Kunert [Ann. Statist. 12 (1984) 1006-1017] proved that, in the class of repeated measurement designs based on t treatments, p = t periods and n = λt experimental units, a balanced uniform design is universally optimal
for direct treatment effects if t ≥ 3 and λ = 1, or if t ≥ 6 and λ = 2. This result is generalized to t ≥ 3 as long as λ ≤ (t −1)/2.Primarily sponsored by NSF Grant DMS-01-03727, National Cancer Institute Grant P01-CA48112-08 and NIH Grant P50-AT00155 ( jointly supported by the National Center for Complementary and Alternative Medicine, the Office of Dietary Supplements, the Office for Research on Women's Health, and the National Institute of General Medicine). The contents are solely the
responsibility of the authors and do not necessarily represent the official views of NIH

Hedayat, A.

Yang, Min, 1970 Oct. 28-

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The Annals of Statistics2003, Vol. 31, No. 3, 978–983© Institute of Mathematical Statistics, 2003UNIVERSAL OPTIMALITY OF BALANCED UNIFORMCROSSOVER DESIGNS1BY A. S. HEDAYAT AND MIN YANGUniversity of Illinois, Chicago and University of Nebraska-LincolnKunert [Ann. Statist. 12 (1984) 1006–1017] proved that, in the classof repeated measurement designs based on t treatments, p = t periods andn = λt experimental units, a balanced uniform design is universally optimalfor direct treatment effects if t ≥ 3 and λ = 1, or if t ≥ 6 and λ = 2. Thisresult is generalized to t ≥ 3 as long as λ ≤ (t − 1)/2.1. Introduction. Repeated measurements designs under the name “crossoverdesigns” have been used in diverse areas of scientific research for many years.A prominent example is the class of studies associated with phase I and phase IIclinical trials, in which patients are randomized to sequences of treatments withthe intention of studying differences between individual treatments or subsets oftreatments.The notation RM(t, n,p) designates a repeated measurements design based onn experimental units, each being used for p periods, to test and evaluate the effectsof t treatments. The class of all such designs is designated by t,n,p . Identifyingand constructing optimal and efficient designs in t,n,p , or in a selected subsetof t,n,p, was initiated by Hedayat and Afsarinejad (1975, 1978). Since then, manyexciting results in this area have been obtained by other researchers, includingCheng and Wu (1980), Kunert (1983, 1984), Stufken (1991), Hedayat and Zhao(1990), Carrière and Reinsel (1993), Matthews (1994) and Kushner (1998). Werefer the readers to the excellent expository review paper by Stufken (1996) foradditional references.Under the traditional model (see Section 2), Kunert (1984), generalizing a resultof Hedayat and Afsarinejad (1978), proved that, when p = t ≥ 3 and n = t thena balanced uniform design in t,n,p is universally optimal for direct treatmenteffects. Unfortunately, a balanced uniform design can lose its universal optimalitywhen n is relatively large compared to t . Counterexamples can be found in Kunert(1984) and Stufken (1991). A natural and intriguing question is: For given tReceived July 2001; revised April 2002.1Primarily sponsored by NSF Grant DMS-01-03727, National Cancer Institute GrantP01-CA48112-08 and NIH Grant P50-AT00155 ( jointly supported by the National Center for Com-plementary and Alternative Medicine, the Office of Dietary Supplements, the Office for Researchon Women’s Health, and the National Institute of General Medicine). The contents are solely theresponsibility of the authors and do not necessarily represent