it is notafionally involved, but instead, discuss a simpler special case which was considered in Majumdar (1992), and earlier in Majumdar (1988b) and Stufken (1991b). The model for the special case is given by (6.1), (6.2), and

E = 0"2Ibk,

B *-1 = O, the null matrix,

B = O'20L--lIb,

(6.9)

i.e., the errors are homoscedastic, the prior on 0 is vague and the 7j's are i.i.d. Also, a E [0, oo); a = 0 corresponds to the case where no prior on 3' is incorporated in the design. While this prior may be limited in scope, our main purpose is to demonstrate the nature of optimal designs in a simple setup. It is expected that the general nature of the optimal design will be similar when the priors are chosen to be more general, for example, those given by (6,8) . Let

k~=k+a. It follows from (6.4) that the inverse of the posterior covariance matrix for 0 is given by, D d 1(a) = D i a g ( r a l , . . . , rdv) -- ~-~NdN~, where Nd is a v × b matrix obtained from the incidence matrix Nd by deleting the row corresponding to the control (the first row). For a > 0, let, A(a) = {(x,z): x = 0 , . . . , k -

T = bk - (bx + z),

[(ks + 1)/2J - 1;z = 0, a , . . . , b } ,

S = bk 2 - 2k(bx + z) + (bx 2 + 2xz + z),

9a(x,z) = k=v(v-1)2[(k~(v

1)-v)T+S]-'+k=v[k~T-S]

-1.

For a = 0, let A(0) = A, the set in given (3.8), and go(x, z) = g(x, z), the function given in (3.7). In addition to Bayes optimal designs we will also consider F-minimax optimal designs. Given a set F, a F-minimax rule (see Berger, 1985) minimizes the maximum risk for the prior parameters in the set F. This is a robust decision rule. A F-minimax

D. Majumdar

1038

optimal design is one that minimizes the maximum risk of the F-minimax rule. When F is the entire space, the P-minimax rule is the minimax rule. DEFINITION 6.2. Given a set/7, do is called a F-minimax optimal design in D(v + 1,

b, k) if, for Dd given by (6.4), Max

E,B,B 6 F

tr(Dd~)=

Min

Max

d6"D(v+l,b,k) E,B,B 6F

tr(Dd).

THEOREM 6.2. Consider the class D(v + 1, b, k) with 2 ~ k <~ v. Let a design do be defined as: (i) if k~ + 1 < 2k, do is a BTIB(v,b,k;t,s) where 9~(t,s) = Min(~,z)ea(~) 9,~(x, z), (ii) if ka + 1 ) 2k, do is a BIB design in D(v, b, k) based on the test treatments. Then the following hold: (A) do is Bayes A-optimal in 79(v + 1, b, k) for the model given by (6.1), (6.2) and (6.9). (B) do is a P-minimax optimal design for the model (6.1) and (6.2), with F = {(E, B, B*): E = a2I, and a 2 a - l I - B is nonnegative definite}. Theorem 6.2 may be found in Majumdar (1992). The following theorem, due to Stufken (1991b), gives families of Bayes A-optimal designs. When a = 0, this is Theorem 3.4. THEOREM 6.3. For 2 <~ k <~ v and a integral, a BTIB(v, b, k; k - q, 0) is Bayes Aoptimal in 79(v + 1, b, k) for the model given by (6.1), (6.2) and (6.9) if (q - 1)2 + 1 <. (krz = q)2v <~q2. EXAMPLE 6.1. A BTIB(v,b, 7; 1,0) is Bayes A-optimal for 7 <~ v < 9 when o~ = 1, i.e., when the prior variance of the 7j's is equal to cr2. What if a was non-integral? A rigorous analysis for this case is difficult, but Stufken (1991b) anticipates that Theorem 6.3 would remain valid. If this is true, then Theorem 6.3 would give, for fixed v, b and k a range of a for which a BTIB(v, b, k; t, 0) is optimal, i.e., an optimal design that is robust against specification of the prior. A robustness result using a different approach is given in Majumdar (1992). THEOREM 6.4. Consider the class D(v + 1, b, k) with 2 <~ k ~ v, and o~ c [0, 0o). Let r* denote the optimal replication of the control given by Theorem 3.2*. For i = 0 , . . . , r* - 1, let 3i+1 be the (unique) solution to the following equation in ~ ~.~. 0~--1 :

g~(ti/bJ, i -

bLi/bJ) = g~([i/bJ,

i - bLi/bj +

1),

and 3o = 0, 3~*+1 = c~. Then, (a) the ~ are increasing. (b) If io is an integer in 0 <. io <~ r* such that a -1 E [~i0,6i0+1), then a BTIB(v, b, k; t, s) with bt + s = io, is Bayes A-optimal for the model given by (6.1),

Optimal and efficient treatment-controldesigns

1039

(6.2) and (6.9), as well as/"-minimax optimal for the model (6.1) and (6.2), with/" as described in Theorem 6.2. Let us denote a BTIB(v, b, k;t, s) with bt + s = i by di. Theorem 6.4 gives a partition of [0, c~) consisting of r* + 1 intervals such that for all a-1 in the interval [5~, 5i+1), the design di is Bayes A-optimal and/"-minimax optimal. Thus, the value of a (and hence the prior variance of the 7j's) need not be specified exactly at the designing stage. The design di will not, however, exist for all i = 0 , . . . , r*. Suppose i* is such that d~. does not exist. Let iland i2 be such that i* E (il, i2), d~l, di2 exist and further, no di with i C (il, i2) exists. Then at least one of the two designs, dii or di2, is likely to be highly efficient (see Section 7 for a general definition of efficiency). Here is a simple example. EXAMPLE 6.2. Let v = 3, b = 24 and k -- 2. This example has been studied in some detail in Majumdar (1992). Here are some observations. A BTIB(3, 24, 2;0, 15) is Bayes A-optimal for a ---- 0.18, i.e., when the prior variance of the 7j's (see (6.9)) is equal to 5.5~r2, as well as for c~ = 0.15, i.e., when the prior variance of the "/j's is equal to 6.5o-2 (hence for all prior variances in the interval [5.5(7z, 6.5a2]). For c~ = 0.1, i.e., when the prior variance of the 7j's is equal to 10ff2, the corresponding di is a BTIB(3, 24, 2; 0, 16) which does not exist; nevertheless the design BTIB(3, 24, 2;0, 15) is highly (at least 99.6%) efficient. An A-optimal design given by Theorem 3.2 is BTIB(3, 24, 2; 0, 18). This design is Bayes A-optimal for all c~ E (0, .033), i.e., whenever the prior variance of the 7j's is greater than 30a 2. Ting and Yang (1994) generalized some of the results in this section to classes D(v + 1, b, k) with k > v.

