3 Optimal crossover designs

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

"1

,31

Optimal Crossover Designs

John Stufken

1. Introduction As for many other designs, the use of crossover designs originated in the agricultural sciences. Crossover designs, also known as change-over or repeated measurements designs, were first used in animal feeding trials. Some early references and a small example, providing only part of the entire data set, are presented in Cochran and Cox (1957). Currently crossover designs have applications in many other sciences and research areas; examples are listed in Kershner and Federer (1981) and Afsarinejad (1990). The use of crossover designs in pharmaceutical studies and clinical trials receives now perhaps more attention than applications in any other area. For some examples and further discussion and references, the reader may want to consult the recent books by Jones and Kenward (1989), Ratkowsky et al. (1992) and Senn (1993). The principal idea associated with crossover designs is to use a number of available units for several measurements at different occasions. We will refer to these units as subjects, and in many applications they are humans, animals or plots of land. The different occasions at which the subjects are used are known as periods. We will assume that the main purpose of the experiment consists of the comparison of t treatments. Each subject will receive a treatment at each of p periods, and a relevant measurement is obtained for each subject in each period. A subject may receive a different treatment in each period, but treatments may also be repeated on a subject. If we denote the number of subjects by n, then we may think of a crossover design as a p x n matrix with entries from { 1 , . . . , t } , where the entry in position ( i , j ) denotes the treatment that subject j receives in the ith period. Corresponding to such a design we will also have an array of pn observable random variables yij, whose values will be determined by the measurements to be made. We will assume that these measurements are of a continuous nature. One possible motive for using a crossover design, as opposed to using each of p n subjects for one measurement, is that a crossover design requires fewer subjects for the same number of observations. This can obviously be an important consideration when subjects are scarce and when including a large number of subjects in the experiment can be prohibitively expensive. Another possible motive for using crossover designs is that these designs provide within subject information about treatment differences. In many applications the different subjects would exhibit large natural differences, and 63

64

J. Stufken

inferences concerning treatment comparisons based on between subject information (available if subject effects are assumed to be random effects) would require a much larger replication of the treatments in order to achieve the same precision as inferences based on within subject information. Indeed, designs are at times chosen based on the within subject information that they provide, and the between subject information is conveniently ignored. There are however also various potential problems with the use of crossover designs. Firstly, compared to using each subject only once, the duration of an experiment when using a crossover design may be considerably longer. Typically, it is therefore undesirable to have a large number of periods. Secondly, we may have to deal with carry-over effects. Measurements may not only be affected by the treatment assigned most recently to a subject, but could also be affected by lingering effects of treatments that were assigned to the same subject in one of the previous periods. Such lingering effects are called carry-over effects. One way to avoid or reduce the problem of carry-over effects is to use wash-out periods between periods in which measurements are made. The idea is that the effect of a previously given treatment can wear out during this wash-out period. Use of washout periods will however further increase the duration of the experiment, and may in some cases meet with ethical objections. (It is hard to deny a pain killer to a suffering patient just because he or she happens to be in a wash-out period!) A third potential problem with crossover designs is that an assumption of uncorrelated error terms may not always be reasonable. It may be more realistic to view the data as n short time series, one for each subject. Different error structures may affect recommendations concerning choice of design. We will return to this issue in Section 5. Fourthly, the use of crossover designs is of course limited to situations where a treatment does not essentially alter a subject. Crossover designs may be fine if the treatments alleviate a symptom temporarily, but not if the treatments provide a cure for the condition that a subject suffers from. This chapter will focus on the choice of design when a crossover design is to be used. Selected results concerning optimal design choices will be discussed. Of course, while the discussion will concentrate on statistical considerations for selecting a design, in any application there may be practical constraints that should be taken into account. The results concerning optimal designs should only be used as a guide to select good designs, or to avoid the selection of very poor designs. The literature on crossover designs is quite extensive. Many different models have been considered, and different models may result in different recommendations concerning design selection. This chapter represents therefore, inevitably, a selection of available results that is biased by the author's personal interest. There are however a number of other recent review papers and book chapters on crossover designs; the interested reader is referred to these sources for further details, additional references, and, possibly, bias in a different direction. Good sources, in addition to the aforementioned recent books, are Afsarinejad (1990), Barker et al. (1982), Bishop and Jones (1984), Matthews (1988, 1994), Shah and Sinha (1989, Chapter 6) and Street (1989).

Optimal crossover designs

65

2. Terminology and notation The approach throughout this chapter will be to assume a linear model for the observable random variables Yij. While different options are possible for such a model, once we settle for a model we will want to address the question of selecting an optimal design for inferences concerning the treatment effects or the carry-over effects. By 1~ and 0~ we will mean the a × 1 vectors of l's and O's, respectively• By f a and 0a x b we will mean the a × a identity matrix and the a x b matrix with all entries equal to 0, respectively• The basic linear model that we will use is the model

y~j = # + a~ + / 3 j + Td(i,j) +

i • {1,2,.•.,p},

"~d(i-l,j)

"~- gij,

j • {1,2,...,n},

where d(i, j ) stands for the treatment that is assigned to subject j in period i under design d. One may think of # as a general mean, of ai as an effect due to the ith period, of [39 as an effect due to the jth subject, of Ta(i,j) as a treatment effect due to treatment d ( i , j ) , and of "Yd(~-l,j) as a carry-over effect due to treatment d(i - 1,j). For the latter, we define 7d(O,j) = O. T h e eij are the non-observable random error terms. In matrix notation we will write our model as

(2.1)

= lZ!pn -~- XlOg q- X2fl__ q- Xd3"l" q- Xd4 ~ + ~,

where y = (Yll,Y21,...,Ypn) t, ~ : (Otl,...,Ogp) t, fl = (/31,...,fin) t, 7" : (~q,...,Tt)', 7_ = ( ' 7 1 , . . . , 7 t ) ' , e__ = ( c l l , e 2 1 , . . . , e p n ) ' , the p n × p and p n x n matrices X1 and X 2 are

!p0~ ...%

g Xl

=

•

=!~ ®Ip,

Xz =

g

:

:

"..

:

= In ® ! p ,

0. o . . . . ! ~

and the p n x t matrices X a 3 and Xa4, which are design dependent, are

Xd31 Xd32 Xd3 =

Xd41 Xd42 ,

Xd3n

Xd4 =

Xd4n

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J. Stufken

where Xd3j stands for the p × t period-treatment incidence matrix for subject j under design d and where Xd4j = L X a 3 j with the p × p matrix L defined as 00...00 10...00 L=

0 1 ... : : ..

00 : :

00...

1 0

For the model in (2.1) we will assume that e follows a multivariate normal distribution with mean 0pn and variance-covariance matrix cr2V, for an unknown scalar ~r2 and a pn × pn positive definite matrix V, to be specified later. Some comments concerning the model in (2.1) are in order. Firstly, all of the effects, including subject effects, are assumed to be fixed effects. While for many applications it may be quite reasonable to take the subject effects as random effects, with a relatively large between subject variability it may also in those cases be quite reasonable to make a design choice based on within subject information only. As explained earlier, in most applications the latter would be by far the more precise source of information. For some references and results on optimal design choice when between subject information is also considered, see Mukhopadhyay and Saha (1983), Shah and Sinha (1989) and Carri~re and Reinsel (1993). Secondly, while the model includes carry-over effects, it only allows for the possibility of first-order carry-over effects; only the treatment that was used in the period immediately preceding the current period is considered to have a possible lingering effect on a measurement in the current period. The model also reflects the assumption that there is no carry-over effect for measurements in the first period. Some have called this a non-circular model (see Shah and Sinha, 1989), in contrast to a circular model (Magda, 1980). A circular model would be a model where there is also a carry-over effect for measurements in the first period, as a result of treatments given to the subjects in a preperiod. But the use of preperiods is rather uncommon and unintelligible in many applications. For some results on optimal choice of design in the presence of a preperiod see Afsarinejad (1990). Thirdly, in some applications additional information on the subjects may be available through concomitant variables. Measurements could, for example, be taken on each subject at the beginning of the experiment or at the beginning of each period. Such socalled baseline measurements could be used in various ways when analyzing data from a crossover design. The model in (2.1) does not include use of such information. Use of baseline measurements can also be incorporated in the design selection problem; see, for example, Laska and Meisner (1985) and Carri~re and Reinsel (1992) for the basic ideas. Fourthly, the model in (2.1) does not include any interactions. It assumes, for example, that the period effects are the same for each of the subjects. It also assumes that the effect of a treatment is the same no matter which treatment contributes the carry-over

