ON OPTIMALITY AND CONSTRUCTION OF CIRCULAR REPEATED-MEASUREMENTS DESIGNS

Transcription

1 Statistica Sinica 27 (2017), oi: ON OPTIMALITY AND CONSTRUCTION OF CIRCULAR REPEATED-MEASUREMENTS DESIGNS R. A. Bailey 1,2, Peter J. Cameron 1,2, Katarzyna Filipiak 3, Joachim Kunert 4 an Augustyn Markiewicz 5 1 University of St Anrews, 2 University of Lonon, 3 Poznań University of Technology, 4 TU Dortmun University an 5 Poznań University of Life Sciences Abstract: The aim of this paper is to characterize an construct universally optimal esigns among the class of circular repeate-measurements esigns when the parameters o not permit balance for carry-over effects. It is shown that some circular weakly neighbour balance esigns efine by Filipiak an Markiewicz (2012) are universally optimal repeate-measurements esigns. These results exten the work of Maga (1980), Kunert (1984b), an Filipiak an Markiewicz (2012). Key wors an phrases: Circular weakly balance esign, repeate-measurements esign, uniform esign, universal optimality. 1. Introuction The problem of universal optimality of repeate-measurements esigns is wiely stuie in the literature. Most of the esigns consiere have the same number of perios as treatments; we also make this assumption. For experiments without a pre-perio, Heayat an Afsarineja (1978) an Cheng an Wu (1980) prove the universal optimality, for the estimation of irect as well as carry-over effects, of some balance uniform repeate-measurements esigns over a restricte class of competing esigns. If the number n of subjects is at most twice the number t of treatments, Kunert (1984a) showe that, for the estimation of irect effects, balance uniform esigns are universally optimal over the class of all esigns. Heayat an Yang (2003) extene this by showing universaloptimalityofbalanceuniformesignsifn t(t 1)/2. Kunert (1984a) also prove that if n is sufficiently large then a balance uniform esign is no longer optimal. Moreover, this esign is not universally optimal for the estimation of carry-over effects when certain other special esigns exist. Stufken (1991) constructe some universally optimal esigns using orthogonal arrays of type I. Jones, Kunert an Wynn (1992) prove universal optimality of some balance uniform esigns uner the moel with ranom carry-over effects.

2 2 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Kunert (1983) consiere repeate-measurements esigns with or without a pre-perio. He prove the universal optimality of some special generalize latin squares an generalize Youen esigns over particular classes of esigns. A repeate-measurements esign is calle circular if there is a pre-perio an, for each subject, the treatment on the pre-perio is the same as the treatment on the last perio. Maga (1980) prove the universal optimality of circular strongly balance uniform esigns (uniform CSBDs) an circular balance uniform esigns (uniform CBDs) over appropriate subclasses of possible esigns. Kunert (1984b) strengthene these results by showing the universal optimality of CBDs over all esigns. Recent constructions of CSBDs an CBDs have been given by Iqbal an Tahir (2009) using cyclic shifts an by Manal, Parsa an Gupta (2016) using integer programming. Universal optimality of some CBDs is also stuie assuming a moel of repeate measurements esigns in which perio effects are negligible. This simpler moel, in which carry-over effects play the role of left-neighbour effects, is known in the literature as an interference moel. Druilhet (1999) consiere optimality of CBDs for the estimation of irect as well as carry-over effects, while Bailey an Druilhet (2004) prove their optimality for the estimation of total effects. Filipiak an Markiewicz (2012) showe universal optimality of circular weakly balance esigns (CWBDs) for the estimation of irect effects only. In this paper we consier circular repeate-measurements esigns uner the full moel an uner two simpler moels. We show universal optimality, for the estimation of irect as well as carry-over effects, of CWBDs an we give methos of constructing some of them. For particular parameter sets, there exists a CWBD using fewer subjects than uniform CBDs. The iea of the possible reuction of number of subjects is suggeste by the results of Filipiak an Markiewicz (2012). 2. Moels an Designs Let D t,n,t be the set of circular esigns with t treatments, n experimental subjects, an t perios, each subject being given one treatment uring each perio. By (l, u), for 1 l t an 1 u n, we enote the treatment assigne to the uth subject in the lth perio. Maga (1980) propose a moel associate with the esign in D t,n,t : y lu = α l + β u + τ (l,u) + ρ (l 1,u) + ε lu, 1 l t, 1 u n, (2.1) where y lu is the response of the uth subject in the lth perio, an α l, β u, τ (l,u), an ρ (l 1,u) are, respectively, the lth perio effect, the uth subject effect, the irect effect of treatment (l, u), an the carry-over effect of treatment (l 1,u),