6.3. Bayes optimal row-column designs Optimal designs for the row-column setup were first considered by Toman and Notz (1991). Suppose 0i = "ri -"r0, 0' = (01,...,0v), /3' = (/31,...,135), and "Y' = (71,... ,'Tk) in (5.1). The model is: Y I To, 0,/3, 7 "~ N(X,d("ro, 0')' + X2fl + )(3"/, cr2I), (To, 0',/3', 7')' "~ N((#o, #~, #~, ##)', Diag(a, T, B, G)), where Diag(a, T, B, G) is a block diagonal matrix. Toman and Notz gave a procedure for obtaining Bayes optimal continuous designs. They focused on the special case where T, B and G were completely symmetric. They also discussed rounding-off strategies for continuous designs. Bayes optimal row-column designs in the exact theory setup was considered in Majumdar (1992). The model was formulated differently. It was (6.1) and (6.2), with 0i :- ~-i - ~'0, 7jr = E(y0fl), 0' = (0t,..., Or) , and 7' = ("/11,..., "ffbk). Exact Bayes optimal designs were derived for a certain class of priors. For a general survey of Bayes optimal designs that is not restricted to treatmentcontrol designs, see Dasgupta (1996).

D. Majumdar

1040

7. On efficiency bounds of designs An (exact) optimal design always exists since the set of designs is finite. Identification of an optimal design may, however, be a very difficult problem. The experimenter, in this case, might look for a design that is close to an optimal design. In other experiments, an optimal design may have a feature that is not desirable to the experimenter, or there may be experimental restrictions that suggest the use of a design that is known to be not optimal. In such cases it would be very useful to compute accurately the efficiency of a design. Given a class D of designs and a criterion function ~(d) (suppose ~ is such that the smaller its value the better the design), the efficiency of a design dl E 79 may be defined as:

e(dl) = Min ¢(d)/~(dl). dE'/9

(7.1)

If a ~-optimal design in D is known then it is easy to compute e(dl). If not, then evaluation of the numerator in (7.1) is a difficult problem. If a lower bound ~L to the numerator is known, i.e., ~(d) ~> ~L,

for all d E 79,

(7.2)

then e(dl) ~> ~SL/~5(dl). This gives us a lower bound to the efficiency, which will be denoted by,

eL(d1) = ~n/~(dl).

(7.3)

In spite of the risk of confusion, eL will also be called the efficiency. Researchers in the area of design have computed the efficiency of designs in various settings (see, e.g., Roy, 1958). In most cases a lower bound ~L is not difficult to find, but clearly, eL is a meaningful measure only when the inequality (7.2) is sharp. Let us start with the model (2.4) in the block design setup and ~(d) = tr(M~-1), i.e., the A-criterion. Theorem 3.2* or 3.7* gives a lower bound ~L for this case. (Even though Theorem 3.2* can be viewed as a special case of Theorem 3.7", we will display the two efficiencies separately.) Thus, for d c D(v + 1, b, k),

eL(d)=

g*(r*)/tr(PCd P' = g*(r*)/tr(Md-1), i f k ~ v .

(7.4)

In view of Theorem 3.6, Cheng et al. (1988) suggested two ways to obtain a design that is highly efficient in 79(v + 1, b, k) in case an optimal design is not known. (1) Find a good BTIB design with rdo close to r*, or (2) find a design combinatorially

Optimal and efficient treatment-control designs

1041

close to

a good BTIB design with rd0 = r*. By a good BTIB design we mean a BTIB(v, b, k; t, s) design, or a BTIB design that is combinatorially close to it. Cheng et al. (1988) also gave a procedure for identifying highly efficient designs, that is based on some techniques developed and discussed in Ture (1982). Some highly efficient designs are also reported in Ture (1985). Here are a few examples of highly efficient designs. EXAMPLE 7.1. Let v = 5, k = 4 and b = 7. This example is studied in some detail in Ture (1982) and in Cheng et al. (1988). Here r* = 7, 9*(r*) = 2.04. There is no BTIB(5, 7, 4; 1,0), so Theorem 3.2 cannot be applied to get an optimal design. Here are two possible designs.

(o00o0o)

dl ~--

0 0 1 1 2 2 2

1 3 3 4 3 4 3 ' 2 4 5 5 5 5 4 0 0 0 0 0 0 0

d2

1 1 1 1 1 2 2 2 2 3 3 4 3 3 4 5 4 5 5 4 5

dl is a BTIB design with rdlo = 8, while d2 is not a BTIB design; rd2o = 7. Also, t r ( M ~ 1) = 2.143, t r ( M ~ 1) = 2.058. Hence, eL(d1) ----95.2% and eL(d2) = 99.2%. Both designs are highly efficient, though d2 is clearly the better of the two. EXAMPLE 712. Let v = 4, b = 4 and k = 7. Here r* = 9 and f*(r*) = 1.3128. There is no design with supplemented balance with nine replications of the control. The design of Pearce (1953) given in Example 2.1 (call it all) has ralo = 8 and t r ( M ~ 1) = 1.3188; hence eL(d1) = 99.5%. Pearce's design is, therefore, highly efficient. An interesting question is: is there a design d2 with r,t20 = 9 that has even higher efficiency? For

