Supplementary materials for this article are available online. Please click the JASA link at http://pubs.amstat.org.
Optimal and Efficient Crossover Designs for Test-Control Study When Subject Effects Are Random A. S. H EDAYAT and Wei Z HENG We study crossover designs based on the criteria of A-optimality and MV-optimality under the model with random subject effects, for the purpose of comparing several test treatments with a standard control treatment. Optimal and efficient designs are proposed, and their efficiencies are also evaluated. A family of totally balanced test-control incomplete crossover designs based on a function of the ratio of the subject effect variance to the error variance are shown to be highly efficient and robust. The results have interesting connections with those in Hedayat and Yang (2005) and Hedayat, Stufken, and Yang (2006). The omitted proofs in the article are included in a supplemental material online. KEY WORDS: A-optimality; Fisher information matrix; Mixed-effects model; MV-optimality.
1. INTRODUCTION
The TBTCI designs were first proposed by Hedayat and Yang (2005) when they studied the A-optimality and MV-optimality for the model with fixed subject effects, that is, θ = ∞ in our setup. They proved that a certain type of TBTCI designs would be optimal in , a mildly restricted subclass of competing designs, and conjectured that does not exclude too many good designs so that the optimal designs in is still highly efficient in , the unrestricted class of designs. As for the question of how efficient are these designs, no investigation was carried out. In this article, we generalize their results and find optimal designs in for general models with random subject effects, that is, for general θ ≥ 0. We also characterize the optimal designs in when θ = 0, which corresponds to the model with no subject effects. Moreover, we give an explicit way to evaluate the efficiency of any design in without finding the optimal designs for any θ . Other selected contributions in this area include, but are not limited to, Hedayat and Afsarinejad (1975, 1978), Cheng and Wu (1980), Kunert (1984), Hedayat and Zhao (1990), Stufken (1991), Kushner (1997, 1998), Kunert and Martin (2000), Kunert and Stufken (2002), and Hedayat and Yang (2003, 2004) among many others. Stufken (1996) is an excellent survey article about crossover designs. However, the majority of the literature deal with universal optimality when subject effects are fixed. The article is organized as follows. Section 2 introduces the model based on which the study of designs is carried out, and further provides some notations which will be used in the subsequent sections. Section 3 introduces the optimality criteria and the corresponding optimal and efficient designs. Section 4 describes the method of deriving the lower bound of the A-efficiency of any design, which will be used to evaluate the efficiencies of the designs proposed in Section 3. Section 5 summarizes the results of this article by comparing them to relevant results in existing literatures and proposes some future directions of research. Section 6 gives the proofs for Theorems 1 and 2 in Section 3 through several lemmas, while further supports for the derivation of these lemmas will be postponed to the supplemental material.
To compare the effects of various treatments we could, if feasible, conduct a crossover design where each subject in the study receives each treatment in succession. Besides the budget consideration, one primary advantage of adopting crossover designs is that the treatment effects are no longer confounded with the subject effects, and hence the systematic bias in estimating the treatment effects could be reduced substantially. However, the period effects and carryover effects could come into the picture at the same time, which makes the design problem more complex. Our interest here is to find optimal or efficient designs for the purpose of estimating direct treatment effects rather than carryover effects. As in Hedayat, Stufken, and Yang (2006), we assume the subject effects to be random, which could be considered as in the middle status between the models with the fixed subject effects and no subject effects, in terms of the information matrix for the direct treatment effects. They studied the universal optimality, for which the totally balanced designs proposed therein exhibit the symmetry among all treatments. In this article, we focus on the A-optimality and MV-optimality, which aim to compare multiple test treatments to a control treatment. We investigate the high efficiency of totally balanced test-control incomplete (TBTCI) crossover designs, which exhibit the symmetry among test treatments. Interestingly, the class of TBTCI designs covers the totally balanced designs as a special case when the control treatment has the same replication as test treatments. The gain in generality for designs is due to our relaxation of the criterion from the universal optimality to one particular optimality either A-optimality or MV-optimality. Note that θ , the ratio between the variance of the subject effects and the variance of the error term, is still critical for the choice of designs. Sam A. Hedayat is Professor (E-mail:
[email protected]) and Wei Zheng is Ph.D. Candidate, Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, IL 60607. Research was primarily sponsored by the National Science Foundation grants DMS-0603761 and DMS-0904125, and the 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. We are very grateful to Professor Min Yang, Department of Statistics, University Missouri at Columbia, for his many valuable and insightful comments and suggestions while this article was under preparation. We also acknowledge with thanks the suggestions we received in improving the article by the Editor, the Associate Editor, and the two referees.