Optimal crossover designs

67

Table 2.1 Notation Symbol

Description

nduj nduj

The The first The The The The

ldul mduv rdu rdu

number of times that treatment u is assigned to subject j number of times that treatment u is assigned to subject j in the p - 1 periods number of times that treatment u is assigned to a subject in period i number of times that treatment u is immediately preceded by treatment v replication of treatment u throughout the design replication of treatment u restricted to the first p - 1 periods

effect. If these assumptions seem unreasonable, recommendations concerning design choices in the next sections may also be quite unreasonable. For some alternative models see, for example, Kershner and Federer (1981). By Tr(A) we will mean the trace of a square matrix A. We will say that an a x a matrix A is completely symmetric, abbreviated as c.s., if A = baIa + b21_~l~ for some constants b~, b2. Following Kunert (1985), for any matrix A we will write w ( A ) to denote the orthogonal projection matrix onto the column space of A, that is, w ( A ) = A ( A ~ A ) - A t. The following basic result on orthogonal projection matrices of partitioned matrices will be quite useful. LEMMA 2.1. For an a x b matrix X = [ Y Z] we have t h a t w ( X ) = w ( Y ) + w ( ( I a -

w(Y))Z). Our notation and terminology will to a large extent follow that of Cheng and Wu (1980). By Y2t,~,p we will denote the class of all crossover designs for t treatments, n subjects and p periods. We will say that a design d E Y2t,n,p is uniform on the periods if d assigns each treatment to n i t subjects in each period. A design d E Y2t,n,p is uniform on the subjects if d assigns each treatment p/~ times to each subject. A design is said to be uniform if it is uniform on the periods and uniform on the subjects. We will also make extensive use of the notation presented in Table 2.1. We also define lduo = O. Among the crossover designs with some desirable properties, as we will see, are those that are balanced or strongly balanced for carry-over effects. A crossover design is said to be balanced for carry-over effects, or balanced for brevity, if no treatment is immediately preceded by itself, and each treatment is immediately preceded by each of the other treatments equally often. A crossover design is called strongly balanced for carry-over effects, or just strongly balanced, if each treatment is immediately preceded by each of the treatments equally often. Orthogonal arrays of Type I form a useful class of designs when searching for optimal crossover designs if p ~< t. These arrays were introduced by Rao (1961). Formally, we define an orthogonal array of Type I and strength 2 as a p x n array with entries from { 1 , 2 , . . . , t} such that any 2 x n subarray contains all t ( t - 1) ordered 2-tuples without repetition equally often. We denote such an array by OAr(n, p, t, 2). Clearly, such an array can only exist if n is a multiple of t(t - 1).

J. Stu.Iken

68

Table 2.2 Examples of orthogonal arrays of Type I

12 21

23 323

13

12 121

1 2 4 3

1 3 2 4

1 4 3 2

2 1 3 4

2 3 4 1

2 4 1 3

3 1 4 2

3 2 1 4

3 4 2 1

4 1 2 3

4 2 3 1

4 3 1 2

Orthogonal arrays of Type I are closely related to orthogonal arrays. Orthogonal arrays were introduced by Rao (1947). An orthogonaI array of strength 2 is a p x n array with entries from { 1 , 2 , . . . ,t} such that any 2 x n subarray contains all t 2 ordered 2-tuples equally often. We denote such an array by OA(n, p, t, 2); a necessary condition for its existence is that n is a multiple of t 2. A forthcoming book by Hedayat, Sloane and Stufken (1996) contains an overview of existence and construction results for orthogonal arrays. For the purpose of this chapter, the following two well known results are sufficient. LEMMA 2.2. A necessary condition for the existence of an OA(t 2, p, t, 2) is that p <~

+ 1. Moreover, i f t is a prime power, such an orthogonal array exists for any such p. LEMMA 2.3. An orthogonal array of Type L O A i ( t ( t - 1), p, t, 2), can be constructed from an orthogonal array OA(t 2, p + 1, t, 2). Combining Lemmas 2.2 and 2.3 leads thus to the result that an OA~(t(t - 1),p, t, 2) exists for any p ~< t if t is a prime power. (See also Rao, 1961.) Examples of orthogonal arrays of Type I for p -- t = 2, p = t = 3 and p = t = 4, each time with n = t(t - 1), are presented in Table 2.2. If smaller values of p are desired, one can simply delete one or more rows from the given arrays; if larger values of n are desired one can simply juxtapose multiple copies of the given arrays. The reader who is less familiar with these combinatorial structures can be assured that Table 2.2 is essentially all that is needed for the remainder of this chapter. Choosing an optimal design is typically based on choosing a design with, in some sense, a large information matrix (see, for example, Shah and Sinha, 1989; or Silvey, 1980). The concept of universal optimality (Kiefer, 1975) has been considered extensively for crossover designs, and we will give a brief definition for completeness. Let 79~ be the class of all nonnegative definite a x a matrices with entries summing to 0 in each row. Let ~ be the class of all functions ¢: 79~ -+ ( - o o , oo] such that (i) ¢ ( Q C Q ' ) = ¢ ( C ) for all 17 E 79~ and all a x a permutation matrices Q (permutation invariance), (ii) ¢(bC) is non-increasing in the scalar b ~> 0, for all C E 7v~, and (iii) ¢(bC1 + (1 - b)C2) <~ b¢(C1) + (1 - b)¢(C2) for all b E (0, 1) and all e l , C2 E "~)a (convexity). If Cd and t~d denote the information matrices for ~- and 7 under design

d E X2t,n,p, respectively, then Ca, Ca E 79t. We say that d* E f2, where (2 C ~2t,n,p, is universally optimal for Z (3') in S2 if, for all ¢ E ~t, ¢(Ca) (¢(Ca)) is minimized over d E g? by d*. It is well-known that a design that is universally optimal in g? is also optimal in ~ under the commonly used A-, D- and E-optimality criteria.

Optimal crossover designs

69

Except for a minor extension, there is only one major tool to establish whether a design is universally optimal. We formulate this result (due to Kiefer, 1975) with reference to the i n.nformation matrix Cd for z, but with the understanding that a similar result holds for Cd and 7. THEOREM 2.1. A design d* E 32 is universally optimal f o r r in [2 if it maximizes Tr(Cd) over d E 32 and if Cd* is c.s. The result in Theorem 2.1 will be used extensively in the next sections. Preferably we would like to take 32 = 32t,n,p, but will not always be able to do this either because we are unable to show that a candidate design d* maximizes Tr(Cd) over that large a class of designs, or because we know that it only maximizes this trace over a subclass of 32~,n,p and not over the entire class.

3. O p t i m a l crossover designs w h e n errors are uncorrelated

3.1. Expressions f o r the information matrices The additional assumption in Section 3 for model (2.1) is that the error variancecovariance matrix a 2 V i s equal to a 2 I , , . If we define C d l l = Xld3(Ipn -Cd22 = X~4(Xpn -

w([X1 w([Xl

X2]))Xd3, X2]))Xd4,

Cdl2 = Xld3(Ipn -- w([X1 X2]))Xd4, Cd21 =

C~I2

then we have, using Lemma 2.1, that the information matrix Cd for r__is equal to C d = Xtd3(Ipn --

= x

w([XI X2

3(g. - w ( [ x l

- - X l d 3 W ( ( l v n --

Xd4]))Xd3

x2]))xd3

w([X1

X2]))Xd4)Xd3

= Calla -- C d 1 2 C £ 2 C ' e 2 1 .