the official views of NIH.AMS 2000 subject classifications. Primary 62K05; secondary 62K10.Key words and phrases. Balanced design, crossover design, carryover effect, repeated measure-ments.978BALANCED UNIFORM CROSSOVER DESIGNS 979and p = t , how far can we increase n without losing the universal optimalityof balanced uniform designs? For studying this question, we always assume thatn = λt , for integral λ, since this is a necessary condition for a balanced uniformdesign.Higham (1998) proved that when t is a composite number, the class t,t,tcontains a balanced uniform design. However, when t is a prime number the classmay lack a balanced uniform design. For example, 3,3,3, 5,5,5 and 7,7,7 donot contain a balanced uniform design. When t,t,t does not contain a balanceduniform design, it is most unlikely that it contains a universally optimal design. Ifn = 2t , another result of Kunert (1984) states that if t ≥ 6 then a balanced uniformdesign in t,2t,t is universally optimal. It is known that this class contains balanceduniform designs [see, e.g., Stufken (1996)]. Surprisingly, it is not known whetherbalanced uniform designs in 3,6,3 and 4,8,4 are universally optimal, althoughStreet, Eccleston and Wilson (1990) showed, by a computer search, that a balanceduniform design in 3,6,3 is A-optimal.The main purpose of this paper is to show that a balanced uniform designin t,λt,t retains its universal optimality as long as λ ≤ (t − 1)/2. Note that forλ = 1, our result is that of Kunert (1984) and for λ = 2, our result extends theresult of Kunert (1984) to t ≥ 5.2. Response model. The model we assume throughout this paper is thetraditional homoscedastic, additive, fixed effects model, which in the notation ofHedayat and Afsarinejad (1975) isYdku = µ + ζk + ηu + τd(k,u) + ρd(k−1,u) + eku, 1 ≤ k ≤ p, 1 ≤ u ≤ n,(1)where Ydku denotes the response from unit u in period k to which treatment d(k,u)is assigned. In this model, µ is the general mean, ζk is the effect due to period k,ηu is the effect due to unit u, τd(k,u) is the direct treatment effect, ρd(k−1,u) is thecarryover (or residual) effect of treatment d(k − 1, u) on the response observedon unit u in period k (by convention, ρd(0,u) = 0), and the eku’s are independentlynormally distributed errors with mean 0 and variance σ 2.In matrix notation, we can write model (1) asYd = µ1 + Pζ + Uη + Tdτd + Fdρd + e,(2)where Yd = (Yd11, Yd21, . . . , Ydpn)′, ζ = (ζ1, . . . , ζp)′, η = (η1, . . . , ηn)′, τd =(τ1, . . . , τt )′, ρd = (ρ1, . . . , ρt )′, e = (e11, e21, . . . , epn)′, P = 1n ⊗ Ip, U =In ⊗ 1p, Td = (T ′d1, . . . , T ′dn)′ and Fd = (F ′d1, . . . ,F ′dn)′. Here Tdu stands forthe p × t period-treatment incidence matrix for subject u under design d andFdu = LTdu with the p × p matrix L defined as0 0 · · · 0 01 0 · · · 0 00 1 · · · 0 0....... . .......0 0 · · · 1 0 .980 A. S. HEDAYAT AND M. YANGThe information matrix for direct effects, Cd , may then be written asCd = T ′dpr⊥(P |U |Fd)Td,(3)where pr⊥(X) denotes the orthogonal projection matrix I − X(X′X)−X′.As in Cheng and Wu (1980), the notations ndiu and n˜diu denote, respectively,the number of times that treatment i is assigned to unit u and the number of timesthis happens in the first p−1 periods. In the whole design, the quantities ldik , mdij ,rdi and r˜di are, respectively, the number of times that treatment i is assigned toperiod k, the number of times treatment i is immediately preceded by treatment j ,the total replication of treatment i and the total replication of treatment i limitedto the first p − 1 periods. Let