© 2010 American Statistical Association Journal of the American Statistical Association Accepted for publication, Theory and Methods DOI: 10.1198/jasa.2010.tm10134 1
2
Journal of the American Statistical Association, ???? 2010
2. MODEL ASSUMPTIONS AND NOTATIONS We denote by t+1,n,p the set of all of designs where n subjects are used in p ≥ 3 occasions, called periods, for the purpose of evaluating and studying one control treatment and t ≥ 2 test treatments. Hereafter, we shall designate the t test treatments by 1, 2, . . . , t and the control treatment by 0. For a continuous response Y, a plausible and useful linear model with random subject effects can be written as Ydku = μ + πk + ςu + τd(k,u) + ρd(k−1,u) + εku ,
(1)
iid
where we assume the subject effect ςu ∼ (0, σς2 ), the error term iid εku ∼ (0, σ 2 ), and {ςu } ⊥⊥ {εku }. The identical condition in the notation iid is not essential here, and hence could be removed. Here, Ydku denotes the response from subject u in period k to which treatment d(k, u) ∈ {0, 1, 2, . . . , t} was assigned by design d ∈ t+1,n,p , k = 1, . . . , p, and u = 1, . . . , n. Furthermore, μ is the general mean, πk is the kth period effect, τd(k,u) is the (direct) treatment effect of treatment d(k, u), and ρd(k−1,u) is the (first-order) carryover or residual effect of treatment d(k − 1, u) that subject u received in the previous period (by convention ρd(0,u) = 0). Finally, σς2 is the subject effects variance, and σ 2 is the error variance. Writing the np × 1 response vector as Yd = (Yd11 , Yd21 , . . . , Ydp1 , Yd12 , . . . , Ydpn ) , we have E(Yd ) = 1np μ + Pπ + Td τ + Fd ρ, var(Yd ) = σ 2 V,
(2)
where V = In ⊗ (Ip + θ Jp ) and θ = σς2 /σ 2 . Here π = (π1 , . . . , πp ) , τ = (τ0 , . . . , τt ) , ρ = (ρ0 , . . . , ρt ) , P = 1n ⊗ Ip , and Td and Fd denote the treatment/subject and carryover/subject incidence matrices. The notation represents the transpose operator; ⊗ represents the Kronecker product; 1s represents the column vector of length s with all its entries as 1; Is represents the s × s identity matrix; and Js = 1s 1s . The information matrix Cd for τ under model (1) can now be expressed as Cd = Td V−1/2 pr⊥ V−1/2 [1np |P|Fd ] V−1/2 Td , (3) where pr⊥ is a projection operator such that pr⊥ A = I − A(A A)− A for any matrix A. Throughout the article, for each design d, we adopt the notation ndiu , n˜ diu , ldik , mdij , rdi , r˜di , to denote the number of times that treatment i is assigned to subject u, the number of times this happens in the first p − 1 periods associated with subject u, the number of times treatment i is assigned to period k, the number of times treatment i is immediately preceded by treatment j, the total replication of treatment i in the n experimental subjects, and the total replication of treatment i limited to the first p − 1 periods in the design. Also, we would like to define the subclass of designs t+1,n,p = d ∈ t+1,n,p |ld0k = rd0 /p, k = 1, . . . , p and mdii = 0, i = 0, 1, . . . , t . Therefore, for any design in , the control treatment appears equally often in all periods and no treatment is allowed to be preceded by itself. Additionally, we have the following convention: For any two square matrices (e.g., A, B) of the same size, the inequality A ≤ B represents the loewner’s ordering
of the matrices; Tr(A) represents the trace of the matrix A; [x] represents the greatest integer that is not greater than x; Bs = pr⊥ 1s = Is − Js /s; {Si , i = 1, 2, . . . , t!} is the set of all t × t permutation matrices and 1 01×t S˜ i = , i = 1, . . . , t!. 0t×1 Si 3. OPTIMAL AND EFFICIENT DESIGNS In comparing two or more test treatments with a control, the most frequently used optimality criteria are A-optimality and MV-optimality. Definition 1. (i) In a class of competing designs, a design is said to be A-optimal if it minimizes ti=1 Vard (τˆi − τˆ0 ), where τˆ = (τˆ0 , τˆ1 , . . . , τˆt ) is the generalized least square estimate of τ . (ii) A design is said to be MV-optimal if it minimizes maxi=1,...,t Vard (τˆi − τˆ0 ). Since the row and column sums of Cd are both 0, we could express Cd as 1t Md 1t −1t Md Cd = . (4) −Md 1t Md The t × t submatrix Md is closely related to the A-optimality and MV-optimality. Let H = (−1t |It ), then var(Hτˆ ) = −1 2 −1 σ 2 HC− d H = σ Md , since one could choose diag(0, Md ) as a generalized inverse of Cd in view of (4). For completeness and convenience, we cite a well-known result as Lemma 1 below, which is similarly done by Hedayat and Yang (2005). Lemma 1. A design d∗ is A-optimal if it minimizes Tr(M−1 d ). Further, if the Md∗ is completely symmetric, then d∗ is also MV-optimal. Lemma 1 indicates that Md contains all the information needed to evaluate the A-optimality and MV-optimality of a design. Since Md in turn is a submatrix of Cd by ignoring the first row and first column, designs with the same Cd should be equivalent in the A-sense and MV-sense. On the other hand, the matrix Cd is a function of the unknown variable θ , hence the determination of optimal designs depends on the value of θ . As pointed out by Hedayat, Stufken, and Yang (2006), two extreme cases are worth mentioning. The case where θ = 0 corresponds to the situation of no subject effect. It is easily seen that lim Cd = Td pr⊥ [1np |P|Fd ] Td , θ→0
which would indeed be precisely the information matrix for τ if we were to ignore the subject effect. Under this special case, Theorem 1 below gives the optimal designs in . The other extreme case corresponds to θ = ∞, and we have lim Cd = Td pr⊥ [1np |P|U|Fd ] Td , θ→∞
where U = In ⊗ 1p . This limit is precisely the information matrix that we would have obtained had we treated the subject effects as fixed. Under this special case, Hedayat and Yang (2005) gave the optimal designs in the subclass , and they conjectured that optimal designs in is still highly efficient in . In this article, we derive optimal designs in for any value of θ , which covers their result as a special case. We also give the explicit way of evaluating the efficiency of any design for any value of θ .