3 CIRCULAR WEAKLY BALANCED DESIGNS 3 where (0,u)=(t, u). The ε lu are uncorrelate ranom variables with common variance an zero mean. In vector notation moel (2.1) can be rewritten as y = Pα + Uβ + T τ + F ρ + ε. (2.2) Here y is the transpose of the vector y =(y 11,y 21,...,y tn ). Also, α, β, τ, an ρ are the vectors of perio, experimental subject, irect, an carry-over effects, respectively. Moreover, ε is the vector of ranom errors, with ε N(0 nt,σ 2 I nt ), where σ 2 is a positive constant, I n enotes the ientity matrix of orer n, an 0 n is the n-imensional vector of zeros. The matrices T an F are the esign matrices for irect an carry-over effects, respectively, while P = 1 n I t an U = I n 1 t are the incience matrices for perio an experimental subject effects, respectively, where 1 n is the n-imensional vector of ones an enotes the Kronecker prouct. Let H t =(h ij ) be the circulant matrix of orer t with h ij = 1 if j i = 1 or i =1,j = t, an h ij = 0 otherwise. Then F =(I n H t )T. In this paper we also consier simpler moels moel (2.2) without perio effects, y = Uβ + T τ + F ρ + ε, (2.3) an moel (2.2) without experimental subject effects, y = Pα + T τ + F ρ + ε. (2.4) In the context of experiments in agriculture an forestry, as iscusse by Azaïs, Bailey an Mono (1993), perios correspon to rows, subjects correspon to columns, an the carry-over effect correspons to the neighbour effect of the treatment to the North. The roles of rows an columns are frequently interchange in such literature, an so moel (2.3) is known as the interference moel with left-neighbour effects; cf., Druilhet (1999), Filipiak an Markiewicz (2012). Following Maga (1980), we say that a esign in D t,n,t is: (i) uniform on perios if all treatments occur equally often in each perio; (ii) uniform on subjects if each treatment occurs exactly once on each subject; (iii) uniform if it is uniform on both perios an subjects; (iv) circular strongly balance (CSBD) if the collection of orere pairs ((l 1,u),(l, u)), for 1 l t an 1 u n, contains each orere pair of treatments (istinct or not) λ 0 times, where λ 0 = n/t; (v) circular balance (CBD) if the collection of orere pairs ((l 1,u),(l, u)), for 1 l t an 1 u n, contains each orere pair of istinct treatments λ 1 times, where λ 1 = n/(t 1), an oes not contain any pair of equal treatments.

4 4 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ We aitionally efine circular weakly balance esigns. Let S = T F = (s ij ) 1 i,j t. The entry s ij is the number of appearances of treatment i precee by treatment j in the esign. Thus the rows an columns of S sum to the vector of treatment replications. Filipiak et al. (2008) calle the matrix S the left-neighbouring matrix. When the number of treatments is equal to the number of perios, Filipiak an Markiewicz (2012) calle a esign in D t,n,t (vi)circular weakly balance (CWBD) if the collection of orere pairs ((l 1,u),(l, u)), for 1 l t an 1 u n, contains each orere pair of istinct treatments λ or λ 1 times, where λ = n/(t 1), an (a) S 1 t = S 1 t = n1 t, so that each treatment has replication n; (b) S S is completely symmetric (all iagonal entries are equal an all offiagonal entries are equal). In this efinition x is the smallest integer greater than or equal to x. Wilkinson et al. (1983) efine partially neighbour balance esigns as esigns with s ij {0, 1} if i =j; however, their esigns are not circular, an they consier neighbours in more than one irection. Some methos of constructing circular partially neighbour-balance esigns are given by Azaïs, Bailey an Mono (1993). If is a CSBD then S = λ 0 J t, where J t = 1 t 1 t; if is a CBD then S = λ 1 (J t I t ). If is a CWBD but not a CBD then S is not completely symmetric but S S is. 3. Existence Conitions A necessary conition for the existence of a CBD with t perios is that (t 1) ivies n: see e.g., Druilhet (1999), while for the existence of a CWBD the expression n(n 2λ + 1) must be ivisible by t 1; cf., Filipiak an Markiewicz (2012). Parameters satisfying the necessary conition for the existence of a CWBD with t 19 an n<3(t 1) are liste in Table 1 of Filipiak an Markiewicz (2012). Let be a CWBD in D t,n,t which is not a CBD. Then λ = n/(t 1). Put k = n (λ 1)(t 1). (3.1) Since is not a CBD, 1 k t 2. Using this notation, the necessary conition for a CWBD given by Filipiak an Markiewicz (2012) is t 1 ivies k(k 2λ + 1). (3.2) Filipiak an Różański (2009) showe that if n = 1, or if t is even an n = 2, then all esigns are isconnecte in the sense that it is not possible to estimate all contrasts between irect effects an all contrasts between carry-over effects