It follows in a similar way that the information matrix Ca for "y is equal to Od : Cd22 -- Cd21CdllCdl2.

Many results on optimality of crossover designs are based on a careful study of the matrices CdU, Cd22 and Cd12, or on a more subtle application of Lemma 2.1. The elements of the matrices C d u , Cd22 and Cdl2 are readily expressed in terms of the quantities introduced in Section 2. Once more calling on Lemma 2.1, and writing Ipn --

w([Xl X2]) = Ivn - w(X2) - w ( ( I p .

- w(X2))X1)

= I v , - w(X2) - w((Xv, - W(lv,~))X,),

J. Stufken

70

it is easily seen that the diagonal elements for Cd11, Cd22 and G'a12 in position (u, u) are equal to n

1

~:

rdu -- -- E P j=l

_ _ Z(z~,

' _ r~,jp):,

1 P

(3.1)

n i=1

n

_

duj

p

1 ~2j_

r d . - p 5=1

1 ~ ( I d . ( i - 1 ) - Vd./P) 2

(3.2)

n ~=1

and 1

p

mduu -- P--j=l ndujnduj -- --n ~(Iduii=l

-- rdu/p)(ldu(i-1) -- ~du/P),

(3.3)

respectively. The off-diagonal elements in position (u, v) for these matrices are P

-- ~ ~'~1"=ndujndvj -- '~-~lE ( l d u i - -

_ 1~

~duj~dvj -- -~ ~ ( l d u ( i - 1 )

P j=l

rdu/p)(ldvi --

-- Ydu/P)(ldv(i-1) -- ~dv/P)

(3.4)

(3.5)

i=1

and 1

n

ma~v - - E

P j=l

1

P

nd~jnavj -- -- E ( l a ~ i

n

-- rd~/p)(ld~(i-1) -- Yd~/P),

(3.6)

i=1

respectively.

3.2. Optimal strongly balanced crossover designs

We will first focus on finding optimal designs for Z. Since Theorem 2.1 will be our major tool, candidate designs d* should have an information matrix Ca* that is c.s. One way to achieve this is by selecting a design for which each of the matrices Cd* I1, Ca.22 and Ca*lz is c.s. Since it is easily seen that all row sums in these matrices are 0, they will be c.s. if all off-diagonal elements are equal. Moreover, if d* also maximizes Tr(Ca11) over all d E f2t,n,p and if Cd*12 = 0, then d* is universally optimal for Z in f2t,,~,p. The simple strategy of searching for such designs d* works surprisingly well. The following results based on this strategy are due to Cheng and Wu (1980). THEOREM 3.1. A uniform strongly balanced crossover design is universally optimal f o r Z in ~t,n,p.

Optimal crossover designs

71

THEOREM 3.2. A strongly balanced crossover design that is uniform on the periods and uniform on the units when restricted to the first p - 1 periods is universally optimal for Z in f2t,n, p. The proofs for the two theorems are based on the same ideas. If d* is a design as in one of the previous theorems, then it is uniform on the periods. The expression in (3.6) reduces therefore to n

md*uv

P j~l'= nd*ujnd*vj.

It is easily verified that this becomes 0 under the conditions in each of Theorems 3.1 and 3.2, so that Cd.12 = Otxt. Moreover, using (3.1) we have that t

t

1 t

u=l

p

1~

u=l j=l

~-~(ldui -- rdu/P) 2.

u=l

i=l

The last term that is subtracted in this expression is 0 for a design that is uniform on the periods. The other term that is subtracted is, following arguments as in Cheng and Wu (1980), minimized over f2t,n,p by designs as described in the theorems. Thus, based on the strategy as outlined previously, the claims in the theorems follow. A strategy analogous to that used for finding universally optimal designs for Z can also be used for finding universally optimal designs for 7. Thus, a design d* that maximizes Tr(Cd22) over Y2t,n,p and for which Cd*12 = Ot×t and Cd-22 is c.s. is universally optimal for 7. This leads to the following results, also due to Cheng and Wu (1980). THEOREM 3.3. A uniform strongly balanced crossover design is universally optimal for 7 in g?t,n,p. THEOREM 3.4. A strongly balanced crossover design that is uniform on the periods and uniform on the units when restricted to the first p - 1 periods is universally optimal for 7_ in £2t,n,p. For the proofs, the argument that Cd,12 = Otxt for designs in Theorems 3.3 and 3.4 is similar as for Theorems 3.1 and 3.2. Further, using (3.2) we have that 1

-

Tr(gd*22) = E ~ ' d ~ -- P u=l

= n(p

-- n u=l

u=l

1 t n 1) - P E E n~uj

-

u=l j=l

1

t

i=1

1~ -

-n

~,,/(p(p-

u=l

P

u=l i=2

1))

1))

J. Stufken

72

The last term that is subtracted in this expression is 0 for a design that is uniform on the first p - 1 periods. The other two terms that are subtracted are, again based on arguments as in Cheng and Wu (1980), minimized over J2t,n,p by designs as described in the theorems. Observe that the designs in Theorems 3.1 and 3.2 are the same as those in Theorems 3.3 and 3.4, respectively. The previous theorems would be rather meaningless if crossover designs as described in the theorems would not exist. For the designs in Theorems 3.1 and 3.3 it is readily seen that necessary conditions for their existence are (i) p _= 0 (mod t) and p / t t> 2, (ii) n ~ 0 (mod t), and (iii) n(p - 1) ~ 0 (mod t2). Therefore, we need that n = A l t 2 and p = A2t for integers A1 ~> 1 and A2 ~> 2. These conditions are actually also sufficient. THEOREM 3.5. A uniform strongly balanced crossover design in g2t,,~,p exists whenever r~ = A l t 2 and p = A2t for integers AI /> 1 and A2 >>-2. PROOE It suffices to show that the designs exist for n = t2; for r~ = A l t 2 one can then simply juxtapose A1 copies of the design for n = t 2. Let A t be an OA(t 2, 3, t, 2) with entries from { 1 , 2 , . . . , t}. Such an orthogonal array exists for any t >~ 2, and can easily be obtained from a Latin square of order t. Let B t be the OA(t 2, 2, t, 2) obtained from A t by deleting the third row in At. For i E { 1 , 2 , . . . ,t - 1], let Ai = A t + i and B i = B t + i, where i is added to each element of At or B t modulo t. Let A and B be defined as the 3t x t z and 2t x t 2 arrays B1

A1

B2

A2 A =

B

=

Bt

At

With p = A2t, Az/> 2, write A2 = 331 + 232 for nonnegative integers 31 and ~2. The p × t 2 array

[ A I ...

A ~ B'

...

BI]1

consisting of 61 copies of A and 32 copies of B is a uniform strongly balanced crossover design in ~t,t2,p, As a small example of the construction in the previous proof, let t = 3, n = 9 and p = 9. Since A2 = 3, we take one copy of A to form the desired crossover design. For A3 we can take the orthogonal array 1 1 1 2 2 2 3 3 3]

I

1 2 3 1 2 3 1 2 3 .