(oooo 0000 1111 dz=

2 2 2 2 , 3333 4444 ~0123

eL(d2) = 99.6%. In view of the very high efficiency of Pearce's design, the additional property of symmetry of this design, together with the IX(Md21) ~--- 1.3181, so

1042

D. Majumdar

very minor improvement achieved by the design d2, the design da would be preferable to d2 in most experiments. More examples of highly efficient designs with supplemented balance in classes

D(v + 1, b, k) with k > v, are available in Jacroux and Majumdar (1989). An alternative approach to identifying efficient designs is to restrict to a subclass of all designs. If the subclass is sufficiently large, then this method is expected to yield designs that are close to an optimal design. The method also has the advantage that the chosen design is guaranteed to possess any property that is shared by all designs in the subclass. Examples of subclasses are BTIB designs or GDTD's in 79@ + 1, b, k). Hedayat and Majumdar (1984) gave a catalog of designs that are A-optimal in the class of BTIB designs for k = 2, 2 ~< v ~ 10 and v ~< b ~< 50. Jacroux (1987b) obtained a similar catalog of designs for the MV-optimality criterion. Stufken (1988) investigated the efficiency of the best reinforced BIB design (Das, 1958; Cox, 1958), i.e., a BTIB(v, b, k; t, 0) which has the smallest value of the A-criterion. He established analytic results, using which he evaluated eL for each k in the interval [3, 10] and all v. Here is an example. EXAMPLE 7.3. Let k = 4. For a b for which it exists, let d denote a BTIB(v, b, 4; 1,0). For 5 ~< v ~< 9, it follows from Theorem 3.3 that d is A-optimal. Stufken (1988) shows that if v = 4, eL(d) >~99.99%, and if v /> 10,

eL(d) >~ (3v -- 1)(v -- 1 + ~/-v-~)2/(4v2(v + 1)). Thus if v = 10, eL(d) ~> 99.98%, if v = 20, eL(d) >>-97.65%, and if v = 100, eL(d) >~ 88%. Stufken's study strengthens the belief that the subclass of BTIB designs in 79(v + 1, b, k) usually contains highly efficient designs, with exceptions more likely when v is large compared to k. The reason seems to be that due to its demanding combinatorial structure, sometimes it is impossible to find a BTIB design d with rd0 in the vicinity of r*. Here is an example. EXAMPLE 7.4. Let v = 10, b = 80 and k = 2. Here r* = 39 and 9*(r*) = 1.896. There is no BTIB design with 39 replications of the control, so Theorem 3.2 does not give an optimal design. Let d 0) be a BTIB(10, 10,2; 1,0) and d (2) be a BIB(10,45,9,2, 1) design in the test treatments only. Further let d (3) E 79(10,40, 2) be a BIB(10,45,9,2, 1) design in the test treatments with the blocks {1,6}, {2, 7}, {3, 8}, {4, 9} and {5, 10} deleted, d (4) C 79(10, 41,2) be the design d (3) U {1,6}, and d (5) be the design d (1) with the block {0, 10} deleted. Hedayat and Majumdar (1984) showed that an A-optimal design in the subclass of BTIB designs within 79(11,80,2) is do = 8d (1). For this design, rdoo = 80, tr(M~0 ~) = 2.5, hence eL(do) = 75.8%. Now consider two other designs in 79(11,80,2): dl = 4d(U tO d (3) and d2 = 3d0) t3 d (4) U d (5). It is easy to see that, /'dl0 40, tr(M~ 1) = 1.904, eL(dr) = 99.6%, and rd20 = 39, tr(M~ 1) 1.905, eL(dz) = 99.5%. Neither dl nor "=

Optimaland efficienttreatment-controldesigns

1043

dz are BTIB designs - dl is a GDTD, d2 is not. Both of these designs are highly efficient, and both are about 24% more efficient that the best BTIB design do in 79(11, 80, 2). Since the class of GDTD's is larger than the class of BTIB designs, the best design in the former class will perform at least as well as the best design in the latter class, and, as in Example 7.4, sometimes substantially better. Stufken and Kim (1992) gave a complete listing of designs that are A-optimal in the class of GDTD's for k = 2, 3, k ~< v ~< 6 and b ~< b0 where b0 is 50 or more. In this context it may be noted that all of the optimal designs for simultaneous confidence intervals in Section 4 were optimal within certain subclasses of designs. A general method of obtaining a lower bound ~bL to • is to use the principle of Theorem 3.1. Consider the models A and B in (3.2). For a design d if ¢J(d) denotes the value of the criterion based on model J ( J -- A, B), and if the criterion • (d) = ¢(Cd) has the property ~b(H - G) /> ¢ ( H ) , whenever H, G, H - G are nonnegative definite, then ¢B(d) ) ~A(d). Since A is a simpler model it is usually easier to obtain a lower bound to the criterion under Model A. This is an old technique in design theory. The 0-way elimination of heterogeneity model is used to compute the efficiency of designs under the l-way elimination of heterogeneity model, the 1-way for the 2-way elimination of heterogeneity model and so on. In Majumdar and Tamhane (1996) this method has been used to compute the efficiency of row-column designs. To describe the results we need some notation. For the class of designs with b blocks of size k each, i.e., 79(v + 1, b, k), let us denote the function f*(~) by the extended notation f~,k(r) and the quantity f*(r*), that was used in Theorem 3.7*, by fb*,k(r~,k). Now consider the class 79RC(v + 1, b, k) of row-column designs. If d c 79Rc(v + 1, b, k), then it follows from the discussion above that tr(Md 1) ~> Max(f~,k(r ), f~,b(r))

>~Max(f~*k(r~,k), f;,b(r~,b)),

which gives a measure of efficiency,

eL(d) =

Max(f

*k

*

*

- 1 ).