z be the sum of all positive xdiu = ndiu − 1. We alsoassociate with unit u the integer n∗d(u) = ndxu if treatment x is assigned to u in thelast period.3. Universally optimal designs for direct treatment effects. From the toolintroduced by Kiefer (1975), a design d in t,n,p is universally optimal, if the traceof Cd is maximal and if in addition Cd is completely symmetric. The main purposeof this paper is to show that a balanced uniform design d∗ in t,λt,t is universallyoptimal when λ ≤ (t − 1)/2. Before we present a proof of this result we needthe following lemma which can be concluded from inequalities (5.6) and (5.7) inKushner (1997).LEMMA 1. For any design d ∈ t,n,p , we have the following inequality:Tr(Cd) ≤ q11(d) − q212(d)q22(d).Here,q11(d) = Tr (BtT ′dpr⊥(U)TdBt ),q12(d) = Tr (BtT ′dpr⊥(U)FdBt),q22(d) = Tr (BtF ′dpr⊥(U)FdBt ),where Bt = I − 1t Jt .We shall now present our main result.THEOREM 1. Assume that t = p > 2, λ ≤ (t − 1)/2. A balanced uniformdesign d∗ ∈ t,λt,t is universally optimal.PROOF. It is easy to see that Cd∗ is completely symmetric. Therefore, if weshow that Tr(Cd) ≤ Tr(Cd∗) for each d in t,λt,t , then the claim is established.From Cheng and Wu (1980), we know that Tr(Cd∗) = n(t − 1) − n(t−1)t2−t−1 . WhenBALANCED UNIFORM CROSSOVER DESIGNS 981z ≥ n, by Proposition 4.3 in Kunert (1984), we have Tr(Cd) ≤ Tr(Cd∗). Whenz < n, if we can show thatTr(Cd) ≤ n(t − 1) − n(t − 1)t2 − t − 1 ,then we have established our result.It can be shown that for any design d ∈ t,n,p ,q11(d) = np − 1pn∑u=1t∑i=1n2diu,q12(d) =t∑i=1mdii − 1pn∑u=1t∑i=1ndiun˜diu,q22(d) = n(p − 1)(1 − 1tp)− 1pn∑u=1t∑i=1n˜2diu.Next, for p = t , we will find the maximum value of q11(d), the minimum valueof q212(d), and the maximum value of q22(d) for a given value z ∈ [0, n).We notice that since the sum of all positive xdiu = ndiu − 1 is z, and∑nu=1∑ti=1 ndiu = nt , so the sum of all negative xdiu = ndiu − 1 is −z, whichmeans that z of the ndiu’s are 0 and the remaining ndiu’s must be greater than 0.Thus,∑nu=1∑ti=1 n2diu is equivalent to∑nt−zj=1 a2j subject to∑nt−zj=1 aj = nt , whereaj is a positive integer, j = 1, . . . , nt −z. It can be verified that the minimum valueof∑nt−zj=1 a2j is nt + 2z. Thus, q11(d) ≤ n(t − 1) − 2z/t .For q212(d), we notice that∑nu=1∑ti=1 ndiun˜diu =∑nu=1∑ti=1 n2diu −∑nu=1 n∗d(u). Since z of the ndiu’s are 0, therefore∑nu=1∑ti=1 n2diu −∑nu=1 n∗d(u)is equivalent to∑nt−zj=1 a2j −∑nj=1 aj subject to∑nt−zj=1 aj = nt , where 1 ≤ aj isan integer, j = 1, . . . , nt − z. We claim that the minimum value of ∑nt−zj=1 a2j −∑nj=1 aj is reached when aj is either 1 or 2, j = 1, . . . , n, and the remainingaj ’s are all 1. Otherwise, there are only two competing alternatives: (1) Supposesome of aj ’s are not 1 when j = n + 1, . . . , nt − z, say, an+1 > 1. Then one ormore of aj ’s must be 1, j = 1, . . . , n, say, a1 = 1, because ∑nt−zj=1 aj = nt . Byexchanging the values of an+1 and a1 and keeping the others unchanged we canobtain a smaller value for∑nt−zj=1 a2j −∑nj=1 aj . (2) Suppose that all aj ’s are 1,j = n + 1, . . . , nt − z, and there exists an aj which is not 1 or 2, j = 1, . . . , n.Without loss of generality, we assume that a1 = 1 and a2 = δ > 2. By changinga1 to 2 and a2 to δ − 1, and keeping the remaining ai’s unchanged, it can be eas-ily verified that the latter case produces a smaller value for∑nt−zj=1 a2j −∑nj=1 aj .So, the minimum value of∑nt−zj=1 a2j −∑nj=1 aj is nt − n + z. On the other hand,∑ti=1 mdii ≤ z. So, 1t∑nu=1∑ti=1 ndiun˜diu −∑ti=1 mdii ≥ (t − 1)(n − z)/t > 0,consequently, q212(d) ≥ (t − 1)2(n − z)2/t2.982 A. S. HEDAYAT AND M. YANGSincen∑u=1t∑i=1n˜diu = n(t − 1),at least n of the n˜diu are 0. Thus,∑nu=1∑ti=1 n˜2diu is equivalent to∑nt−nj=1 a2j subjectto∑nt−nj=1 aj = nt − n, where aj is a nonnegative integer, j = 1, . . . , nt − n. Theminimum value of∑nt−tj=1 a2j is nt − n. So q22(d) ≤ n(t − 1)(1 − 1/t − 1/t2).Therefore, by Lemma 1, we haveTr(Cd) ≤ q11(d) − q212(d)q22(d)≤ n(t − 1) − 2zt− (t − 1)2(n − z)2/t2n(t − 1)(1 − 1/t − 1/t2)= n(t − 1) − 2zt− (t − 1)(n − z)2n(t2 − t − 1) .