1 2 3 2 3 1 3 1 2

Optimal crossoverdesigns

73

The desired uniform strongly balanced crossover design looks now as follows: 2 2 2 3 3 3 1 1 1 2 3 1 2 3 1 2 3 1 2 3 1 3 1 2 1 2 3 3 3 3 1 1 1 2 2 2 3 1 2 3 1 2 3 1 2 3 1 2 1 2 3 2 3 1 1 1 1 2 2 2 3 3 3 1 2 3 1 2 3 1 2 3 1 2 3 2 3 1 3 1 2 Necessary conditions for the existence of crossover designs as described in Theorems 3.2 and 3.4 are (i) p - 1 ~ 0 (mod t) and (ii) n - 0 (rood t). Cheng and Wu (1980) observed a simple way to construct some of these designs from uniform balanced designs. We will therefore return to the construction of these designs in Subsection 3.4, after a discussion on the existence of uniform balanced designs. []

3.3. Optimal and efficient crossover designs when p <<.t Theorems 3.1 through 3.4 all require that the number of periods is larger than the number of treatments. This may not be desirable or possible in many practical problems. The larger the number of periods, the longer the experiment will have to last. For p ~< t the knowledge about universally optimal designs for ~- in/2t,n,p is rather limited. This is in part because the strategy that resulted in Theorems 3.1 and 3.2 does not work for this case. When p ~< t and n ~ 0 (mod t) we would have to take nd~j E {0, 1} for all u and j in order to maximize Tr(Cdll); but this would also imply that m d ~ = 0 for all u, and hence, using (3.3), that Gd12 cannot be equal to 0t×t. A result on universally optimal designs for ~- in f2t,r~,p is however available for the case p = 2. Hedayat and Zhao (1990) prove the following result. THEOREM 3.6. Any crossover design d* E g2t,m2 with (i) rd*~ ---- 0 (modt) for all u c { 1 , 2 , . . . ,t} and (ii) md*uv = Fd.v/tfor all u,v E { 1 , 2 , . . . ,t} is universally

optimal for ~ in [2t,n,2. Theorem 3.6 implies in particular that a design that uses only treatment 1 in the first period and that is uniform on the second period is a universally optimal design for Z. (Of course, such a design would provide absolutely no information about %) As another consequence of Theorem 3.6, a design in the form of an OA(n, 2, t~2) is also universally optimal for __T.Designs as in Theorem 3.6 exist if and only if n - 0 (mod t). For more discussion on the case p = 2 see also Carri~re and Reinsel (1993); they consider a model in which subject effects are assumed to be random

74

J. Stu]ken

effects, and find that some of the designs in Theorem 3.6 are not very efficient under that model. Designs in the form of an orthogonal array, however, are still universally optimal for T under their model. For t = 2 and t = 3, in both cases with n = t z, universally optimal designs in the form of orthogonal arrays are: 1 1 2 2] 1212

and

[1 1 1 2 2 2 3 3 3]. 1 2 3 1 2 3 1 2 3

Stufken (1991) studies crossover designs for p ~< t and n = At(t - 1). The designs studied there exist if an OAi(t(t - 1), p, t, 2) exists (see the discussion after Lemmas 2.2 and 2.3). While these designs are only shown to be universally optimal for T__ in a small subclass of ff2t,n,p, all indications are that they are at least very efficient and possibly universally optimal for T__in the entire class f2t,n,p. T h e designs are constructed as follows. Let A be an OAi(t(t - 1),p, t, 2) and let the p × t(t - 1) array B be obtained from A by repeating period p - 1 in A as period p, thereby deleting the original pth period in A. With n = At(t - 1), compute )~/(t(p - 1)). Let 5 be the nearest integer to this ratio, or either one of the two nearest integers if there is a tie. Form a crossover design in Ot,n,p by juxtaposing )~ - (f copies of A and 5 copies of B . For more discussion and precise results concerning the optimality for T_of these designs see Stufken (1991). Observe that the designs suggested by Stufken (1991) are, as a result of Theorem 3.6, universally optimal for T_in [2t,mp if p = 2 and if A / t is an integer. The following result (Kunert, 1984; Stufken, 1991) identifies universally optimal designs for _7 when p ~< t. THEOREM 3.7. A crossover design f o r which the first p - 1 periods are in the f o r m of an OAi(n, p - 1, t, 2) and f o r which the pth period is identical to period p - 1 is universally optimal f o r 7__in f2t,n, p.

The proof for this result is analogous to those for Theorems 3.3 and 3.4. As previously explained, the strategy employed in these proofs fails for identifying universally optimal designs for Z when p ~< t, but, as shown by Theorem 3.7, remains successful for 7_.The existence of crossover designs as described in Theorem 3.7 has already been discussed briefly after Lemmas 2.2 and 2.3. A necessary condition for the existence of such crossover designs if p ~> 3, but certainly not a sufficient condition, is that n - 0 (mod t(t - 1)). If n = t(t - 1) the design exists if and only if there are p - 1 mutually orthogonal Latin squares of order t. If p = 2, all that is needed in Theorem 3.7 is that the first period is uniform on the treatments and that the second period is identical to the first. (See also Hedayat and Zhao, 1990.) While these designs for p --- 2 are universally optimal for 7, they provide no information at all for Z. Referring back to the discussion before Theorem 3.7, note that (A - 5)/5 is approximately equal to t(p - 1) - 1, so that the suggested efficient crossover designs for Z use many more copies of A than of B . Theorem 3.7 implies that designs that are universally optimal for 7_ only use copies of B .

Optimal crossover designs

75

3.4. Optimal and efficient balanced crossover designs

Uniform balanced designs form a class of designs that have been in use for a long time. Existence of these designs was already studied by Williams (1949). Optimality properties of these designs were first studied by Hedayat and Afsarinejad (1978), and later by Cheng and Wu (1980) and Kunert (1984). While uniform balanced designs are universally optimal for Z and -7 in subclasses of ~2t,n,p, generally this is not true in the entire class. The following results are due to Cheng and Wu (1980). THEOREM 3.8. A uniform balanced crossover design with n = )~lt and p = ~2t is universally optimal f o r "r in the subclass o f $2~,n,p that consists o f all designs d with (i) r a a ~ = Of o r all u, (ii) d is uniform on the units, and (iii) d is uniform on the last period. THEOREM 3.9. A uniform balanced crossover design with n = ~1 t and p = )~2t is universally optimal f o r 7 in the subclass o f £2t,n,v that consists o f all designs d with (i) mduu = 0 f o r all u, and (ii) if ~2 ~> 2, then ~du is the same f o r all u. Compared to Theorems 3.1 through 3.4 and 3.7, the additional difficulty in Theorems 3.8 and 3.9 is that for a uniform balanced crossover design the matrix Cdl 2 is not 0t×t. In applying Theorem 2.1, this makes it tremendously difficult to show that Tr(Ga) or Tr(Ca) are maximized by uniform balanced crossover designs. This leads to the restrictions for the subclasses as defined in Theorems 3.8 and 3.9. And, indeed, these restrictions are essential because the two traces are in general not maximized over the entire class J2t,,~,p by uniform balanced designs. For example, if p = t and n -- 0 (rood t(t - 1)), the universally optimal designs in Theorem 3.7 have a larger value for Tr(~'a) than uniform balanced designs. Under the same conditions, the efficient designs for Z suggested in Stufken (1991), as discussed in Subsection 3.3, have a larger value for Tr(Ca) than uniform balanced designs. Uniform balanced designs for p = t have, as an immediate consequence of the definition, all mdu~ values equal to 0. The designs that improve on uniform balanced designs suggest that this is not ideal. They also suggest that good designs for 7 tend to have larger values for maul, than do good designs for 7- (see Subsection 3.3). As a result, the efficiency of uniform balanced designs for p = t is smaller for 7 than it is for "r. For some further discussion see Kunert (1984), who also shows the following result. N

THEOREM 3.10. A uniform balanced crossover design with n = )qt and p = t is universally optimal f o r Z in ~2t,n,p if (i)/~1 = 1 and t >~ 3, or (ii) ~1 = 2 and t >~ 6. Thus for small values of n, some uniform balanced design are universally optimal for "r in ~t,n,p. This is also consistent with the recommendations in Stufken (1991). Necessary conditions for the existence of a uniform balanced design in g2t,n,p are (i) n = A l t for a positive integer A1, (ii) p = A2t for a positive integer A2, and (iii) A1 (A2 - 1) -~ 0 (rood t - 1). The first two conditions are needed for the uniformity, while the last condition is a consequence of n(p - 1) = 0 (mod t(t - 1)) which