5 CIRCULAR WEAKLY BALANCED DESIGNS 5 without bias. If is isconnecte then it cannot be consiere to be universally optimal; in fact, the proof of Theorem 3.1 of Filipiak an Markiewicz (2012) breaks own in this case. If n = 2 then (3.1) an (3.2) show that the only CWBD is a CBD for t = 3. From now on, we assume that n 3 an is connecte. Let A = S (λ 1)(J t I t ). Then A is a t t matrix whose iagonal entries are all zero an whose other entries are all in {0, 1}. Moreover, each row an column of A has k non-zero entries. Hence A J t = J t A = kj t. Therefore S S = [ (λ 1)(J t I t )+A ] [(λ 1)(Jt I t )+A ] =(λ 1) 2 [(t 2)J t + I t ]+2(λ 1)kJ t + A A (λ 1)(A + A ). Thus S S is completely symmetric if an only if A A (λ 1)(A + A ) is completely symmetric. (3.3) If it satisfies (3.3), we shall say that esign has Type I if A + A is completely symmetric; Type II if A + A is not completely symmetric an λ = 1; Type III if A + A is not completely symmetric an λ>1. If has Type I or II then A A is completely symmetric. The off-iagonal entries in each row of A A sum to k(k 1), so in this case k(k 1) is ivisible by t 1. If has Type I then k =(t 1)/2 an A + A = J t I t. Then t 1 ivies (t 1)(t 3)/4, an so t 3 mo 4. If k = 1 then A A = I t : thus cannot have Type III, an (3.1) shows that if has Type II then n = 1, which we exclue. Theorem 1. Suppose that is a CWBD in D t,n,t an is a CBD in D t,m,t, for some values of n an m. Then the esign in D t,n+m,t which juxtaposes an is a CWBD if an only if has Type I. Proof. If is a CBD then m is a multiple of t 1 an S is completely symmetric. Hence S = S + S an so A = A. Put λ = m/(t 1). Conition (3.3) for esign says that A A (λ + λ 1)(A + A ) is completely symmetric. (3.4) If has Type I then A A an A + A are both completely symmetric, an so conition (3.4) is satisfie an is a CWBD. Conversely, if is a CWBD then conitions (3.3) an (3.4) are both satisfie. Hence A + A is completely symmetric an so has Type I.

6 6 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Lemma 1. Suppose that is a CWBD in D t,n,t which has Type III. (a) If k =(t 1)/2 then λ k/2. (b) If k<(t 1)/2 then λ k. (c) If k>(t 1)/2 then λ t k 1. Proof. Put m 0 = max{0, 2k t}. If i j then m 0 (A A ) ij k (A +A ) ij. Let m 1 an m 2 be the smallest an largest off-iagonal entries in A + A. The entries in the corresponing positions of A A (λ 1)(A + A ) lie in the intervals [m 0 m 1 (λ 1),k m 1 λ]an[m 0 m 2 (λ 1),k m 2 λ] respectively. If the latter entries are equal then k m 2 λ m 0 m 1 (λ 1). (a) If k =(t 1)/2 but A + A is not completely symmetric then m 0 = 0, m 1 = 0 an m 2 = 2. Hence k 2λ 0. (b) If k<(t 1)/2 then m 0 = 0, m 1 = 0 an m 2 1. Hence k λ 0. (c) If k>(t 1)/2 then k t/2 an so m 0 =2k t. Also, m 2 = 2 an m 1 1. Hence k 2λ 2k t (λ 1). Theorem 2. If is a CWBD in D t,n,t an has Type II or III then is not uniform on the perios. Proof. If is uniform on the perios then t ivies n. If has Type II then n = k t 2, an so this is not possible. If t ivies n then equation (3.1) shows that t ivies k λ + 1. Lemma 1 shows that if has Type III then 0 <λ k<t 1 an so 0 <k λ +1<t, thus t cannot ivie k λ Optimality 4.1. Preliminaries Kunert (1984b) showe that any CBD which is uniform on subjects is universally optimal for the estimation of irect as well as carry-over effects uner moel (2.3) over the class D t,n,t. Druilhet (1999) extene this to esigns where the number of perios is any multiple of t. Filipiak an Markiewicz (2012) efine circular weakly neighbour balance esigns to be CWBDs for t perios which are uniform on subjects; they showe their universal optimality for the estimation of irect effects uner moel (2.3) over the class D t,n,t with n t 1, an over the class of equireplicate esigns without self-neighbours if n>t 1. One aim of this paper is to prove universal optimality for the estimation of irect as well as carry-over effects of uniform CWBDs, CWBDs uniform on subjects, an CWBDs uniform on perios uner moels (2.2), (2.3) an (2.4), respectively. We are intereste in etermining esigns with minimal (in some sense) variance of the best linear unbiase estimator of the vector of parameters. Kiefer

7 CIRCULAR WEAKLY BALANCED DESIGNS 7 (1975) formulate the universal optimality criterion in terms of the information matrix, which is the inverse of the variance-covariance matrix; cf., Pukelsheim (1993). Therefore, following Proposition 1 of Kiefer (1975), we suppose that a class C = {C : D t,n,t } of non-negative efinite information matrices with zero row an column sums contains a matrix C which is completely symmetric an has maximal trace over D t,n,t. Then the esign is universally optimal in Kiefer s sense in the class D t,n,t. For a κ 1 κ 2 matrix K efine ω (K) =I κ1 K(K K) K = I κ1 ω(k) as the orthogonal projector onto the orthocomplement of the column space of K, where (K K) is a generalize inverse of K K. Then the information matrix for the least squares estimate of τ uner moel (2.g), g =2, 3, 4, is given by C (g) = T ω (Z (g) )T with zero row an column sums, where Z (g) is a block matrix containing the esign matrices of nuisance parameters, Z (2) =(P : U : F ), Z (3) =(U : F ), an Z (4) =(P : F ); cf., e.g., Kunert (1983, 1984a,b). Since ω((a : B)) = ω(a)+ω(ω (A)B), we may rewrite the matrix C (g) as C (g) = T ω (F )T T ω(ω (F )W (g) )T, with W (2) =(P : U), W (3) = U, an W (4) = P. Similarly, Kunert (1984b) showe that the information matrix for the least squares estimate of ρ uner moel (2.g), g =2, 3, 4, is C (g) = F ω ( Z (g) )F = F ω (T )F F ω(ω (T )W (g) )F, with Z (2) =(P : U : T ), Z(3) =(U : T ), an Z (4) =(P : T ) Optimality results Filipiak an Markiewicz (2012) showe that for a CWBD, S S = φi t + ξj t with φ = n(2λ 1) λ(λ 1)t n(n 2λ+1)/(t 1) an ξ = λ(λ 1)+n(n 2λ+ 1)/(t 1). Since S is nonsingular an commutes with J t, pre-multiplying by S an post-multiplying by (S ) 1 we get S S = S S ; cf., Raghavarao (1971, Theorem 5.2.1), Filipiak an Markiewicz (2016). Moreover, since for a CWBD T ω (F )T = ni t n 1 S S = F ω (T )F, the following hols. Proposition 1. Assume is a CWBD. Uner moels (2.2), (2.3) or (2.4), if is uniform, uniform on subjects, or uniform on perios, respectively, then is universally optimal for the estimation of irect effects if an only if is universally optimal for the estimation of carry-over effects.