Hedayat and Zheng: Optimal and Efficient Crossover Designs
3
Theorem 1. If θ = 0, then for any n, t, p, a design d is simultaneously A-optimal and MV-optimal in n,t,p if it satisfies: mdii = rdi r˜di /pn, i = 0, 1, . . . , t mdij , md0i , mdi0 are constants across all 1 ≤ i = j ≤ t dik1 = dik2 , i = 0, 1, . . . , t, k1 , k2 = 1, 2, . . . , p rdi = rdj , i, j = 1, 2, . . . , t rd0 = arg minh∈{1,2,...,np} (t/(np − h) + 1/h) √ Condition 5 is equivalent to rd0 = np/(1 + t) when the latter is an integer. 1. 2. 3. 4. 5.
We would like to give an intuitive explanation for the conditions imposed in Theorem 1. Conditions 2–4 impose some structure of symmetry among test treatments and periods. Condition 5 directly determines the replication of each treatment in conjunction with Condition 4. Finally, Condition 1 requires the exact relationship between two type of variables, which is too strong in√general. When n = 18, t = 4, p = 3, we need rd0 = np/(1 + t) = 18. Under this situation, the design below constructed by Yang and Stufken (2008) is optimal in 18,4,3 when θ = 0: 111222333444000000 d1 : 123234340400001012 4 0 0 0 1 2 2 1 2 3 4 0 3 3 0 4 0 1. For general θ , it is hard to compare all designs in . However, we succeeded in finding optimal designs in the subclass . In order to introduce the result, it is necessary to give the definition of a class of designs proposed by Hedayat and Yang (2005). Definition 2. A design d is said to be a TBTCI design if: 1. 2. 3. 4. 5. 6. 7.
mdii = 0 for all 0 ≤ i ≤ t md0i , mdi0 , and mdij are constants across all 1 ≤ i = j ≤ t dik1 = dik2 , i = 0, 1, . . . , t, k1 , k2 = 1, 2, . . . , p rdi = rdj , i, j = 1, 2, . . . , t ndiu = 0 or 1 for i = 1, 2, . . . , t, u = 1, 2, . . . , n |n − nd0v | ≤ 1 and |˜nd0u − n˜ d0v | ≤ 1 for all 1 ≤ u, v ≤ n n n d0u n n n , n n , ˜ d0u n˜ diu , u=1 n nu=1 d0u diu n u=1 diu dju n n ˜ n ˜ , n n ˜ , n ˜ n , and diu dju d0u diu d0u diu u=1 u=1 u=1 n n n ˜ , are constants across all 1 ≤ i = j ≤ t. u=1 diu dju
Theorem 2. (i) Suppose p ≥ 3 and t ≥ max(p − 1, 3), then for any value of θ a design d is simultaneously A-optimal and MV-optimal in t+1,n,p if d is a TBTCI design and rd0 =
arg min
f (h, θ),
(5)
h∈{1,2,...,np−1}
where f (rd0 , θ) = t(t − 1)2 (α1 − β12 /γ1 )−1 + t(α2 − β22 /γ2 )−1 , α1 = t(1 − λp )(np − rd0 ) − η(np − rd0 )2 2 − rd0 + λp χ1 + ηrd0 ,
β1 = λp t(n(p − 1) − r˜d0 ) + η(n(p − 1) − r˜d0 )(np − rd0 ) − λp χ2 − ηrd0 r˜d0 , γ1 = (t + 1 − 2/p − λp t)(n(p − 1) − r˜d0 ) − n(p − 1)2 /p 2 − η(n(p − 1) − r˜d0 )2 + η˜rd0 + λp χ3 , 2 α2 = rd0 − λp χ1 − ηrd0 ,
β2 = λp χ2 + ηrd0 r˜d0 , 2 2 γ2 = r˜d0 − (np2 − np)−1 r˜d0 − λp χ3 − η˜rd0 ,
λp = θ (1 + θ p)−1
and
η = λp (θpn)−1 ,
χ1 = rd0 + (2rd0 − n)rd0 /n − nrd0 /n2 , χ2 = r˜d0 + (rd0 + r˜d0 − n)˜rd0 /n − n˜rd0 /n2 × 1{rd0 /n−˜rd0 /n t + 1, for which we have not seen any work carried out yet. We believe some designs with similar structures as TBTCI designs would be highly efficient in this case.
7
6. MAIN PROOFS Lemma 1 in Section 3 gives an explicit relationship between the optimality criteria and the information matrix Cd . In order to find optimal designs, we first need to find a function 0 (θ ) such that Tr(Md (θ )−1 ) ≥ 0 (θ )
(18)
for any (d, θ). If at the same time we have Tr(Md∗ (θ ∗ )−1 ) = 0 (θ ∗ ), then the design d∗ would be optimal when θ = θ ∗ . To establish (18), we can start with maximizing Cd in the Loewner’s sense. Lemma 3. For any design d, we have Cd ≤ Td V−1/2 pr⊥ 1np |V−1/2 Fd V−1/2 Td .