J. Stuflazn

76

is required for the balance. We will first address the existence question of uniform balanced designs for )`1 = ),2 = 1 and will then return to the more general case. A uniform balanced crossover design in g2t,t,t exists for all even values of t, but only for some odd values. A construction for even values of t is due to Williams (1949). Take the first column, say a t, to be ( 1 , t , 2 , t - 1 , . . . , t / 2 - l , t / 2 + 2 , t / 2 , t / 2 + 1)', and define a t, 1 = 1 , 2 , . . . ,t - 1, by a t = a t + I, where l is added to every entry of a t modulo t. The t x t array

At=

[a 1 ...

at ]

(3.7)

is then the desired uniform balanced crossover design. For odd values of t, it is, among others, known that a uniform balanced design in J'-2t,t, t does not exist for t = 3, 5, 7 and does exist for t = 9. (See Afsarinejad, 1990.) The existence problem is thus much more difficult for odd t than for even t. What is known for every odd value of t is that a uniform balanced design exists in J2t,2t,t. To see this, let bt = (1, t, 2, t - 1 , . . . , ( t + 5 ) / 2 , ( t - 1)/2, ( t + 3 ) / 2 , ( t + 1)/2)'. For I = 1 , 2 , . . . , t - 1, let _bt = b t + l, and for l = 1 , . . . , t, let c t = - b t, where computations are again modulo t. The t x 2t array

B t = [_bl "

b,

""a]

(3.8)

is then the desired uniform balanced crossover design. More general results concerning the existence of uniform balanced crossover designs are formulated in the following theorem. THEOREM 3.11. The necessary conditions for the existence of a uniform balanced

crossover design in ~2t,n,p are also sufficient if t is even. For odd t, a uniform balanced crossover design exists in ~2t,n,p if in addition to the necessary conditions, n i t is even. PROOF. Let )`1 = n / t and )`2 = p / t be integers with ) , I ( A a - 1) = 0 (mod t - 1). I f t is even, let D = (d~j) be a )`2 x )`1 matrix with entries from { 1 , 2 , . . . , t} such that, when computed modulo t, among the )`1(A2 - 1) differences dij - d(i-l)j, i --- 2 , . . . , )~2, j = 1,...,)`1, the numbers 0 , 1 , . . . , t / 2 - 1 , t / 2 + 1 , . . . , t - 1 all appear equally often. (Clearly, a matrix D as required exists for all positive integers )`1, )`2 and even t as in this proof.) With A t as in (3.7), define AL = A t + l for 1 = 1 , 2 , . . . , t - 1, where the addition is modulo t. Define a p x n array A by

Ad,l A=

:

" " " Adl~,l "..

:

Adx2, "'" Ad:~2:~, Using the definition of Az and the properties of D , it is easily verified that A is a uniform balanced crossover design. If t is odd and )`1 is even, construct a ),2 × ),1/2 matrix D = (d~j) with entries from { 1 , 2 , . . . ,t} such that, when computed modulo t, among the ( ) q / 2 ) ( ) ` 2 - 1)

Optimal crossover designs

77

differences d~j - d ( i - 1 ) j , i = 2 , . . . ,A2, j = 1 , . . . , ) q / 2 , the ( t - 1)/2 numbers (t 4- 3)/2, (t + 7 ) / 2 , . . . , ( t - 3)/2, reduced modulo t where needed, all appear equally often. It is again obvious that such a matrix D exists. With B t as in (3.8), let Bz, 1 = 1 , 2 , . . . ,t - 1, be defined as B t 4- l, with addition again modulo t. The p x n array B defined by

B

=

Bd11

"""

:

"..

Bd~(:~/2) :

Bd~,21

"..

Bd),2(~,1/2)

is then a uniform balanced crossover design. Verification of this result is straightforward, and therefore omitted. As pointed out earlier, the condition that A1 is even when t is odd is not a necessary condition for the existence of uniform balanced crossover designs with n = A l t . We now return to the existence of strongly balanced crossover designs that are uniform on the periods and uniform on the units when restricted to the first p - 1 periods. Cheng and Wu (1980) observed that such strongly balanced designs can be constructed if p = t + 1 from a uniform balanced design with p = t by repeating the tth period in the balanced design as period t 4- 1 in the strongly balanced design. The following is a simple extension of their idea. [] THEOREM 3.12. Let n = Air and p = A2t 4- 1, where A1 and A2 are positive integers with AI(A2 - 1) --- 0 ( m o d t - 1). Then there exists a strongly balanced crossover design in [2t,n,p that is uniform on the periods and uniform on the units when restricted to the first p - 1 periods if (i) t is even, or (ii) t is odd and A1 is even. PROOE For even t, let A t be defined as in the proof of Theorem 3.11. Use A t and At~2 to form a A2t x Alt array as follows. For the first t rows juxtapose Aa copies of At. For the next t rows juxtapose A1 copies of At~2. Continue like this, alternating between A t and At~2, until A2t rows are obtained. Add one more row to this array, identical to row A2t. This gives the desired strongly balanced crossover design. I f t is odd and A1 is even, let B t be defined as in (3.8). We start by making a A2tx Alt array as follows. For the first t rows juxtapose Al/2 copies of Bt. For the next t rows do the same thing, but permute the columns within each copy of B t such that the first row of this juxtaposition is identical to the last row in the previous juxtaposition. Continue like this until Azt rows are obtained, each time, through permutations of columns within copies of Bt, making sure that periods It and It + 1, l = 1 , . . . , A2 - 1 are identical. Add one more period to this array, identical to period A2t. This gives the desired strongly balanced crossover design. []

4. O p t i m a l crossover designs w h e n errors are uncorrelated: The special case o f two treatments

While the previous section presents results on optimal crossover designs for uncorrelated errors and for general values of the number of treatments t, the special case

J. Stufl:en

78

t = 2 deserves some extra consideration. Firstly, it is quite common that the number of treatments to be compared in a crossover design is small, including t = 2. Secondly, while the results in the previous section for general t apply also for t = 2, the problem of finding universally optimal designs for t = 2 can be simplified considerably. The considerations in this section will also reveal that we have, typically, a choice among several optimal designs. This choice can be important when practical constraints may make one optimal design preferable over another. The basic ideas in this section appear in Matthews (1990); for our notation and presentation we will however heavily rely on the ideas and development in the previous section. The model assumptions in this section are the same as those in Section 3. The two treatments will be denoted by 1 and 2, and the 2 p possible p x 1 vectors with entries from { 1,2} represent the sequences of treatments that can be assigned to the subjects. A universally optimal design for z in f22,n,p is now simply one that minimizes the variance of the best linear unbiased estimator of T2 - ~-1 over all designs in ~2,n,p. An analogous interpretation holds for universally optimal designs for 7_. For a p × 1 treatment sequence T_T_we will say that 31v - T is its dual sequence; the dual sequence of a treatment sequence T__is thus obtained by permuting l's and 2's in T_T.We will call a design dual balanced if every sequence is used equally often as its dual. If a design is considered to be a probability measure on all possible sequences, then it is easily seen that among the optimal designs, whether for Z or for % there is one that is dual balanced (Matthews, 1987). Thus, there is no loss of generality by restricting attention to dual balanced designs. The strategy that we will employ to find universally optimal designs is analogous to that in Section 3. But because t = 2, the information matrices C d and C d are 2 x 2 matrices, and, since their column and row sums are 0, will automatically be c.s. Hence, to find a universally optimal design for Z we will search among the dual balanced designs for a design d* that maximizes Tr(Cd11) over all designs, and for which Ca. 12 : 02×2. Since we can restrict our attention to dual balanced designs, the expression in (3.1) reduces to

np 2

1 ~ n2 p duj,

which implies that Tr(Call) is maximized if and only if for each sequence in the design the difference in replication for the two treatments in the sequence is at most 1. Thus, for example, if p = 4 only the sequences that replicate treatment 1 twice should be included; if p = 5 only the sequences in which treatment 1 appears twice or thrice should be included. How often a sequence should be used is determined by the requirement that Cd* 12 = 0zx2. The expression in (3.3) reduces for a dual balanced design to l

n

?Tt'duu -- -- E

P j=l

N

ndujnduj.