(4)The right-hand side of (4) can be maximized when z = λt−1 , but notice thatλ ≤ (t − 1)/2 and z must be nonnegative integers, so the maximum value of theright-hand side of (4) is n(t − 1) − n(t−1)t2−t−1 . Therefore we have established thetheorem. Acknowledgments. We greatefully acknowledge numerous constructive com-ments that we received from the Associate Editor and the two referees. Their com-ments helped to make the presentation more concise and clear.REFERENCESCARRIÈRE, K. C. and REINSEL, G. C. (1993). Optimal two-period repeated measurement designswith two or more treatments. Biometrika 80 924–929.CHENG, C.-S. and WU, C.-F. (1980). Balanced repeated measurement designs. Ann. Statist. 81272–1283. [Corrigendum. Ann. Statist. 11 (1983) 349.]HEDAYAT, A. and AFSARINEJAD, K. (1975). Repeated measurements designs. I. In A Survey ofStatistical Design and Linear Models (J. N. Srivastava, ed.) 229–242. North-Holland,Amsterdam.HEDAYAT, A. and AFSARINEJAD, K. (1978). Repeated measurements designs. II. Ann. Statist. 6619–628.HEDAYAT, A. and ZHAO, W. (1990). Optimal two-period repeated measurements designs. Ann.Statist. 18 1805–1816. [Corrigendum. Ann. Statist. 20 (1992) 619.]HIGHAM, J. (1998). Row-complete Latin squares of every composite order exist. J. Combin. Des. 663–77.KIEFER, J. (1975). Construction and optimality of generalized Youden designs. In A Survey ofStatistical Design and Linear Models (J. N. Srivastava, ed.) 333–353. North-Holland,Amsterdam.KUNERT, J. (1983). Optimal design and refinement of the linear model with applications to repeatedmeasurements designs. Ann. Statist. 11 247–257.BALANCED UNIFORM CROSSOVER DESIGNS 983KUNERT, J. (1984). Optimality of balanced uniform repeated measurements designs. Ann. Statist.12 1006–1017.KUSHNER, H. B. (1997). Optimal repeated measurements designs: The linear optimality equations.Ann. Statist. 25 2328–2344. [Corrigendum. Ann. Statist. 26 (1998) 2081.]KUSHNER, H. B. (1998). Optimal and efficient repeated-measurements designs for uncorrelatedobservations. J. Amer. Statist. Assoc. 93 1176–1187.MATTHEWS, J. N. S. (1994). Modeling and optimality in the design of crossover studies for medicalapplications. J. Statist. Plann. Inference 42 89–108.STREET, D. J., ECCLESTON, J. A. and WILSON, W. H. (1990). Tables of small optimal repeatedmeasurements designs. Austral. J. Statist. 32 345–359.STUFKEN, J. (1991). Some families of optimal and efficient repeated measurements designs.J. Statist. Plann. Inference 27 75–83.STUFKEN, J. (1996). Optimal crossover designs. In Design and Analysis of Experiments. Handbookof Statistics 13 (S. Ghosh and C. R. Rao, eds.) 63–90. North-Holland, Amsterdam.DEPARTMENT OF MATHEMATICS,STATISTICS AND COMPUTER SCIENCEUNIVERSITY OF ILLINOIS851 SOUTH MORGAN STREETCHICAGO, ILLINOIS 60607-7045E-MAIL: hedayat@uic.eduDEPARTMENT OF MATHEMATICSAND STATISTICSUNIVERSITY OF NEBRASKA810 OLDFATHER HALLLINCOLN, NEBRASKA 68588-0323E-MAIL: myang@math.unl.edu

Universal Optimality in Balanced Uniform Crossover Design

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