8 8 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Let Λ t,n,t be the class of esigns in D t,n,t with no treatment precee by itself. Using the above proposition we can exten Theorem 3.1 an Theorem 3.2 of Filipiak an Markiewicz (2012), in which optimality of CWBDs for the estimation of irect effects was shown, as follows. Theorem 3. If there exists a CWBD in D t,n,t which is uniform on subjects, then it is universally optimal for the estimation of carry-over effects uner moel (2.3) over the collection of esigns in D t,n,t if n t 1, an over the collection of equireplicate esigns in Λ t,n,t otherwise. If a esign is uniform on perios then t ivies n an so n>t 1. The following theorem can be prove in the same way as Theorem 3.2 of Filipiak an Markiewicz (2012) using aitionally Proposition 1 of this paper. Theorem 4. Assume that t>2 an n>t 1. If there exists a CWBD in Λ t,n,t which is uniform on perios, then it is universally optimal for the estimation of irect as well as carry-over effects uner moel (2.4) over the collection of equireplicate esigns in Λ t,n,t. For the two moels with subject effects, we now show optimality over a broaer class of esigns than in Theorem 4. Theorem 1 shows that if is a CWBD which is not a CBD, then we can make larger CWBDs by juxtaposing with one or more CBDs only if has Type I. Therefore, we restrict attention to the case that k =(t 1)/2, where t 3 mo 4, an n is an o multiple of (t 1)/2. It follows from Theorem 2 that a uniform CWBD which is not a CBD can only exist if, aitionally, n is an o multiple of t(t 1)/2. For such esign parameters, λ = n/(t 1)+1/2. We enote by n iu the number of times that treatment i appears in the uth subject (T U = F U =(n iu)), an by r i the number of times that treatment i appears in the esign. As shown by e.g., Kunert (1984b), if g = 2 or g = 3 then t tr C (g) r i 1 t n t t n 2 iu t (s ij (1/t) n n iun ju ) 2. (4.1) r j i=1 i=1 i=1 j=1 We begin with a technical lemma, an omit the straightforwar proof. Lemma 2. If x 1,x 2,...,x b satisfy b i=1 x i = c then b i=1 x2 i c2 /b. Proposition 2. If j is a treatment in esign in Λ t,n,t then t ( s ij 1 n ) 2 rj 2 n iu n ju t t(t 1). i=1

9 CIRCULAR WEAKLY BALANCED DESIGNS 9 Proof. For all competing esigns all s jj = 0. Therefore, t (s ij 1 n n iu n ju )= 1 n n 2 ju t t + (s ij 1 t i=1 i =j It follows that (s ij 1 t i =j n n iu n ju )= 1 t Applying Lemma 2, we conclue that t (s ij 1 n n iu n ju ) 2 = 1 ( n t t 2 i=1 1 t 2 ( n = n 2 ju n 2 ju 1 ( n t(t 1) n n 2 ju. ) 2 + (s ij 1 t i =j n n iu n ju ). ) ( n + t 1 t 2 n 2 ju n n iu n ju ) 2 ) 2 rj 2 t(t 1). Proposition 3. For a esign Λ t,n,t efine a = t n i=1 max{n iu 1, 0}. Then t n n 2 iu nt +2a. i=1 Proof. For 1 i t an 1 u n, efine e iu = n iu 1. Then all e iu are integers an, therefore, e 2 iu e iu. Since t n i=1 n iu = nt, we have eiu = 0 an, since the sum of all positive e iu equals a, we conclue that eiu =2a. In all, we get t n n 2 iu = i=1 t i=1 = nt +2 n (e iu + 1) 2 t i=1 n e iu + t i=1 n 2 ju ) 2 n e 2 iu nt +2a. We immeiately get a first boun for the trace of the information matrix which epens on a. Proposition 4. For any esign Λ t,n,t an g =2, 3 we have tr C (g) n(t 1 1 t 1 ) 2a t. Proof. The boun is well-known; it was use by Kunert (1984a,b). If a is small, we get a sharper boun, erive in the next proposition.