(19)
The equality in (19) holds for any design d in which ldik = rdi /p, i = 0, . . . , t. Proof. It is sufficient to prove Td V−1/2 pr⊥ (1np |V−1/2 Fd ) × V−1/2 P = 0 when ldik = rdi /p, i = 0, . . . , t. Let 0 0 A = 1n ⊗ . 0 Bp−1 Then, we have pr⊥ (1np |V−1/2 Fd )V−1/2 (P − A) = 0 and 1np × V−1/2 A = 0. Plus, ldik = rdi /p, i = 0, . . . , t, implies Td V−1 A = 0 and Td V−1 A = 0. The lemma is established. Lemma 4. (i) For any design d ∈ t+1,n,p , we have Tr(M−1 d )≥
t(t − 1)2 t + , x0 y0
(20)
where x0 = α1 −
β12 , γ1
and 2 rd0 θ 2 nd0u − 1 + θp (1 + θp)pn u=1
n θ − (n(p − 1) − r˜d0 ) md00 − nd0u n˜ d0u 1 + θp n
y0 = rd0 −
u=1
2
1 rd0 r˜d0 (1 + θ p)pn
n θ rd0 + r˜d0 nd0u n˜ d0u − ld01 − md00 + p 1 + θp −
+
1 rd0 r˜d0 (1 + θ p)pn
× n(p − 1) r˜d0 −
−
2 r˜d0 (1 + θ p)pn
−
u=1
2
θ 2 n˜ d0u 1 + θp
2 r˜d0 p
n
−1
u=1
8
Journal of the American Statistical Association, ???? 2010
with α1 = t 1 −
t θ t 2 rdi − rd0 (np − rd0 ) − 1 + θp (1 + θ p)pn i=1
+
β1 =
θ 1 + θp
n2d0u +
u=1
n t
θt 1 + θp −
n
2 rd0 , (1 + θ p)pn
i=1 u=1
θ 1 + θp t
n
t rdi r˜di (1 + θ p)pn t
ndiu n˜ diu +
Proof of Theorem 1. By Lemma 4(i), we have Tr(M−1 d )≥ t(t − 1)/(np − rd0 ) + tnp/(rd0 (np − rd0 )) = t/rd0 + t2 /(np − rd0 ) for any design with the equality holds when the design satisfies Conditions 1–4. Hence the A-optimality is established. Furthermore, these conditions are sufficient for Md to be completely symmetric.
i=1
nd0u n˜ d0u −
u=1
The proof of Lemma 4 is similar to lemma 4 of Hedayat and Yang (2005), and we postpone it to the Supplemental Materials. The merit of Lemma 4 is that we settle down with a very special subclass of designs such that the A-criterion of designs therein are easy to be calculated directly and at the same time the optimal design is guaranteed to be included in this subclass. For the special case of θ = 0, we are ready to prove Theorem 1 in Section 3.
rd0 r˜d0 (1 + θ p)pn
rd0 + ld01 + md00 , p i=1 2 θt γ1 = t + 1 − − (n(p − 1) − r˜d0 ) p 1 + θp
For general value of θ , we first need the following four preliminary lemmas:
n t 2 − (p − 1)2 − r˜di p (1 + θ p)pn
Proof. If ld0k > [ n2 ], it will conflict with the condition that md00 = 0. The other two inequalities follow immediately by noting that ld0k = ld0k for any 1 ≤ k = k ≤ p.
−t
mdii −
Lemma 5. For any d ∈ t+1,n,p , we have rd0 ≤ p[ n2 ], r˜d0 ≤ (p − 1)[ n2 ], and ld0k ≤ [ n2 ].
t
i=1
+
2 r˜d0 θ + (1 + θp)pn 1 + θ p
n
Lemma 6. For any design d ∈ t+1,n,p , we have
n˜ 2d0u .
n
u=1
Further, the equality in (20) holds when the following three conditions hold: 1. ndiu is either 0 or 1, 1 ≤ i ≤ t, 0 ≤ u ≤ n 2. ldik = rdi /p, i = 0, . . . , t 3. Td V−1/2 pr⊥ (1np )V−1/2 Td , Td V−1/2 pr⊥ (1np )V−1/2 Fd , and Fd V−1/2 pr⊥ (1np )V−1/2 Fd are invariant under any permutation of test treatments. (ii) For any design d ∈ t+1,n,p in which rpd0 − ld01 = mdii = 0, 0 ≤ i ≤ t, we have (20) with α1 and γ1 therein keep unchanged, however β1 and y0 have the following simpler forms θt t ndiu n˜ diu + rdi r˜di 1 + θp (1 + θ p)pn n
t
β1 =
t
i=1 u=1
−
θ 1 + θp
n
i=1
nd0u n˜ d0u −
u=1
rd0 r˜d0 , (1 + θ p)pn
y0 = α2 −
β22 , γ2
α2 = rd0 −
2 rd0 θ 2 nd0u − , 1 + θp (1 + θ p)pn n
u=1
β2 =
θ 1 + θp
γ2 = r˜d0 −
n
nd0u n˜ d0u +
u=1 2 r˜d0
(p − 1)pn
−
rd0 r˜d0 , (1 + θ p)pn
θ 1 + θp
n u=1
n˜ 2d0u −
1 nd0u n˜ d0u ≤ np(p − 1). (21) 4 u=1 Consequently, we have nu=1 nd0u n˜ d0u ≤ t(n(p − 1) − r˜d0 ) as long as 2t ≥ p. Proof. To maximize nu=1 nd0u n˜ d0u , it is necessary for rd0 to attain its maximum, which is p[ n2 ] by Lemma 5. (i) When n is an even number, then rd0 = np/2 and the distribution of control treatment in the design fall into one type: n/2 of the subjects take the control treatment at all even periods; the remaining half of the subjects take the control treatment at all odd periods. Then the inequality trivially holds. (ii) When n is an odd number, the maximum of nu=1 nd0u n˜ d0u will be attained when one of the subjects does not take the control treatment while the remaining n − 1 subjects take the control treatment in the same way as in (i). To see this, suppose Subject 1 has the smallest value of n˜ d0u . If n˜ d01 > 0, without loss of generality let us suppose Subject 1 takes the control treatment at the second period. There always exist a subject (say, 2) who takes test treatments in the second period as well as in the neighboring periods, that is, periods 1 and 3, since n is odd. Then we can exchange the treatments between these two subjects at the second period so that md00 is still 0. By this exchange, the decrement of nd01 n˜ d01 is at most 2˜nd01 , while the correspondingincrement of nd02 n˜ d02 is at least 2˜nd02 + 1. Since n˜ d02 ≥ n˜ d01 , nu=1 nd0u n˜ d0u is increased. Hence we have, by the argument in (i), that n u=1
2 r˜d0
(1 + θ p)pn
The equality in (20) still holds under the same conditions.