Here is an example. EXAMPLE 7.5. Let v = 4, b = k = 5. The design d E 79gc(5, 5, 5) in Example 5.5 (the last design in the sequence in that example), has efficiency eL(d) = 1.5/1.568 = 95.7%. Several researchers have computed efficiency of designs in various settings. They include, Pigeon and Raghavarao (1987) for repeated measurements designs, Angelis et al. (1993) and Gupta and Kageyama (1993) for block designs with unequal-sized blocks, Gerami and Lewis (1992) and Gerami et al. (1993) for factorial designs. These will be briefly discussed in the next section.

D. Majumdar

1044

8. Optimal and efficient designs in other settings Research on treatment-control designs has progressed in various directions to accommodate different experimental situations and models. In this section we will briefly outline some of these.

8.1. Repeated measurements designs Consider the problem of designing experiments where subjects receive some or all of the treatments in an ordered fashion over a number of successive periods. The model may include the residual effect of the treatment applied in the previous period, in addition to the direct effect of the treatment applied in the current period. Suppose there are b subjects and k periods. We can say that d(1,j) = i if design d assigns treatment i (0 ~< i ~< v) to subject j (1 ~< j ~< b) in period l (1 ~< 1 ~< k). We will denote the class of connected designs by 79nM(v + 1, b, k). The model for an observation in period l of subject j is:

Yjl = P q- Td(l,j) q- Pd(l-l,j) -}- flj q- ")'l q- ~ijl, where /3j and 3't are the subject and period effects, Tdq,j) is a direct effect of the treatment applied in period 1 and Pd(t-l,j) is a residual effect of the treatment applied in period l - 1. Since there is no residual effect in the first period, we can write Pd(o,j) = 0. See Stufken (1996) for a general treatise of repeated measurement designs. Pigeon (1984) and Pigeon and Raghavarao (1987) studied this problem. They called a design a control balanced residual effects design if the information matrix of the treatment-control direct effect contrasts is completely symmetric, and the information matrix of the treatment-control residual effect contrasts is completely symmetric. They characterized control balanced residual effects designs, gave several methods for construction of these designs, and also gave tables of such designs along with their efficiencies. A-optimal designs for the direct effect of treatments versus control contrasts were obtained in Majumdar (1988a). The main tool used in this paper was Theorem 3.1. Hedayat and Zhao (1990) gave a complete solution for A- and MV-optimal designs for the direct effect of treatments versus control contrasts for experiments with only two periods, k = 2. Let v be a perfect square and d (~) E 7~nM(v + 1,bi,2) be a design with treatment i in the first period of each subject and the in the second period, treatments distributed according to Theorem 2.1. Then a result of Hedayat and Zhao (1990) asserts that the union d (~1) t_J.. • U d (i'0 for any i i , . . . , ira, each chosen from {0, 1 , . . . , v} is A- and MV-optimal in 79RM(v + 1, ~t=lm bit, 2). EXAMPLE 8.1. Let v = 4, k = 2 and b = 18. The following design with subjects represented by columns and periods represented by rows is optimal: 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 0 0 1 2 3 4 0 0 1 2 3 4 0 0 1 2 3 4

Optimaland efficienttreatment-controldesigns

1045

Koch et al. (1989) did an extensive study of comparing two test treatments among themselves, as well as with a control, in a two-period repeated measurements design (thus v = 2, k = 2). The design they considered consisted of the treatment sequences (1,2), (2, 1), (0, 1), (1,0), (0,2), (2,0) distributed on subjects in the ratio r e : m : 1 : 1 : 1 : 1, with special emphasis on the case m = 3. They used several models, each consisting of subject effects, period effects and direct effects of treatments; in addition, some of the models had residual effects. The efficiencies of the estimators of the contrasts were computed and a simulated example was studied.

8.2. Designs for comparing two sets of treatments A natural extension of the problem of comparing several treatments with a control is the problem of comparing several treatments with more than one control. For instance, in drug studies, there may be a placebo (inactive control) and a standard drug (active control). In general, the two controls may be of different importance, e.g., the test treatment-placebo contrasts and the test treatment-standard treatment contrasts may be of different significance in an experiment. Thus these contrasts may have to be weighted differently in the design criterion. This will lead to a somewhat complicated criterion that depends on the weights chosen. If the weights are the same, the criterion simplifies to some extent and some results on optimal and efficient designs are available in the literature. Suppose there are v ÷ u treatments divided into two groups, G and H of v and u treatments each. Then the contrasts of primary interest are: "ra -~-h, where 9 E G and h c HI Thus there are, in all, vu contrasts of interest. Consider the block design setup with model (2.4), and suppose G = { 1 , . . . , v } and H = {v + 1 , . . . , v + u}. It was shown in Majumdar (1986) that the information matrix of the contrasts of interest is completely symmetric if the Fisher Information matrix for the treatments of a design d E 79(v + u, b, k) is of the form,

Cd= ( pdIv + qdJvv tdJvu ) tdJ~,~ rdI~ + SdJ~ '

(8.1)

for some Pd, qd, rd, 8d, td, where J,~v is a u × v matrix of unities. Condition (8.1) is the generalization of condition (2.5) for the case of several controls. In this setup a design is called A-optimal if it minimizes ~gcC ~heH Var(?d9 -?dh) among all designs in 79(v + u, b, k). Some results on A-optimality were given in Majumdar (1986). A generalization of Theorem 3.2 was given for the case k ~< min(v, u). Another result in this paper, obtained by applying Theorem 3.1, gave A-optimal designs for design classes with k -= 0 (mod (v + v/v--~)) and k - 0 (mod (u ÷ v / ~ ) ) . Christof (1987) generalized some of the results in Majumdar (1986) in the setup of approximate theory.