10 10 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Proposition 5. If t 5 an esign has a < (t 1)/2 then, for g =2, 3, ( tr C (g) n t 1 1 ) 2a ( t 1 (t 2a ) t 1 t 4n 2a ). nt Proof. There are at most 2a of the n iu not equal to 1. Since a < (t 1)/2, we conclue that there must be at least t 2a treatments j such that n ju =1 for 1 u n. Define J as the set of all such treatments. Assume without loss of generality that treatment t is in J. Then r t = n. Since we consier the circular neighbour structure, t i=1 s it = r t = n. Without loss of generality we can assume that the treatments are labelle in such a way that s 1t s 2t s,t 1,t. Recall that 2k = t 1 an λ = n/(t 1)+1/2. If s kt λ 1 then t 1 i=k+1 s it ks kt = k(λ 1) = (n k)/2. Since t 1 i=1 s it = n, this implies that k i=1 s it (n+k)/2. Otherwise, s kt λ an so k i=1 s it kλ =(n+k)/2 again. Hence Now put c = k i=1 (s it r i /t). The efinition of a gives k i=1 r i kn+a. c = i=1 k s it 1 t i=1 k i=1 r i n + k 2 1 ( ) (t 1)n + a t 2 It follows that c > n/(2t) because a <kan t>2. Furthermore, t ( 0= s it r ) t 1 ( i = c+ s it r ) i +s tt r t 1 ( t = c+ t t t i=1 i=k+1 = n 2t + k 2 a t. (4.2) i=k+1 an therefore t 1 i=k+1 (s it r i /t) =n/t c. Then Lemma 2 gives t ( s it r ) i 2 n 2 t t [ ( n ) ] 2 c 2 + k t c. i=1 s it r i t Since c > n/(2t), this boun is increasing in c. Therefore (4.2) gives t ( s it r ) i 2 n 2 t t ( n 2 k 4t 2 + k2 4 ka ) + a2 t t 2 n2 t(t 1) + k 2 2a t. Since n tu = 1 for 1 i n, this shows that t ( s it 1 n ) 2 t ( n iu n tu = s it r i t t i=1 i=1 ) 2 n 2 t(t 1) + k 2 2a t. ) n t The same boun applies when treatment t is replace by any treatment j in J. For all other treatments j, we use the boun in Proposition 2. Inserting these, an the boun in Proposition 3, into (4.1), we get, for g = 2, 3,

11 tr C (g) CIRCULAR WEAKLY BALANCED DESIGNS 11 nt 1 t (nt +2a ) = nt 1 t (nt +2a ) j J t j=1 r j t(t 1) j J ( r j t 1 t(t 1) J 4n ( n t(t 1) + t 1 ) 4n 2a nt ) 2a, nt where we have use the fact that r j = n for all j J, an where J is the number of elements of J. Due to the restrictions that a < (t 1)/2 an t 5, we observe that (t 1)/(4n) 2a /(nt) > (t 1)(t 4)/(4nt) > 0. Since J t 2a, it follows that, for g =2, 3, tr C (g) nt 1 t (nt +2a ) n ( t 1 t 1 (t 2a ) 4n 2a ), nt which implies the esire inequality. Now we can prove our main optimality result. Theorem 5. Assume that t 5 an that n t(t 1)/2. Assume that t is o an that n is an o multiple of (t 1)/2. If is a uniform CWBD in Λ t,n,t then is universally optimal for the estimation of irect as well as carry-over effects over the esigns in Λ t,n,t uner moel (2.2). If is a CWBD in Λ t,n,t which is uniform on subjects, then is universally optimal for the estimation of irect as well as carry-over effects over the esigns in Λ t,n,t uner moel (2.3). Proof. If the esign has a = 0, we get from Proposition 5 that, for g =2, 3, ( tr C (g) n t 1 1 ) t(t 1) t 1 4n, which is the trace of the information matrix of the CWBD. Consiering the simple boun erive in Proposition 4, we see that any esign Λ t,n,t can only perform better than if t(t 1) 4n 2a t. Since we restrict to the case n t(t 1)/2, the left-han sie is less than or equal to 1/2. If, however, a (t 1)/2, then the right-han sie is at least (t 1)/t > 1/2. Therefore, we only have to consier esigns with a < (t 1)/2 an the boun in Proposition 5 applies. Taking the erivative of ( f(a) =n t 1 1 ) 2a ( t 1 t 1 t (t 2a) 4n 2a ) nt

12 12 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ with respect to a, we get f (a)= 2 t +2 ( t 1 4n = 2 t + 2(t 1) 4n 2a ) ( (t 2a) 2 ) nt nt + 2t nt 4a nt 4a nt 8n +2t(t 1)+8t 32a = 4nt 4t(t 1)+2t(t 1)+8t = 4nt (t 1)+4 2n 0. This, however, implies that the boun from Proposition 5 is largest for a = 0, an for any esign Λ t,n,t we have tr C (g) tr C (g), for g =2, Constructions In this section we suppose that is a CWBD in D t,n,t which is not a CBD. For each type of CWBD, we give constructions for a suitable matrix A an then search for a esign with A = A. By Theorem 2, only Section 5.1 inclues uniform CWBDs Designs of Type I For a esign of Type I, we have t 3 mo 4 an k =(t 1)/2. We nee a t t matrix A which has zero entries on the iagonal, k entries equal to 1 in each row an column, an all other entries zero; it must also satisfy (a) A + A = J t I t an (b) A A = φi t + ξj t with φ =(t + 1)/4 an ξ =(t 3)/4. The matrix A can be regare as the ajacency matrix of a irecte graph Γ on t vertices: there is an arc from vertex i to vertex j if an only if A ij = 1. This irecte graph is calle a oubly regular tournament precisely when the matrix A satisfies the foregoing conitions, see Rei an Brown (1972). For a esign which is a CWBD, is uniform on subjects, an has λ = 1, we nee a ecomposition of a oubly regular tournament Γ into Hamiltonian cycles. One construction of oubly regular tournaments uses finite fiels. If t is a power of an o prime then there is a finite fiel GF(t) of t elements. If t is prime then GF(t) is the same as Z t, which is the ring of integers moulo t. Let S be the set of non-zero squares in GF(t), an N the set of non-squares. If t 3mo4 then 1 N; in this case, if we label the vertices of Γ by the elements of GF(t) an efine the ajacency matrix A by putting A ij = 1 if an only if j i S, then Γ is a oubly regular tournament, see Lil an Nieerreiter (1997). By reversing all the eges of Γ, we obtain another oubly regular tournament, which can be mae irectly by using N in place of S.