.
1 1 nd0u n˜ d0u ≤ (n − 1)p(p − 1) ≤ np(p − 1). 4 4
Lemma 7. For any design d ∈ t+1,n,p , we have n t i=1 u=1
ndiu n˜ diu ≥ n(p − 1) − r˜d0 .
Hedayat and Zheng: Optimal and Efficient Crossover Designs
9
Proof. n t
ndiu n˜ diu ≥
i=1 u=1
n t i=1 u=1
n˜ diu =
t
cannot be minimized simultaneously, we need the following decomposition: r˜di = n(p − 1) − r˜d0 .
t
i=1
Lemma 8. For any design d ∈ t+1,n,p , we have β1 + γ1 ≥ 0 for any value of θ . Proof. If we consider β1 + γ1 as a function of θ , then by the simple derivative equations θ 1 d , = dθ 1 + θ p (1 + θ p)2 (22) d 1 p =− dθ 1 + θ p (1 + θ p)2 we have Kd d(β1 + γ1 ) , = dθ (1 + θ p)2
(23)
where Kd is a constant which depends on the design d only. That means β1 + γ1 is either nondecreasing or nondecreasing with respective to θ when the design d is fixed. Hence, it is enough to prove the inequality for θ = 0 and θ = ∞. For θ = 0, we have, in view of Lemma 5, that 2 (n(p − 1) − r˜d0 ) β1 + γ1 = t + 1 − p r˜d0 ld0p n t − (p − 1)2 + r˜di ldip − p pn pn i=1 1 ≥ t− n(p − 1) p 2 ld0p − t+1− + r˜d0 p pn t
tp − 1 n(p − 1) p 2 1 n(p − 1) − t+1− + p 2p 2 1 n(p − 1) p(t − 1) − ≥ 0. = 2p 2 ≥
For θ = ∞, we have, in view of Lemmas 5 and 7, that 2 n 1 (n(p − 1) − r˜d0 ) − (p − 1)2 − r˜d0 β1 + γ1 ≥ t + 1 − p p p 1 1 ≥ t− n(p − 1) − t + 1 − r˜d0 p p n(p − 1) ≥ (p(t − 1) − 1) ≥ 0. 2p In the following, we shall search for designs which minimizes the right-hand side n of (20), where t the ncomponents n n 2 , 2 , n n ˜ n n ˜ , ˜ diu , i=1 u=1 ndiu n d0u u=1 d0u u=1 d0u d0u u=1 t t t 2 2 i=1 rdi , i=1 r˜di , and i=1 rdi r˜di are related to each other. The latter four components are investigated by Lemmas 9 and 10 while the remaining three by Lemma 11. are2 investigated t 2 , and , ti=1 r˜di r r˜di Since ti=1 nu=1 ndiu n˜ diu , ti=1 rdi di i=1
rdi r˜di =
i=1
t
2 r˜di +
i=1
t
2 rdi
=
i=1
t
t
r˜di ldip ,
i=1 2 r˜di
+2
i=1
t
r˜di ldip +
i=1
t
t
t
(24) 2 ldip .