D. Majumdar

1046

8.3. Trend-resistant designs Jacroux (1993) considered the problem of comparing two sets of treatments, G and H of sizes v and u using experimental units that are ordered over time, where it is assumed that the observations are affected by a smooth trend over time. (The special case u = 1 corresponds to a single control.) If the unit at the jth time point (j = 1 , . . . , n), receives treatment i then the model is:

Yij = # + "ri + ~ l j + " " + ~pjP +eij,

(8.2)

where ~lj + " " + flpJP is the effect of a trend that is assumed to be a polynomial of degree p. A design is a sequence or run order. We will denote the set of all connected designs by 79to(V, u; n). For a design d E 79~o(v, u; n), if the least square estimator of ~-g - ~-h under model (8.2) is the same as the least square estimator under model (2.1) (i.e., the model with all fit = 0), then this design is called p-trend free. Thus in a p-trend free design, treatments and trends are orthogonal. Jacroux (1993) studied trend free designs, and obtained designs that are A-optimal and/or MV-optimal as well as p trend-free. EXAMPLE 8.2. The following run order is A-optimal and 1-trend free in 79(2, 5; 29). Here G = {1,2} and H = {3,4,5}. (12121267673455344537676212121)

8.4. Block designs with unequal block sizes and blocks with nested rows and columns Consider the class D(v + 1, b, k l , . . . , kb) of designs defined in (6.5), and the 1-way elimination of heterogeneity model given by (2.4). What are good designs for estimation in this class? Angelis and Moyssiadis (1991) and Jacroux (1992) extended Theorem 3.2 to this setup. All of these authors, as well as Gupta and Kageyama (1993), characterized designs in D(v + 1, b, k-l,..., kb) for which the information matrix of treatment-control contrasts is completely symmetric, i.e., the matrix Md in (3.1) is completely symmetric. Gupta and Kageyama (1993) gave several methods for constructing such designs, and tables of designs along with their efficiencies. Angelis and Moyssiadis (1991) gave methods to construct these designs, as well as an algorithm to find A-optimal designs. They also gave tables of optimal designs. Angelis et al. (1993) gave methods for constructing designs with Cd as above that are efficient. Jacroux (1992) gave a generalization of Theorem 3.6, as well as infinite families of A- and MV-optimal designs. Next consider experiments where the units in each of the b blocks are arranged in a p × q array, i.e., the nested row-column setup. For a design, if the unit in row l and column m of block j receives treatment i, then the model is:

y~jt,-. = # ÷ T~ + flj + Pt(y) + 7,~(j) + ~ijlm.

Optimal and efficient treatment-control designs

1047

Gupta and Kageyama (1991) considered the problem of finding good designs for treatment-control contrasts in this setup. Extending Pearce's (1960) definition of designs with supplemented balance, they sought designs for which the information matrix for the treatment-control contrasts is completely symmetric. Let n~dit, n~i h and nd~j denote respectively the number of times treatment i appears in row l, column h and block j. Let A~dii, ---- AX-'v r r l, "~dii' e . ~ / = I ?Zdilndi' = ~ - - ~ = 1 ?~dihndi c h and )~dii' : ~-~=1 •dijndi'j" Gupta and Kageyama (1991) showed that the information matrix for the treatment-control contrasts is completely symmetric if for s and so with so ~ 0 and so + (v - 1)s ~ 0, the following is true for all ( i , i ' ) e { 1 , . . . ,v} × { 1 , . . . ,v}, i # i':

and p)Cdi e + q)~ii' -- Adii" = S.

P)~dio + qA~diO -- )~diO = SO

These designs were called type S designs by Gupta and Kageyama, who also gave several methods for constructing such designs. Here is an example. EXAMPLE 8.3. For v = 3, b = 3, p = 2 and q = 3, here is a type S design with s0=6, s=3. Blockl

Block2

Block3

012

013

023

120

130

230

8.5. Block designs when treatments have a factorial structure

Gupta (1995) considered block designs when each treatment is a level combination of several factors. Suppose that there are t factors F 1 , . . . , Ft, where Fi has m i levels, 0, 1 , . . . , m i - 1 for / = 1,... ,t. For each factor the level 0 denotes a control. A treatment is written as X l X 2 , . . . ,Xn, with xi E {0, 1,... , m i - 1}. Gupta extended the concept of supplemented balance to this setup, and called such designs type S-PB designs. He also gave methods for constructing such designs. Since the description of the factorial setup involves elaborate notation, we refer the reader to Gupta's paper for details, but give an example here to illustrate his approach. EXAMPLE 8.4. Let t -----2, ml = m2 ---- 3. Thus we have two factors at three levels each. Let r be the vector of treatment effects with the treatments written in lexicographic order. Let,

Ull

= 'U,21 =

U13 =

~23 ~

v1(1) ~

~

-1 0

1 1

,

U12 = U22 =

v1(1) ~

0

-1

,

1048

D. Majumdar

and for (/,l') 7~ (3,3), let u(1,l') = T'(U,t ® U2V), where @ denotes Kronecker product. The two contrasts u(1,3) and u(2, 3) will together represent the main effect of F1, the two contrasts u(3, 1) and u(3, 2) represent the main effect of F2, while the remaining four contrasts u(l, l'), l ¢ 3, l' ¢ 3, represent the FIF2 interaction. Note that the set of contrasts that represent a main effect or interaction are not orthogonal, as is the case with the traditional definition when the factors do not have a special level such as the control. Gupta (1995) defined a type S-PB design as one in which the estimators for all contrasts of any effect (main effect of F1, main effect of F2 or the interaction F1F2) have the same variance. The idea is that the design for each effect has the properties of a design with supplemented balance. For k = 5 and b = 6 here is an example of a type S-PB design, where treatments are denoted by pairs x l x 2 , (xi = 0, 1,2; i = 1, 2) and, blocks are denoted by columns. 01 01 01 01 00 00 02 02 02 02 01 11 10 10 10 10 02 12 20 20 20 20 10 21 11 12 21 22 20 22 Motivated by a problem in drug-testing, Gerami and Lewis (1992) also considered block designs when treatments have a factorial structure, but their approach was different. Consider the case of two factors, i.e., t = 2. Suppose each factor is a different drug and the level 0 is a placebo. Each treatment, therefore, is a combination of two drugs. Gerami and Lewis considered experiments where it is unethical to administer a double placebo, i.e., the combination (0, 0). There are, therefore, v = rnlrn2 - 1 treatments. The object is to compare the different levels of each factor with the control level, at each fixed level of the other factor. The contrasts of interest are, Tie - "rio, the difference in the effects of level i' and level 0 of factor F2 when F1 is at level i, and "rii, - "roe, the difference in the effects of level i and level 0 of factor F1 when F2 is at level i', for i = 1 , . . . , m l - 1 and i' = 1 , . . . ,rn2 - 1. A designs do E D ( v , b, k) is called A-optimal if it minimizes Tn, 1 - - 1 rr~2-- 1