13 CIRCULAR WEAKLY BALANCED DESIGNS 13 If t is itself prime, then there is an obvious Hamiltonian ecomposition of Γ: the circular sequences have the form (0, s, 2s,...,(t 1)s) for s in S. Construction 1. Suppose that t 3 mo 4 an t is prime with t>3. Put n =(t 1)/2. Label the t treatments an the t perios by the elements of Z t, an the n subjects by the elements of S. Define the esign by (l, u) =lu for l in Z t an u in S. Then is a CWBD which is uniform on the subjects with λ = 1. Example 1. When t = 7 we have S = {1, 2, 4}. We obtain the esign in Figure 1(a), where the entries are integers moulo 7. (In every figure, the rows enote perios an the columns enote subjects.) Example 2. When t = 11 we have S = {1, 3, 4, 5, 9}. This gives the esign in Figure 1(b), where the entries are integers moulo 11. For n>1, Construction 1 eals with t = 7, 11, 19, 23 an 31 for t<35. Suitable matrices A also exist for many other values of t. Rei an Brown (1972) showe that the (t + 1) (t + 1) matrix [ 1 1 t ] 1 t J t 2A is a skew-haamar matrix if an only if A is the ajacency matrix of a oubly regular tournament. Skew-Haamar matrices of orer t + 1 are conjecture to exist whenever t + 1 is ivisible by 4. This has been verifie for t < 187: see Craigen (1996). Rei an Brown (1972) give the following oubling construction. If A 1 is the ajacency matrix of a oubly regular tournament Γ 1 on t vertices an A 2 = A 1 0 t A 1 + I t 1 t 0 0 t A 1 1 t A 1, (5.1) then A 2 is the ajacency matrix of a oubly regular tournament Γ 2 on 2t +1 vertices. Example 3. Let t = 15. Take Γ 1 to be the oubly regular tournament use in Example 1. The oubling construction (5.1) gives the ajacency matrix A 2 of a oubly regular tournament Γ 2 on 15 vertices. Label the vertices, in orer, 0, 1, 2, 3, 4, 5, 6,, 0,1,2,3,4,5 an 6. For x in GF(7), there is an arc from to x an an arc from x to. For x an y in GF(7), there is an arc from x to y if y x N; an arc from x to y if x = y or y x S; an arc from x to y if y x S; an an arc from x to y if y x S.

14 14 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ (a) (b) (c) Figure 1. Three CWBDs for t treatments on n subjects in t perios which are uniform on the subjects: (a) t = 7 an n = 3; (b) t = 11 an n = 5; (c) t = 15 an n = 7. To fin a CWBD which is uniform on subjects, we use GAP (2014) to fin a irecte cycle ϕ of length 15 starting (, 0,...)inΓ 2 with the extra property that if i is any non-zero element of GF(7), then the cycles ϕ an ϕ + i have no arc in common. Here we use the conventions that if ϕ =(ϕ 1,...,ϕ 15 ) then ϕ + i =(ϕ 1 + i,...,ϕ 15 + i), where + i = an x + i =(x + i) for x an i in GF(7). GAP (2014) foun all such cycles. There are 120, an they come in groups of three because if ϕ is such a cycle an s Sthen sϕ is also such a cycle (here the convention is that sϕ =(sϕ 1,...,sϕ 15 ), where s = an s x =(sx) for s an x in GF(7)). For each such cycle ϕ, the collection of cycles ϕ, ϕ +1,..., ϕ + 6 gives a CWBD for 15 treatments on 7 subjects in 15 perios which is uniform on subjects an for which A = A 2. One of these is shown in Figure 1(c). Alternatively, the function FinHamiltonianCycles in Mathematica 9.0 can be use to fin a Hamiltonian ecomposition of Γ 2. For t = 3, Construction 1 gives a esign with n = 1 that is isconnecte. In orer to obtain a connecte CWBD which is not a CBD, we nee to use one of the sequences (0, 1, 2) an (0, 2, 1) twice, an the other one once. If esign is mae by Construction 1 then a uniform esign with t(t 1)/2 subjects may be obtaine by replacing the sequence ϕ for each subject by the sequences ϕ+i for all i in GF(t). However, this has the effect that S = ts, so is not a CWBD, because the off-iagonal entries of S inclue both 0 an