i=1
t 2, 2 By (24), we transform i=1 rdi i=1 r˜di , and i=1 rdi r˜di into t t t 2 , and 2 . The advantage is that the r ˜ l , r ˜ l i=1 di dip i=1 di i=1 dip latter two terms are independent of each other, and thus we can first find the minimum of ti=1 r˜di ldip for fixed values of the two independent terms as shown in Lemma 9. Lemma 9. For any design t d2∈ t+1,n,p , when a and b are the 2 and values of ti=1 r˜di i=1 ldip , respectively, we have t
r˜di ldip
i=1
1 ≥ (n(p − 1) − r˜d0 )(n − ld0p ) t (n − ld0p )2 (n(p − 1) − r˜d0 )2 − b− , (25) a− t t with the equality holds when r˜di = r˜dj and ldip = ldjp for any 1 ≤ i = j ≤ t. Proof. For notational simplicity we will let xi = r˜di and yi = ldip . The proof reduces to minimizing ti=1 xi yi under the re strictions of ti=1 xi = n(p − 1) − r˜d0 , ti=1 xi2 = a, ti=1 yi = n − ld0p , and ti=1 y2i = b. Let x¯ = ti=1 xi /t = (n(p − 1) − t r˜d0 )/t and y¯ = i=1 yi /t = (n − ld0p )/t, then ti=1 xi yi = t¯xy¯ + t t i=1 ui vi = (n(p − 1) − r˜d0 )(n − ld0p )/t + i=1 ui vi , where ui = xi − x¯ and vi = yi − y¯ . However,
t 1/2 t 1/2 t ui vi ≥ − u2i v2i i=1
i=1
i=1
(n − ld0p )2 (n(p − 1) − r˜d0 )2 =− b− a− t t
with the equality holds if and only if xi = −c0 yi , i = 1, . . . , t, with ta − (n(p − 1) − r˜d0 )2 c0 = . tb − (n − ld0p )2 Hence the lemma is established. Lemma 10. If p ≤ t + 1 and θ ≥ 0, we have, for any design d ∈ t+1,n,p , that t t(t − 1)2 t t(t − 1)2 + ≥ + , x0 y0 x˜ 0 y0
(26)
where x˜ 0 = α˜ 1 −
β˜12 γ˜1
(27)
10
with
Journal of the American Statistical Association, ???? 2010
α˜ 1 = t 1 −
θ (np − rd0 )2 (np − rd0 ) − − rd0 1 + θp (1 + θ p)pn
n n θ θ 2 nd0u n˜ d0u + n˜ d0u , 1 + θp 1 + θp u=1 u=1 2 θt 2 = t + 1 − − (n(p − 1) − r˜d0 ) p 1 + θp
−
θ 2 1 nd0u + r2 , 1 + θp (1 + θ p)pn d0 n
+
u=1
β˜1 =
θt (n(p − 1) − r˜d0 ) 1 + θp +
(n(p − 1) − r˜d0 )2 n − (p − 1)2 − p (1 + θ p)pn
(n(p − 1) − r˜d0 )(np − rd0 ) (1 + θ p)pn
θ rd0 r˜d0 nd0u n˜ d0u − , 1 + θp (1 + θ p)pn u=1 2 θt γ˜1 = t + 1 − − (n(p − 1) − r˜d0 ) p 1 + θp (n(p − 1) − r˜d0 )2 n − (p − 1)2 − p (1 + θ p)pn 2 r˜d0 θ 2 n˜ d0u . + + (1 + θp)pn 1 + θ p n
u=1
The equality in (26) holds when all ndiu ’s are binary and r˜di = r˜dj , ldip = ldjp for any 1 ≤ i = j ≤ t. t 2 2, Proof. Let a, b, and c to be the values of ti=1 r˜di i=1 ldip , t and i=1 r˜di ldip , respectively. In view of the decomposition (24), and Lemma 8, we have 2 ∂x0 = − (β1 + γ1 ) ≤ 0. ∂c γ1 Thus, we have x0 ≤ x0 , where x0 is obtained from x0 by t replacing i=1 r˜di ldip by its lower bound in the right-hand side of (25). Similarly, denote by β1 the new term obtained from β1 by the same replacement. Obviously, we have β1 ≥ β1 . In the following, the notation u ∝ v stands for u = f (θ, t, n, p, b, ld0p , γ1 )v with the function f > 0 for any d ∈ t+1,n,p and θ ≥ 0, and the explicit form of f may vary from line to line. By direct calculation, we have ∂x0 β1 ta − (n(p − 1) − r˜d0 )2 −1 ∝ 1+ ∂b γ1 tb − (n − ld0p )2 ∝ (β1 + γ1 ) ta − (n(p − 1) − r˜d0 )2 − γ1 tb − (n − ld0p )2 = 1 ta − (n(p − 1) − r˜d0 )2 − 2 tb − (n − ld0p )2 , (28) where
2 θt 1 = t + 1 − − (n(p − 1) − r˜d0 ) p 1 + θp θt n ndiu n˜ diu − (p − 1)2 1 + θp p t
+
n
i=1 u=1
2 r˜d0 θ 2 n˜ d0u . + (1 + θp)pn 1 + θ p n
+
n
−
(n(p − 1) − r˜d0 )(n − ld0p ) ld0p r˜d0 − (1 + θ p)pn (1 + θ p)pn
+
u=1
Let r = 1 /2 , then (28) implies ∂x0 =0 iff ∂b tb − (n − ld0p )2 = r ta − (n(p − 1) − r˜d0 )2 ,
(29)
∂x0 >0 iff ∂b 2 tb − (n − ld0p ) < r ta − (n(p − 1) − r˜d0 )2 ,
(30)
∂x0 r ta − (n(p − 1) − r˜d0 )2 .