+ i=1

it=l

For the case rn2 = 2, Gerami and Lewis determined bounds on the efficiency of designs, described designs that have a completely symmetric information matrix for the contrasts of interest and discussed methods of constructing such designs. Gerami et al. (1993) continued the research of Gerami and Lewis (1992) by identifying a class of efficient designs. A lower bound to the efficiency of designs in that

Optimal and efficient treatment-control designs

1049

class was obtained to determine the performance of the worst design in the class. Tables of designs and their efficiencies were also given. Here is an example. EXAMPLE 8.5. For ml -- 3, m2 = 2, b = 5 and k : 8, here is a design that is at least 94% efficient. 01 01 01 01 01 01 01 01 01 01 10 10 10 10 10 20 20 20 20 20 11 I1

11 10 20

11 11 11 11 11 21 21 21 11 21 21 21 21 21 21

8.6. Block designs when errors are correlated

Consider the class of designs D ( v + 1, b, k) and the model (2.4), with one difference - instead of being homoscedastic, the errors, eifl, are possibly correlated. The problem of finding optimal designs in this setup has been studied by Cutler (1993). He assumed that errors have a stationary, first-order, autoregressive correlation structure. The estimation method is the general least square method. Cutler established a general result on optimality for treatment-control contrasts which generalizes Theorem 3.2. He also suggested two families of designs and studied their construction and optimality properties.

Acknowledgements I received an extensive set of comments from the referee on the first draft of this paper, which ranged from pointing out typographical errors to making several excellent suggestions regarding the style and contents. These led to a substantial improvement in the quality of the paper. For this, I am extremely grateful to the referee. I am also grateful to Professor Subir Ghosh for his patience and encouragement.

References Agrawal, H. (1966). Some generalizations of distinct representatives with applications to statistical designs. Ann. Math. Statist. 37, 525-528. Angelis, L., S. Kageyama and C. Moyssiadis (1993). Methods of constructing A-efficientBT|UB designs. Utilitas Math. 44, 5-15. Angelis, L. and C. Moyssiadis (1991). A-optimal incomplete block designs with unequal block sizes for comparing test treatments with a control. J. Statist. Plann. Inference 28, 353-368.

1050

D. Majumdar

Bechhofer, R. E. (1969). Optimal allocation of observations when comparing several treatments with a control. In: E R. Krishnalah, ed., Multivariate Analysis, Vol. 2. Academic Press, New York, 463-473. Bechhofer, R. E. and C. Dunnett (1988). Tables of percentage points of multivariate Student t distribution. In: Selected Tables in Mathematical Statistics, Vol. 11. Amer. Math. Soc., Providence, RI. Bechhofer, R. E. and D. J. Nocturne (1972). Optimal allocation of observations when comparing several treatments with a control, II: 2-sided comparisons. Technometrics 14, 423-436. Bechhofer, R. E. and A. C. Tamhane (1981). Incomplete block designs for comparing treatments with a control: General theory. Technometrics 23, 45-57. Bechhofer, R. E. and A. C. Tamhane (1983a). Design of experiments for comparing treatments with a control: Tables of optimal allocations of observations. Technometrics 25, 87-95. Bechhofer, R. E. and A. C. Tamhane (1983b). Incomplete block designs for comparing treatments with a control (II): Optimal designs for p = 2(1)6, k = 2 and p = 3, k = 3. Sankhya Ser. B 45, 193-224. Bechhofer, R. E. and A. C. Tamhane (1985). Selected Tables in Mathematical Statistics, Vol. 8. Amer. Math. Soc., Providence, RI. Bechhofer, R. E. and B. W. Tumbull (1971). Optimal allocation of observations when comparing several treatments with a control, III: Globally best one-sided intervals for unequal variances. In: S. S. Gupta and J. Yackel, eds., Statistical Decision Theory and Related Topics. Academic Press, New York, 41-78. Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York. Cheng, C. S., D. Majumdar, J. Stufken and T. E. Ture (1988). Optimal step type designs for comparing treatments with a control. J. Amer. Statist. Assoc. 83, 477-482. Christof, K. (1987). Optimale blockpl~e zum vergleich yon kontroll- und testbehandlungen. Ph.D. Dissertation, Univ. Augsburg. Constantine, G. M. (1983). On the trace efficiency for control of reinforced balanced incomplete block designs. J. Roy. Statist. Soc. Ser. B 45, 31-36. Cox, D. R. (1958). Planning of Experiments. Wiley, New York. Cutler, R. D. (1993). Efficient block designs for comparing test treatments to a control when the errors are correlated. J. Statist. Plann. Inference 36, 107-125. Das, M. N. (1958). On reinforced incomplete block designs. J. Indian Soc. Agricultural Statist. 10, 73-77. DasGupta, A. (1996). Review of optimal Bayes designs. In: Handbook of Statistics, this volume, Chapter 29. Dunnett, C. W. (1955). A multiple comparison procedure for comparing several treatments with a control. J. Amer. Statist. Assoc. 50, 1096-1121. Dunnett, C. W. (1964) New tables for multiple comparisons with a control. Biometrics 20, 482-491. Dunnett, C. W. and M. Sobel (1954). A bivariate generalization of Student's t-distribution, with tables for certain special cases. Biometrika 41, 153-169. Fieler, E. C. (1947). Some remarks on the statistical background in bioassay. Analyst 72, 37-43. Finney, D. J. (1952). Statistical Methods in Biological Assay. Haffner, New York. Freeman, G. H. (1975). Row-and-column designs with two groups of treatments having different replications. J. Roy. Statist. Soc. Ser. B 37, 114-128. Gerami, A. and S. M. Lewis (1992). Comparing dual with single treatments in block designs. Biometrika 79, 603-610. Gerami, A., S. M. Lewis, D. Majumdar and W. I. Notz (1993). Efficient block designs for comparing dual with single treatments. Tech. Report, Univ. of Southampton. Giovagnoli, A. and I. Verdinelli (1983). Bayes D- and E-optimal block designs. Biometrika 70, 695-706. Giovagnoli, A. and I. Verdinelli (1985). Optimal block designs under a hierarchical linear model. In: J. M. Bemardo, M. H. DeGroot, D. V. Lindley and A. E M. Smith, eds., Bayesian Statistics, Vol. 2. North-Holland, Amsterdam, 655-662. Giovagnoli, A. and H. P. Wynn (1981). Optimum continuous block designs. Proc. Roy. Soc. (London) Ser. A 377, 405-416. Giovagnoli, A. and H. E Wynn (1985). Schur optimal continuous block designs for treatments with a control. In: L. M. LeCam and R. A. Olshen, eds., Proc. Berkeley Conf. in Honor ofJerzy Neyman and Jack Kiefer, Vol. 2. Wadsworth, Monterey, CA, 651-666. Gupta, S. (1989). Efficient designs for comparing test treatments with a control. Biometrika 76, 783-787. Gupta, S. (1995). Multi-factor designs for test versus control comparisons. Utilitas Math. 47, 199-210. Gupta, S. and S. Kageyama (1991). Type S designs for nested rows and columns. Metrika 38, 195-202.