(4.1)

Optimal crossover designs

79

Table 4.1 Pairs of treatment sequences for p = 4: Searching for optimal designs for r

vl

Pair 1

Pair 2

Pair 3

1 1 2 2

1 2 1 2

l 2 2 1

.5

2 2 1 I

2 1 2 1

-1.5

2 l 1 2

-.5

We only need to evaluate this expression for u = 1, and require that it reduces to 0. Since the design will be dual balanced, we will evaluate (4.1) for n = 2 for each of the pairs of dual sequences, only using those pairs with sequences that replicate treatment 1 as close to p/2 as possible. If there are s such pairs of dual sequences, and if Vl,. • •, vs denote the values in (4.1) for these s pairs, then a universally optimal design for _r is obtained by using the lth pair nft/2 times, where the fz's are nonnegative numbers that add to 1 and for which ~ t ftvz = 0. The latter condition is required in order that Cdl2 = 02×2.

Of course, nfz/2 will only give integer values for all 1 for certain values of n; but that is inevitable with the approach in this section. As an example, consider the case of p = 4. There are only three pairs of dual sequences for which treatment 1 is replicated twice in each sequence. These pairs, with the corresponding values for the vt's, are presented in Table 4.1. Since computation of the vz's is trivial, so is finding optimal designs. With fz, I = 1,2, 3, denoting the proportion of time that pair l is used, any design with ~ t fzvt = 0, or equivalently fl - 3f2 - f3 = 0, is optimal for z in ~2,n,4. ThUS, we can take f2 E [0, 1/4] and fl = 1/2 + f2, f3 = 1/2 - 2f2. A popular solution is the one with f2 = 0, and fl = f3 = 1/2, using pairs 1 and 3 equally often; but, for example, fl = 3/4, f2 = 1/4 and f3 = 0 is another solution that gives an optimal design for r. The optimal designs for r in/22,8,4 corresponding to these two solutions are as follows: 1 1 2 2 1 1 2 2

1 1 1 2 2 2 1 2

1 1 2 2 2 2 1 1

1 1 1 2 2 2 2 1

2 2 1 1 2 2 1 1

2 2 2 1 1 1 1 2

2 2 1 1 1 1 2 2

2 2 2 1 1 1 2 1

Finding optimal designs for "7 can proceed in a similar way. The only difference is that the set of pairs of dual sequences that can possibly be included in an optimal design will be different. The pairs that need to be considered now are those in which each sequence replicates treatment 1 in the first p - 1 periods as close to (p - 1)/2 times as possible. Using only such pairs will lead to a design that maximizes Tr(Ca22) ....

80

J. Stufiwn Table 4.2 Pairs of treatment sequences for p = 4: Searching for optimal designs for "/

v~

Pair 1

Pair 2

Pair 3

Pair 4

Pair 5

Pair 6

1 1

2 2

1 2

2 1

1 2

2 1

1 1

2 2

1 2

2 1

1 2

2 1

2 2

1 1

1 2

2 1

2 1

1 2

2 1

1 2

1 1

2 2

2 2

1 1

.5

-1.5

-.5

-.75

-.75

.25

Table 4.3 Pairs of sequences for optimal two-treatment designs, 4 <~ p <~ 6 p

Pairs

vz

Z or 7_

p

Pairs

4

1122

1/2 -3/2 -1/2 -3/4 -3/4 1/4

both both both '7 only 7_ only 7 only

6

111222 112122

0 1 -2 -1 0 -1 4/5 -6/5 -6/5 -1/5

both both both both both both zonly Z only Z only Z only

1212 1221

1121 l 211 1222

112212

112221 121122

121212 121221

11221 11222 12121

12122 12211 12212 11122 11212 12112 12221

122112 122121 122211 111221 112121

112211 121121 121211 122111 112222 121222 122122 122212

vt

z o r 7_

3/2 -1/2 -1/2 1/2 -1/2 -5/2 -3/2 -1/2 -3/2 1/2 1/3 -5/3 1/3 -5/3 -5/3 1/3 4/3 -2/3 -2/3 -2/3

both both both both both both both both both both "7 only "7 only 7. only 7_only 7_ only 7_only 7 only 7_ only ")' only '7 only

Other steps are the same as in the case of finding optimal designs for Z. The values of vt are again c o m p u t e d for each pair of sequences from (4.1). As an example, Table 4.2 presents the pairs of dual sequences that can be used w h e n searching for optimal designs for 7 for p = 4. With the n o n n e g a t i v e n u m b e r s f t , . . . , f6 associated with the six pairs as before, and with ~ ft = 1, any design with 2 f t - 6f2 - 2f3 - 3f4 - 3f5 + f6 = 0 is optimal for 7__.

If p is even, every pair of sequences that can be used to construct optimal dual balanced designs for Z can also be used in the construction of optimal designs for 3'. Thus, any optimal design for ~- constructed by the method in this section is also optimal for 7 if p is even. If p is odd, the roles for Z and 7 in the above statement

Optimal crossover designs

81

need to be reversed: any optimal design for 77_constructed by the method in this section is also optimal for __rif p is odd. As already alluded to in Section 3, the strategy used in this section does not work for r when p = 2. For p = 3 the unique optimal design for z and 7 uses only one pair of dual sequences, namely the one consisting of (1,2, 2)' and (2, 1, 1)'. Table 4.3 presents the sequences that can be used to construct optimal designs with t = 2 for 4 ~< p ~< 6. To save space, each pair is represented by only one of the sequences in the pair. The values for vt are given for each pair. Information whether a pair can be used in an optimal design for r__or for 7 is also provided in the table. Optimal designs are obtained by finding nonnegative ft's that add to 1 for which ~t fzvt = 0, only using eligible pairs of sequences. As an example, if we are interested in an optimal design for _r when p = 6, we can only use the 10 pairs for p = 6 that are labeled as 'both' in Table 4.3. With ( f l , . . . ,fl0) = (1/4, 1/4, 1/4,0, 1 / 4 , 0 , 0 , 0 , 0 , 0 ) or

( f l , . - . ,fl0) = ( 1 / 2 , 0 , 0 , 0 , 0 , 0 , 1/2,0,0,0), for example, we have two vectors of nonnegative constants that add to 1 and for which El fzvz = 0. Hence, designs corresponding to these vectors of constants are optimal for _r_ (and also for 7_). For n = 8 these designs arc 1 2 1 2 1 2 1 2

1 1 2 2 1 1 2 2

1 2 1 2 1 2 2 1

11222211

1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2

and

1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1

2 1 2 1 1 2 2 1

2 2 1 1 2 2 1 1

2 1 2 1 2 1 2 1

2 2 1 1 1 1 2 2

respectively.