(31)
Note that r is a positive constant with respect to the values of a and b. (29)–(31) imply that the optimal vector (a, b) ∈ R2 maximizing x0 has to satisfy the right-hand side of (29). By replacing b as a function of a using the equality in (29), we denote the updated expression of β1 and x0 by β1 and x0 , respectively. Note that the latter two could also be obtained directly by plugging the following equations into β1 and x0 , respectively: t (n(p − 1) − r˜d0 )2 rdi r˜di = (1 − r) a − t i=1
(np − rd0 )(n(p − 1) − r˜d0 ) , t t (np − rd0 )2 (n(p − 1) − r˜d0 )2 2 rdi = (1 − r)2 a − + . t t +
i=1
Notice that β1 , γ1 , and x0 are functions of a without b involved. By direct calculation, we have ∂x0 β1 2 ≤ 0. (32) ∝− 1−r+ ∂a γ1 Hence, x0 is maximized when a equals to its minimum, (n(p − 1) − r˜d0 )2 /t. Then, by (25) and (29), the point of (a, b, c) ∈ R3 given by the following equations will attain the maximum of x0 : a = (n(p − 1) − r˜d0 )2 /t, b = (n − ld0p )2 /t, c = (n(p − 1) − r˜d0 )(n − ld0p )/t.
(33)
Hedayat and Zheng: Optimal and Efficient Crossover Designs
11
Note that the preceding equations will be satisfied if r˜di = r˜dj
and
ldip = ldjp
for any 1 ≤ i = j ≤ t. (34)
By plugging the equations of (33) into β1 and x0 , we denote the new terms derived by β1 and x0 , respectively. Obviously, x0 ≤ x0 with equality holds when (34) holds. By Lemmas 6 and 7 in the supplemental material, we have
t n n θ ndiu n˜ diu − nd0u n˜ d0u β1 = t 1 + θp i=1 u=1
+
H(ξ1 , ξ2 , ξ3 , θ) =
1 (ξ1 , ξ2 , ξ3 , θ) =
t n ˜ diu That indicates that x0 increases when i=1 u=1 ndiu n decreases. Notice that x˜ 0 is obtained from x0 by replacing t n ˜ diu by its minimum, n(p − 1) − r˜d0 , in view i=1 u=1 ndiu n of Lemma 7. The conclusion is obtained. Remark 1. In maximizing the x0 in (20), t the 2 conditions 2 and r˜di = r˜dj , ldip = ldjp minimize ti=1 rdi i=1 r˜di , however not ti=1 rdi r˜di . Suppose (n(p − 1) − r˜d0 )/(t − 1) is an integer, let rd1 = 0, ld1p = n − ld0p , and r˜di = (n(p − 1) − r˜d0 )/(t − 1) for i = 2, . . . , t, then we have (35) (36)
for any value of ξi , i = 1, 2, 3, and θ , it is enough to show ∂H(
To work on the right-hand side of(26) we now only need to investigate how it is influenced by nu=0 n2d0u , nu=0 nd0u n˜ d0u , and nu=0 n˜ 2d0u . Lemma 11. (i) When p ≥ 3 and t ≥ max(p − 1, 3), for any design d ∈ t+1,n,p , we have
x0∗
(37)
y∗0
and are derived from x˜ 0 and y0 respectively by where replacing nu=0 n2d0u , nu=0 nd0u n˜ d0u , and nu=0 n˜ 2d0u therein by their minimum with given rd0 . Automatically, the equality in (37) holds for a design d∗ which minimizes those three terms. (ii) When p = 3 and t = 2, the above conclusion is still valid if rd0 /n ≥ 0.6306. Proof. By Lemma 6, we have n
(40)
In the following, we will omit the variables ξ1 , ξ2 , ξ3 , θ for the functions defined when there is no ambiguity. We have θt (t − 1)2 1 ∂H = − , ∂ξ1 1 + θ p y20 x˜ 02 2θ t 2 (t − 1)2 1 ∂H = − , ∂ξ2 1 + θ p y20 x˜ 02 2 2 (t − 1)2 21 θt ∂H = − . ∂ξ3 1 + θ p y20 x˜ 02
for d ∈ whenever p < t.
t t(t − 1)2 t t(t − 1)2 + ≥ + ∗, x˜ 0 y0 x0∗ y0
(39)
Note that the derivative ∂H ∂ξi increases with ξj for any 1 ≤ i, j ≤ 3. Now to establish ∂H ≥ 0, i = 1, 2, 3, (41) ∂ξi
i=1
< (np − rd0 )(n(p − 1) − r˜d0 )/t
β˜1 (ξ1 , ξ2 , ξ3 , θ) , γ˜1 (ξ1 , ξ2 , ξ3 , θ)
β2 (ξ1 , ξ2 , ξ3 , θ) 2 (ξ1 , ξ2 , ξ3 , θ) = . γ2 (ξ1 , ξ2 , ξ3 , θ)
(n(p − 1) − r˜d0 )(np − rd0 ) − rd0 r˜d0 (1 + θ p)pn
rdi r˜di = (n(p − 1) − r˜d0 )2 /(t − 1)
t(t − 1)2 t + . x˜ 0 (ξ1 , ξ2 , ξ3 , θ) y0 (ξ1 , ξ2 , ξ3 , θ)
For notational convenience, we also define
u=1
≥ 0.
t
We can write α˜ 1 as a function α˜ 1 (ξ1 , ξ2 , ξ3 , θ) with the value rd0 fixed. The same argument could be applied to β˜1 , γ˜1 , α2 , β2 , γ2 , x˜ 0 , y0 , and we define
nd0u n˜ d0u ≤ t(n(p − 1) − r˜d0 )
u=1
˜ by ξ1 , ξ2 , and ξ3 the which implies any θ ≥ 0. Denote β1 ≥ 0 for values of nu=1 n2d0u , nu=1 nd0u n˜ d0u , and nu=1 n˜ 2d0u . Among the restrictions by the nature of the design, we have 2 rd0 ξ1 ≥ max , rd0 , n (38) t(n(p − 1) − r˜d0 ) ≥ ξ2 ≥ r˜d0 , ξ3 ≥ r˜d0 .