Optimal and efficient treatment-control designs

1051

Gupta, S. and S. Kageyama (1993). Type S designs in unequal blocks. J. Combin. Inform. System Sci. 18, 97-112. Hall, P. (1935). On representatives of subsets. J. London Math. Soc. 10, 26-30. Hayter, A. J. and A. C. Tamhane (1991). Sample size determination for step-down multiple test procedures: Orthogonal contrasts and comparisons with a control. J. Statist. Plann. Inference 27, 271-290. Hedayat, A. S., M. Jacroux and D. Majumdar (1988). Optimal designs for comparing test treatments with controls (with discussions). Statist. Sci. 3, 462-491. Hedayat, A. S. and D. Majumdar (1984). A-optimal incomplete block designs for control-test treatment comparisons. Technometrics 26, 363-370. Hedayat, A. S. and D. Majumdar (1985). Families of optimal block designs for comparing test treatments with a control. Ann. Statist. 13, 757-767. Hedayat, A. S. and D. Majumdar (1988). Model robust optimal designs for comparing test treatments with a control. J. Statist. Plann. Inference 18, 25-33. Hedayat, A. S. and E. Seiden (1974). On the theory and application of sum composition of latin squares and orthogonal latin squares. Pacific J. Math. 54, 85-112. Hedayat, A. S. and W. Zhao (1990). Optimal two-period repeated measurements designs. Ann. Statist. 18, 1805-1816. Hoblyn, T. N., S. C. Pearce and G. H. Freeman (1954). Some considerations in the design of successive experiments in fruit plantations. Biometrics 10, 503-515. Hochberg, Y. and A. C. Tamhane (1987). Multiple Comparison Procedures. Wiley, New York. Jacroux, M. (1984). On the optimality and usage of reinforced block designs for comparing test treatments with a standard treatment. J. Roy. Statist. Soc. Ser. B 46, 316-322. Jacroux, M. (1986). On the usage of refined linear models for determining N-way classification designs which are optimal for comparing test treatments with a standard treatment. Ann. Inst. Statist. Math. 38, 569-581. Jacroux, M. (1987a). On the determination and construction of MV-optimal block designs for comparing test treatments with a standard treatment. J. Statist. Plann. Inference 15, 205-225. Jacroux, M. (1987b). Some MV-optimal block designs for comparing test treatments with a standard treatment. Sankhyd Set B 49, 239-261. Jacroux, M. (1989). The A-optimality of block designs for comparing test treatments with a control. J. Amer. Statist. Assoc. 84, 310-317. Jacroux, M. (1992). On comparing test treatments with a control using block designs having unequal sized blocks. Sankhya Ser. B 54, 324-345. Jacroux, M. (1993). On the construction of trend-resistant designs for comparing a set of test treatments with a set of controls. J. Amer. Statist. Assoc. 88, 1398-1403. Jacroux, M. and D. Majumdar (1989). Optimal block designs for comparing test treatments with a control when k > v. J. Statist. Plann. Inference 23, 381-396. Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In: J. Srivastava, ed., A Survey of Statistical Design and Linear Models. North-Holland, Amsterdam, 333-353. Kim, K. and J. Stufken (1995). On optimal block designs for comparing a standard treatment to test treatments. Utilitas Math. 47, 211-224. Koch, G. G., I. A. Amara, B. W. Brown, T. Colton and D. B. Gillings (1989). A two-period crossover design for the comparison ef two active treatments and placebo. Statist. Med. 8, 487-504. Krishnaiah, P. R. and J. V. Armitage (1966). Tables for multivariate t-distribution. Sankhyd Ser. B 28, 31-56. Kunert, J. (1983). Optimal design and refinement of the linear model with applications to repeated measurements designs. Ann. Statist. 11, 247-257. Lindley, D. V. and A. E M. Smith (1972). Bayes estimates for the linear model (with discussion). J. Roy. Statist. Soc. Ser. B 34, 1-42. Magda, G. M. (1980). Circular balanced repeated measurements designs. Comm. Statist. Theory Methods 9, 1901-1918. Majumdar, D. (1986). Optimal designs for comparisons between two sets of treatments. J. Statist. Plann. Inference 14, 359-372.