15 CIRCULAR WEAKLY BALANCED DESIGNS 15 t. Thus we nee a ifferent construction for uniform CWBDs. Again, we use GF(t), where t 3 mo 4. If x an y are both in S or N then xy S; if one is in S an the other in N then xy N: see Lil an Nieerreiter (1997). If ϕ is any sequence (ϕ 1,...,ϕ m ) of elements of GF(t), we enote by ϕ δ the sequence (ϕ 2 ϕ 1,ϕ 3 ϕ 2,...,ϕ m ϕ m 1,ϕ 1 ϕ m ) of successive circular ifferences in ϕ. Further, let f 0 (ϕ δ ), f S (ϕ δ ) an f N (ϕ δ ) be the number of entries of ϕ δ which are in {0}, S an N, respectively. Definition 1. Let ϕ be a sequence of length t whose entries are in GF(t), where t is a prime power congruent to 3 moulo 4. Then ϕ is beautiful if the entries in ϕ are all ifferent an f S (ϕ δ )=f N (ϕ δ ) ± 1. If all of the entries of ϕ are ifferent then f 0 (ϕ δ ) = 0. Thus if ϕ has length t then it is beautiful if an only if f S (ϕ δ ) {k, k +1}. Construction 2. Given a beautiful sequence ϕ =(ϕ 1,...,ϕ t ) of all the elements of GF(t), form the t(t 1)/2 sequences sϕ + i for all s in S an all i in GF(t). Create the esign by using each of these sequences for one subject. Theorem 6. Suppose t 3mo4an t is a prime power. If ϕ is beautiful then the esign given by Construction 2 is a uniform CWBD. Proof. The entries in ϕ are all ifferent, so the entries in sϕ + i are all ifferent for each value of s an i. Therefore each treatment occurs once on each subject, so is uniform on subjects an no treatment is precee by itself. For each fixe s in S, every element of GF(t) occurs once in each perio among the t sequences sϕ + i, as i varies in GF(t). Therefore is uniform. Consier perio j. Put ϕ δ j = v. Let i GF(t) an s S. Treatment i occurs in perio j of the sequence sϕ + i sϕ j. The treatment in perio j +1 of this sequence is sϕ j+1 + i sϕ j = i + sv. If v Sthen {sv : s S} = S, an so every orere pair of treatments of the the form (i, i + q), for i in GF(t) an q in S, occurs exactly once in perios j an j + 1. Otherwise, if v N then {sv : s S} = N, an so every orere pair of treatments of the the form (i, i + q), for i in GF(t) an q in N, occurs exactly once in perios j an j + 1. Thus if w i Sthen (i, w) occurs f S (ϕ δ ) times in the esign, while if w i N then (i, w) occurs f N (ϕ δ ) times. If ϕ is beautiful then the offiagonal entries of S are in {k, k +1} an A is the ajacency matrix of one of the oubly regular tournaments efine by S or N. Hence is a CWBD. Example 4. Let t = 7 an ϕ = (3, 1, 0, 2, 6, 4, 5), where the entries are the integers moulo 7. Then ϕ δ = (5, 6, 2, 4, 5, 1, 5). Here S = {1, 2, 4} an N = {3, 5, 6}, an so f S (ϕ δ )=3anf N (ϕ δ ) = 4. Thus ϕ is beautiful. Hence Construction 2 gives a uniform CWBD for 7 treatments on 21 subjects in 7 perios.

16 16 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Now let x be any primitive element of GF(t); that is, x is a generator of the cyclic group (GF(t) \{0}, ). The even powers of x constitute S, while the o powers constitute N. Let ψ be the sequence (1, x, x 2,...,x t 2 ). Then ψ contains each non-zero element of GF(t) exactly once. The entries in ψ δ are x 1, x(x 1),...,x t 3 (x 1) an 1 x t 2, which is x t 2 (x 1). These are again all the non-zero elements of GF(t) exactly once, an so f S (ψ δ )=f N (ψ δ )=k. Theorem 7. Let t be a prime power congruent to 3 moulo 4 with t>3. If x is a primitive element of GF(t) an ϕ is obtaine from ψ by replacing (1,x) with (x, 1, 0), then ϕ is beautiful. Proof. If t>3, the substitution removes 1 x 1, x 1 an x 2 x from ψ δ, an replaces them in ϕ δ by x x 1,1 x, 1 an x 2. None of these is zero if t>3. Now, 1 N an x 2 S. Since x N, one of x 1 an x(x 1) is in S an the other is in N. Since 1 x 1 =( x 1 )(1 x) an x 1 S, the entries 1 x 1 an 1 x are either both in S or both in N. Thus f S (ϕ δ )=f S (ψ δ )+1=k +1 if x x 1 S, while f S (ϕ δ )=f S (ψ δ )=k if x x 1 N. If t = 7 then 3 is a primitive element. The construction in Theorem 7 gives the beautiful sequence ϕ in Example 4. Theorems 6 7 show that there is a uniform CWBD for t treatments on t(t 1)/2 subjects in t perios whenever t is a prime power congruent to 3 moulo 4 an t>3. This covers t = 7, 11, 19, 23, 27 an 31 for t< Designs of Type II For a esign of Type II, we have n = k, where 2 k t 2. Also, conition (3.2) shows that t 1 ivies k(k 1). We nee a t t matrix A which has k entries equal to 1 in each row an column, an all other entries zero, in such a way that A A = φi t + ξj t with φ = k(t k)/(t 1) an ξ = k(k 1)/(t 1). The matrix A can be regare as the incience matrix of a symmetric balance incomplete-block esign (BIBD) : treatment i is in block j if an only if A ij = 1. Given such a esign, Hall s Marriage Theorem (Bailey (2008), Cameron (1994), Hall (1935)) shows that the treatments an blocks can be labelle in such a way that the iagonal entries of A are all zero. Now our strategy is to fin a known BIBD of the appropriate size, label its blocks in such a way that the iagonal entries of A are all zero, an then try to fin a CWBD which is uniform on subjects for which λ = 1 an S = A = A. The conition that t 1 ivies k(k 1) is not sufficient to guarantee the existence of a BIBD. The Bruck Ryser Chowla Theorem shows that some pairs (t, k) have no BIBD: see Cameron (1994). For t<35, the following pairs are exclue by this theorem: (22, 7), (22, 15), (29, 8), (29, 21), (34, 12) an (34, 22).