2 rd0 n , r˜d0 , r˜d0 , θ)
≥ 0,
i = 1, 2, 3,
(42)
∂H(rd0 , r˜d0 , r˜d0 , θ) ≥ 0, ∂ξi
i = 1, 2, 3,
(43)
∂ξi or
for any value of θ . Propositions 1–5 in the Supplemental Materials finish the proof. Proof of Theorem 2. Combining Lemmas 1, 4(ii), 10, and 11, it is enough to prove that a totally balanced test-control incomplete crossover design satisfies: • ndiu is either 0 or 1, 1 ≤ i ≤ t, 0 ≤ u ≤ n • ldik = rdi /p, i = 0, . . . , t • Td V−1/2 pr⊥ (1np )V−1/2 Td , Td V−1/2 pr⊥ (1np )V−1/2 Fd , and Fd V−1/2 pr⊥ (1np )V−1/2 Fd are invariant under any permutation of test treatments • r˜di = r˜dj , ldip = ldjp for any 1 ≤ i = j ≤ t n n 2 • ˜ d0u , and nu=0 n˜ 2d0u are minimized u=0 nd0u , u=0 nd0u n with respect to fixed rd0 . By comparing these conditions to the seven conditions in Definition 2, we conclude the theorem. SUPPLEMENTAL MATERIALS Supplemental Proofs: All the details of the proofs that were omitted in this article are given in this material. (Hedayat and Zheng- Supplementalt.pdf) [Received March 2010. Revised August 2010.]
12
Journal of the American Statistical Association, ???? 2010
REFERENCES Cheng, C. S., and Wu, C. F. (1980), “Balanced Repeated Measurements Designs,” The Annals of Statistics, 8, 1272–1283. [1] Hedayat, A. S., and Afarinejad, K. (1975), “Repeated Measurements Designs, I,” A Survey of Statistical Design and Linear Models, ed. J. N. Srivastava, Amsterdam: North-Holland, pp. 229–240. [1] (1978), “Repeated Measurements Designs, II,” The Annals of Statistics, 6, 619–628. [1] Hedayat, A. S., and Yang, M. (2003), “Universal Optimality of Balanced Uniform Crossover Designs,” The Annals of Statistics, 31, 978–983. [1] (2004), “Universal Optimality for Selected Crossover Designs,” Journal of the American Statistical Association, 99, 461–466. [1] (2005), “Optimal and Efficient Crossover Designs for Comparing Test Treatments With a Control Treatment, The Annals of Statistics, 33, 915– 943. [1-3,7,8] (2006), “Efficient Crossover Designs for Comparing Test Treatments With a Control Treatment,” Discrete Mathematics, 306, 3112–3124. [1,7] Hedayat, A. S., and Zhao, W. (1990), “Optimal Two-Period Repeated Measurements Designs,” The Annals of Statistics, 18, 1805–1816. Corrigendum: 20 (1992), 619. [1] Hedayat, A. S., and Zheng, W. (2010), “Totally Balanced Test-Control Incomplete Crossover Designs and Their Statistical Applications,” Contemporary Mathematics, to appear. [3,5,7] Hedayat, A. S., Stufken, J., and Yang, M. (2006), “Optimal and Efficient Crossover Designs When Subject Effects Are Random,” Journal of the American Statistical Association, 101, 1031–1038. [1,2,7]
Kunert, J. (1984), “Optimality of Balanced Uniform Repeated Measurements Designs,” The Annals of Statistics, 12, 1006–1017. [1] Kunert, J., and Martin, R. J. (2000), “On the Determination of Optimal Designs for an Interference Model,” The Annals of Statistics, 28, 1728–1742. [1] Kunert, J., and Stufken, J. (2002), “Optimal Crossover Designs in a Model With Self and Mixed Carryover Effects,” Journal of the American Statistical Association, 97, 898–906. [1] Kushner, H. B. (1997), “Optimal Repeated Measurements Designs: The Linear Optimality Equations,” The Annals of Statistics, 25, 2328–2344. [1] (1998), “Optimal and Efficient Repeated-Measurements Designs for Uncorrelated Observations,” Journal of the American Statistical Association, 93, 1176–1187. [1] Stufken, J. (1991), “Some Families of Optimal and Efficient Repeated Measurements Designs,” Journal of Statistical Planning and Inference, 27, 75–83. [1] (1996), “Optimal Crossover Designs,” in Design and Analysis of Experiments. Handbook of Statistics, Vol. 13, eds. S. Ghosh and C. R. Rao, Amsterdam: North-Holland, pp. 63–90. [1] Yang, M., and Park, M. (2007), “Efficient Crossover Designs for Comparing Test Treatments With a Control Treatment When p = 3,” Journal of Statistical Planning and Inference, 137, 2056–2067. [7] Yang, M., and Stufken, J. (2008), “Optimal and Efficient Crossover Designs for Comparing Test Treatments to a Control Treatment Under Various Models,” Journal of Statistical Planning and Inference, 138, 278–285. [3]