1052

D. Majumdar

Majumdar, D. (1988a). Optimal repeated measurements designs for comparing test treatments with a control. Comm. Statist. Theory Methods 17, 3687-3703. Majumdar, D. (1988b). Optimal block designs for comparing new treatments with a standard treatment. In: Y. Dodge, V. V. Fedorov and H. P. Wynn, eds., Optimal Design and Analysis of Experiments. NorthHolland, Amsterdam, 15-27. Majumdar, D. (1992). Optimal designs for comparing test treatments with a control utilizing prior information. Ann. Statist. 20, 216-237. Majumdar, D. (1996). On admissibility and optimality of treatment-control designs. Ann. Statist., to appear. Majumdar, D. and S. Kageyama (1990). Resistant BTIB designs. Comm. Statist. Theory Methods 19, 2145-2158. Majumdar, D. and W. I. Notz (1983). Optimal incomplete block designs for comparing treatments with a control. Ann. Statist. 11, 258-266. Majumdar, D. and A. C. Tamhane (1996). Row-column designs for comparing treatments with a control. J. Statist. Plann. Inference 49, 387-400. Miller, R. G. (i966). Simultaneous Statistical Inference. McGraw Hilll New York. Naik, U. D. (1975). Some selection rules for comparing p processes with a standard. Comm. Statist. Theory Methods 4, 519-535. Notz, W. I. (1985). Optimal designs for treatment-control comparisons in the presence of two-way heterogeneity. J. Statist. Plann. Inference 12, 61-73. Notz, W. I. and A. C. Tamhane (1983). Incomplete block (BTIB) designs for comparing treatments with a control: Minimal complete sets of generator designs for k = 3, p = 3(1)10. Comm. Statist. Theory Methods 12, 1391-1412. Owen, R. J. (1970). The optimal design of a two-factor experiment using prior information. Ann. Math. Statist. 41, 1917-1934. Pearce, S. C. (1953). Field experiments with fruit trees and other perennial plants. Commonwealth Agric. Bur., Famham Royal, Bucks, England. Tech. Comm. 23. Pearce, S. C. (1960). Supplemented balance. Biometrika 47, 263-271. Pearce, S. C. (1963). The use and classification of non-orthogonal designs (with discussion). J. Roy. Statist. Soc. Ser. B 126, 353-377. Pearce, S. C. (1983). The Agricultural Field Experiment: A Statistical Examination of Theory and Practice. Wiley, New York. Pesek, J. (1974). The efficiency of controls in balanced incomplete block designs. Biometrische Z. 16, 2i-26. Pigeon, J. G. (1984). Residual effects for comparing treatments with a control. Ph.D. Dissertation, Temple University. Pigeon, J. G. and D. Raghavarao (1987). Crossover designs for comparing treatments with a control. Biometrika 74, 321-328. Roy, J. (1958). On efficiency factor of block designs. Sankhy(~ 19, 181-188. Smith, A. F. M. and I. Verdinelli (1980). A note on Bayes designs for inference using a hierarchical linear model. Biometrika 67, 613-619. Spurrier, J. D. (1992). Optimal designs for comparing the variances of several treatments with that of a standard. Technometrics 34, 332-339. Spurrier, J. D. and D. Edwards (1986). An asymptotically optimal subclass of balanced treatment incomplete block designs for comparisons with a control. Biometrika 73, 191-199. Spurrier, J. D. and A. Nizam (1990). Sample size allocation for simultaneous inference in comparison with control experiments. J. Amer. Statist. Assoc. 85, 181-186. Stufken, J. (1986). On-optimal and highly efficient designs for comparing test treatments with a control. Ph.D. Dissertation, Univ. Illinois at Chicago, Chicago, IL. Stufken, J. (1987). A-optimal block designs for comparing test treatments with a control. Ann. Statist. 15, 1629-1638. Stufken, J. (1988). On bounds for the efficiency of block designs for comparing test treatments with a control. J. Statist. Plann. Inference 19, 361-372. Stufken, J. (1991a). On group divisible treatment designs for comparing test treatments with a standard treatment in blocks of size 3. J. Statist. Plann. Inference 28, 205-211.

Optimal and efficient treatment-control designs

1053

Stufken, J. (1991b). Bayes A-optimal and efficient block designs for comparing test treatments with a standard treatment. Comm. Statist. Theory Methods 20, 3849-3862. Stufken, J. (1996). Optimal crossover design. In: Handbook of Statistics, this volume, Chapter 3. Stufken, J. and K. Kim (1992). Optimal group divisible treatment designs for comparing a standard treatment with test treatments. Utilitas Math. 41, 211-227. Ting, C.-R and W. I. Notz (1987). Optimal row-colurml designs for treatment control comparisons. Tech. Report, Ohio State University. Ting, C.-E and W. I. Notz (1988). A-optimal complete block designs for treatment-control comparisons. In: Y. Dodge, V. V. Fedorov and H. R Wynn, eds., Optimal Design and Analysis of Experiments, NorthHolland, Amsterdam, 29-37. Ting, C.-E and Y.-Y. Yang (1994). Bayes A-optimal designs for comparing test treatments with a control. Tech. Report, National Chengchi Univ., Taiwan. Toman, B. and W. I. Notz (1991). Bayesian optimal experimental design for treatment-control comparisons in the presence of two-way heterogeneity. Z Statist. Plann. Inference 27, 51-63. Ture, T. E. (1982). On the construction and optimality of balanced treatment incomplete block designs. Ph.D. Dissertation, Univ. of California, Berkeley, CA. Ture, T. E. (1985). A-optimal balanced treatment incomplete block designs for multiple comparisons with the control. Bull. lnternat. Statist. Inst.; Proe. 45th Session, 51-1, 7.2-1-7.2-17. Ture, T. E. (1994). Optimal row-column designs for multiple comparisons with a control: A complete catalog. Technometrics 36, 292-299. Verdinelli, I. (1983). Computing Bayes D- and A-optimal block designs for a two-way model. The Statistician 32, 161-167.