5. O p t i m a l crossover designs w h e n errors are autocorrelated

While there are many ways in which the assumption of uncorrelated errors can be violated, in the context of crossover designs it is often reasonable to assume that error terms are only correlated if they correspond to measurements on the same subject. One way to model this is by assuming that the errors follow for each subject a first-order autoregressive process. The assumption of a first-order autoregressive error process for each subject has been made by various authors. It is not only intuitively appealing, but also just shy of being too complicated to address the question of optimal design choice. Among

82

J. Stufl~en

those that have considered such a structure for the error variance-covariance matrix are Azzalini and Giovagnoli (1987), Berenblut and Webb (1974), Bora (1984, 1985), Kunert (1985, 1991), Laska and Meisner (1985) and Matthews (1987). The discussion in this section is based on model (2.1). The results in this section are due to Matthews (1987), but we will again rely heavily on notation and concepts introduced in the previous sections. For some additional results under the same model, resulting in efficient crossover designs for t = 2 and larger values of p, see Kunert (1991). The first-order autoregressive error process for each of the subjects leads to a variance-covariance matrix a2V for the error vector g that is most conveniently expressed by using a Kronecker product: cr2V = ((r2In) @ W , where W = (wij) is t h e p x p m a t r i x defined by wij = pl~-Jl/(1-p2) and p E ( - 1 , 1) is a constant. With U as a p x p matrix such that U W U ' = Ip, consider the non-singular data transformation z = (In ® U)y. The model for _z induced by (2.1) is then the Ganss-Markov model

_z= ~(!,~ ® u i p ) + (l,~ ® u)~_ + (In ® u 4 ) ~ - + (z,~ ® u)x,~3z

+ (In ® U)Xd4 2 + (L ® V)~,

(5.1)

with Var(z__) : ~2Ip,~ as a consequence of the choice of U. The matrix U depends of course on p; while p will be an unknown in applications, to study its effect on optimal design choice we will for the time being assume that it is known. Let Ca(p) and Ca(P) denote the information matrices under model (5.1) for "r and 7, respectively. With

A:(I~@U')(Ip~-w([!~®U I~®UIp]))(I~®U) it is then easily seen that

Cd(p) = X~a3AXa3 - Xtd3AXd4(Xtd4AXd4)- Xtd4AXd3,

(5.2)

and ~a(p)

,

= Xd4AXd4

--

, , Xd4AXd3(Xd3AXd3 ) _ X d 3, A X d 4 .

(5.3)

By observing that

:

nU'U l I .±~l;U u (l~U'Ulp)In

][

! t I,~ ® l_pU

]

Optimalcrossoverdesigns

83

[ I ( W - l_plZp/(l_pU'UIp) 0pXn [in ®U In GUlp] I !v!~) 1 Ld(!y -L Onxp x~ ®!;u'

x

= I n ® . . U , Ulp.+

)

U!v!pU

!.~l__~ ® Ip

& u u&,

it is easily seen that

A = (I.~-lln!~) ® (U'(Ip-w(Ul_v))U).

(5.4)

If t = 2, and it is to this case that we will restrict our attention henceforth, we can, as in the previous section, assume without loss of generality that the competing I Therefore, designs are dual balanced. In that case ~ j = l Xd3j = ~n i p l 2.

n

j=l

- ~

[(u'(,~

_

Observe that

U'U

-

~(u&~))u)

~

'

(5.5)

=

= W -1, and that 1

-p

0

...

0

0

0

-p

1 + p2

_p

.. •

0

0

0

0

-p

0

0

0

0

0

0

...

l+p 2

-p

0

0

0

0

....

0

0

0

-..

1 + p2 . . .

W -1 p 0

1 + p2 _ p -p

1

F r o m (5.4) and (5.5) it follows now easily that

Xtd3AXd3 = Xtd3 (In • (Ut(Ip - zo(Ulp) )U) ) Xd3 = f~= X~3j(W-' -- W-%;~w-l'~ !tpW-11p ; Xd33"

(5.6)

J. Stuflwn

84 It follows in a similar way that

j=l

-

I~W -1 l_p

Xd43"

Furthermore, for a dual balanced design we have that ~ j = x L is as defined in Section 2. Using this, we obtain that

(5.7)

Xd4j = -~Llplz, where

Xtd4 [(ll_.nltn) @ (Ut(Ip-w(Ul_.p))U)] Xd4 =- (j=~lX~d4j) (gt(Ip-w(glp))g)

(~ld4jl V =1

n (

= ~ I•L'W-'LI_p-

(1;w-l-Lip)2 ~

-~_

_

/

t

j 1212.

With (5.4) this results in

Xtd4AXd4 "=Ej=l Xtd4j W-' --

1--;W-ll-p

Xdaj

n (1;LtW_ILlp_ (lpW-1L_lp)2~

gw-%

The matrices X~3AXd3, Xtd3AXd4and Xtd4AXd4a r e all 2 x 2 matrices with zero row and column sums. Hence it suffices to compute just one entry of each of these matrices to know the entire matrices. To this purpose, let x__jdenote the first column of Xa3j. For some jl, a subject who receives the dual of the sequence that subject j receives, we have that X-j, = _lp - X-j. Moreover, we have that the first column of Xa4j is LX-j. The contribution that these two subjects make to the entry in position (1,1) of the matrix X'd3AXa3 is thus equal to

~3w-lx-j

=

2

(x-tjW-l!p) 2 l'pW! llp + ( l p - -xj)'W-l(lp( (lp -- x-j)tW--1].p)2

X-}w-lxj

l p! W - I lp J

"

x_j)

(5.8)

If we use ft, as in Section 4, to denote the proportion of time that the lth pair of dual sequences is used, where fl's are nonnegative numbers that add to 1, and if we

Optimal crossover designs

85

use s to denote the number of such pairs (s = 2p-l), then we obtain from (5.8) that the entry in position (1, 1) of the matrix X~3AXa3 is given by

s

n~f,

(

(~_~w-~!~)~ xtlW-lxl

g !W -

1lp

] '

(5.9)

l=l

where we have made a slight change of notation by using x_l to denote the column that would appear as first column in a matrix X d 3 j if subject j would receive the sequence in pair l that starts with treatment 1; we will refer to this sequence as the first sequence in the pair. In a similar way we obtain the entries in position (1, 1) for the matrices X~d3AXd4 and X~d4AXd4 . These are

7

:

fl 2 z'lW-lLz_.l- (z-IW-~lP)(!;W-~Lz--t) !;w-%

-

x_IW-~L!~ +

(z~w -lip)(i;W -~L!p)] 1~W_11_p

j

(5.10)

and

n ( -~ 1 ; L ' W - I L l p + n~

fz

(lpW-lLlp)2~ - ~ ]

- ~_~L'W -~L(±~ -

~)

l=l

( I_' W - I Lz~)(!; W - l Z( I_p + !;W-'!p

_~))] ]'

(5.11)

respectively. Equations (5.9), (5.10) and (5.11) all depend on scalars of the form _z~W - l z 2 , for some vectors _z1 and _z2. These expressions are perhaps easiest evaluated by defining N as the p × p matrix with a 1 in positions (i,i) for i E { 2 , . . . , p - 1}, and a 0 elsewhere, and by observing that

W -I = Ip + p~N - pL - pL r. Using this expression, all the relevant scalars of the form z__~W-lz2 can be further evaluated and the results are presented in Table 5.1. The notation used in this table is as follows: By rt we denote the replication of treatment 1 for the first sequence in pair l; by r(i)t and r(i,i,)t we denote this same replication after deleting period i and periods i and i', respectively; by mz we denote the number of times that treatment 1 is immediately preceded by itself for the first sequence in pair l; by m(p)z we denote

J. Stufken

86

Table 5.1 Searching for optimal designs for t = 2 when errors are autocorrelated No.

Expression

Evaluation

1

!~w-l!p

(1 - p ) ( p - p ( p - 2 ) )

2 3

!;W-ILlp !;L'W-lL±p

(1 - o)2(p - 2) + a

4

l p W - 1 _zt

rt + p2r(1,p)l -- p(r(1)l + r(p)l )

5

l_pW-ILxz

(1 -- p)(r(p)t -- pr(p_l,p)l)

6

x~W-1Llp

r(1)/ q- p2r(1,p)l -- p(r(1,2)l + r(p)/)

7

~'lLtW-XLl_p

r(p)l -I- p2r(p_l,p) l -- p(r(1,p)l q- r(p_l,p)l )

8

~W-lxt

rl + p2r(l,p)l -- 2pint

(1 - p ) ( p -

1 - p(p-

2))

9

~l W - l L x z

ml + pam(p)l -- p(ml 2) + r(p)l)

10

~ILtW-1Lxl

r(p)/ -t- p21"(p_l,p) l -- 2pm(p)t

a similar quantity, but now counted after deleting period p; finally, by m} 2) we denote the number of periods i, i E { 1 , . . . ,p - 2}, such that treatment 1 appears in period i and period i + 2 in the first sequence of pair I. As an example of the computation, !~W-1Lz__z=

l~pLz_z + pZltpNLxt_

- pl_'pL' 2 z_l _ pl~pL,

= r(p)l + p 2 r ( p - l , p ) l -- p r ( p _ l , p ) l -- pr(p)l =

(1

-

-

Other expressions in Table 5.1 are evaluated in a similar way. For p = 3 we will demonstrate how the preceding computations can be used to obtain optimal designs. First evaluate, as a function of p, the expressions numbered as 1 through 3 in Table 5.1 for p = 3. These expressions do not depend on the choice of design. Then, for each possible pair of treatment sequences, or equivalently for each 3 x 1 vector x z with the first entry 1 and all other entries 0 or 1, evaluate the expressions numbered as 4 through 10 in Table 5.1. The results of this for p = 3 are given in Table 5.2, with each possible pair represented by the first sequence in the pair. The numbering in Table 5.2 corresponds to that in Table 5.1. With the values in Table 5.2, the expressions in (5.9), (5.10) and (5.11) can be evaluated, all as a function of p and f l , . . . , f4. If we call these three expressions v33 , v34 and v44, respectively, then we should maximize

V24 V33 -- _ _ V44

(5.12)

Optimal crossover designs

87

Table 5.2 Searching for optimal designs with t = 2 and p = 3 Pairs No. 1

2 3 4 5 6 7 8 9 l0

over fl,..-,

111

112

121

122

(1 - p ) ( 3 - p)

(1 - p ) ( 3 - p)

(1 - p ) ( 3 - p)

(1 - p ) ( 3 - p)

(1 (1 (1 (1 (l (1 (1 (1

(1 (1 (1 (1 (1 (1 (1 (1

(1 - p)(2 - p) (1 - p)2 + 1 2(1 - p) (1 - p)2 1 - 2p 1 + 02 _ p 2 --2p

(1 - p)(2 - p) (1 - p)2 _[_ I 1- p (1 - p)2 -p 1 + p2 _ p 1 -p

1 + p2

1 + p2

-

p)(2 - p) p)2 + 1 p)(3 - p) p)(2 - p) 0)(2 - p) p)2 + 1 p)(3 - p) p)(2 - p)

(1 -- p) 2 + 1

--

p)(2 - p) p)2 + 1 0)(2-- p) p)(2 - p) p)2 p)2 + 1 p)2 + 1 p)2

(I -- 0)2 + 1

f4 to o b t a i n p r o p o r t i o n s for an o p t i m a l d e s i g n for _r_, and

v44 - - ~J33

(5.13)

to o b t a i n p r o p o r t i o n s for an o p t i m a l d e s i g n for 7. T h e results in T a b l e s 5.1 and 5.2 and e q u a t i o n s (5.9), (5.10) and (5.11) g i v e that 2n ~)33 --- - 3--p

[k +f4+

(1

+p)f3],

n

/)34 -- 2(3 -- p) [--(1 -- P)f2 -- 3(1 + p ) f , -- 2 0 f 4 ] , and 72

V44 m m 2(3 -- p)

n

+ 3_-77 [(1 + p)(f, + f4)].

T h e e x p r e s s i o n s in (5.12) and (5.13) are then e q u a l to rz [4f2 + 4f4 + 4(1 + P)f3 2 ( 5 - - p) and

n[

2(3--p)

l+2(l+p)(f3+f4)-

((1 - P)f2 + 3(1 + P)f3 + 2 0 f 4 ) 2] 1 + 2(1 + P ) ( f 3 + f4) ]

((1-p)f2+3(1+p)f3+2pf4)

aT277f; a(57)Sf

2]

j,

r e s p e c t i v e l y . To o b t a i n o p t i m a l designs, for any g i v e n v a l u e o f p it is rather s i m p l e to m a x i m i z e the a b o v e q u a n t i t i e s o v e r f l , f2, f3 a n d f4, s u b j e c t to the f z ' s b e i n g

88

J. Stu3~en

nonnegative and adding to 1; in fact, explicit formulas for optimal ft's as a function of p can be found for this simple case of p = 3 (see Matthews, 1987). While the procedure described in this section works nicely for small values of p, it becomes quite cumbersome for larger values of p. See Kunert (1991) for efficient designs in that case. Matthews gives optimal values for the f t ' s as a function of p for the cases p = 3 and p = 4, both for ~- and for 7. One important final consideration is that we will typically not know what the correct value for p is. In such case, instead of choosing a design that is optimal for a particular value of p we may want to choose a design that is at least efficient over an interval that most likely contains the true value for p. The efficiency of a dual balanced design may be computed, for various values of p in such an interval, as the ratio of quantities as in (5.12) and (5.13), using the value for the design under consideration in the numerator and the optimal value in the denominator. Matthews (1987) shows that some of the commonly used designs for p = 3 and p = 4 have excellent efficiencies for T_ or for 3' over large intervals for p, but that some other designs can have extremely poor effi~encies.

6. Summary and discussion One of the early incentives for using uniform balanced and strongly balanced crossover designs was the relative simplicity of analysis of data obtained by using these designs, based on a model as in Section 3. While this is no longer a strong incentive in our era, Theorems 3.1 through 3.4 show that the classes of balanced and strongly balanced crossover designs remain appealing. Section 3 also identifies optimal and efficient designs for the case p ~< t based on orthogonal arrays of Type I. The case t = 2 has received special attention in the literature, for good reason, and results on optimal designs for this case under different assumptions for the error variance-covariance matrix are presented in Sections 4 and 5. The many recent review articles, books, and book chapters on crossover designs are a reflection of the increased attention that these designs have received since the late seventies. Research continues to flourish, and several interesting and challenging problems require additional attention. Following are a few thoughts on directions for further research. The results in the previous sections provide universally optimal designs for ~ or 7 for selected values of t, n and p. The severity of the constraints on t, n and p is in part a problem that is inextricably tied to exact design theory, but in part also due to the use of universal optimality as the optimality criterion. For many values of t, n and p a universally optimal design will simply not exist; less stringent optimality criteria could be considered for such cases. When theoretical results are harder to obtain due to the increased mathematical difficulties under such an approach, development of efficient algorithms or general guidelines for obtaining good designs under specified optimality criteria for arbitrary values of t, n and p would be quite useful. As alluded to previously, results from optimal design theory should be used as a guide towards obtaining good designs or to avoid use of extremely poor designs. It

Optimal crossover designs

89

would be foolish to insist on using optimal designs only; the specified model under which an optimal design for £ or_7 is optimal represents at best an approximation of the true relationship between the response variable and the independent variables. In the worst case, the model may be entirely misspecified, and the optimal design under this misspecified model may be hopelessly inefficient under a better model. It is therefore important to identify designs and classes of designs that are highly efficient under various models. Alternative models to the one used in Section 3 could include models with some interactions, with random subject effects, or with different error variancecovariance matrices. While there are some results pertaining to identification of designs that possess a certain robustness to model misspecification (see, for example, the discussions concerning p = 2 and t = 2 in Sections 3 and 5, respectively), this direction of research is still in its infancy. Extension of results for t = 2 as discussed in Sections 4 and 5 to other values of t is also desirable. Further, attention to designs for small values of p, possibly through additional considerations for p ~< t, is required. Some directions of research not addressed in this chapter include designs for special treatment structures (for example when one of the treatments is a control treatment or when the treatments possess a factorial structure) and optimal crossover designs for binary or discrete response variables. These are also areas that require further attention. Finally, a question of a more philosophical nature that has recently been raised is whether the models that were originally developed for use in areas of agricultural sciences are still adequate in the many different fields where they are used nowadays. Are the traditional model assumptions reasonable in all these fields, or are they merely used because nobody has seriously thought about more appropriate alternatives?

Acknowledgement Research for this chapter was partially supported by NSF grant DMS-9504882.

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