17 CIRCULAR WEAKLY BALANCED DESIGNS 17 Some BIBDs can be constructe from ifference sets, see Hall (1986). If (G, +) is a finite Abelian group an P G, then P is calle a ifference set if every non-zero element of G occurs equally often among the ifferences x y for x an y in P with x y. When G is the aitive group of GF(t) an t 3mo4 then S an N are both ifference sets. If P is a ifference set then so is its complement P, an so is the set P + i = {x + i : i P} for each i in G. If i/ P then P i is a ifference set that oes not contain 0. In particular, when t is a prime power an t 3 mo 4 then S 1 is a ifference set with k =(t + 1)/2 that oes not contain 0. To obtain a BIBD from the ifference set P, label the treatments an blocks by the elements of G, an put A ij = 1 if an only if j i P. As above, we can assume that 0 / P, an then the iagonal entries of A are all zero. Difference sets give a generalization of Construction 1. Construction 3. Suppose P is a ifference set of size k in Z t, that 0 / P, an that all elements of P are coprime to t. Label the t treatments an the t perios by the elements of Z t, an the k subjects by the elements of P. Define the esign by (l, u) =lu for l in Z t an u in P. Then is a CWBD which is uniform on subjects with λ = 1. Example 5. When t = 7 an k = 4 we have the ifference set S 1 ={2, 4, 5, 6}. Then Construction 3 gives a CWBD for 4 subjects which is uniform on subjects. Difference sets exist for many other values of t an k satisfying the ivisibility conitions, see Baumert (1971) an Table 2 of Filipiak an Markiewicz (2012). For example, when t = 13 then {1, 2, 5, 7} an {2, 3, 5, 7, 8, 9, 10, 11, 12} are both ifference sets in Z 13. Construction 3 gives CWBDs that are uniform on subjects, one for 4 subjects an one for 9 subjects. When t = 31, {1, 2, 4, 9, 13, 19} is a ifference set in Z 31. Thus Construction 3 gives CWBDs uniform on subjects, one for 6 subjects an one for 25 subjects. A result of Mann (1964) shows that there is no ifference set of size 9 or 16 for Z 25. Theorems of Laner (1983) rule out ifference sets of size k or t k for Z t when (t, k) is (16, 6), (27, 13) or (31, 10). There is a ifference set of size 8 for Z 15, but its elements are not all coprime to 15, so Construction 3 cannot be use. The same problem occurs for k = 5 an k = 16 when t = 21. If A is symmetric then it can also be regare as the ajacency matrix of an unirecte graph. If A A is completely symmetric then every pair of istinct vertices have the same number of common neighbours. Such graphs were stuie by Ruvalis (1971). If such a graph has a Hamiltonian ecomposition then using each cycle once in each irection gives a CWBD which is uniform on subjects.

18 18 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Example 6. The smallest such graph is the square lattice graph L 2 (4), which has 16 vertices an valency 6. Every pair of istinct vertices has exactly two common neighbours. The vertices form a 4 4 gri. There is an ege between i an j if i j but i an j are in the same row or i an j are in the same column. Label the vertices row by row, so that the first row is (1, 2, 3, 4), an so on. Let π be the permutation (2, 3, 4)(5, 9, 13)(6, 11, 16)(7, 12, 14)(8, 10, 15) of the vertices, which is an automorphism of L 2 (4). There is a Hamiltonian ecomposition of L 2 (4) which is invariant uner π. Using each of these cycles in both irections gives the esign in Figure 2(a). The Shrikhane graph is another graph with 16 vertices, valency 6, an the common-neighbour property, see Seiel (1968). Using GAP (2014), we foun that it has a large number of Hamiltonian ecompositions. Each gives a CWBD that cannot be obtaine from the one in Figure 2(a) by renaming treatments. Example 7. The Clebsch graph Ω is another such graph with 16 vertices, see Seiel (1968). It has valency 10, an every pair of istinct vertices has exactly 6 common neighbours. The vertices are the vectors of length 5 over GF(2) of even weight (equivalently, the treatments in the factorial esign with efining contrast ABCDE = I); two vertices are joine if they iffer in precisely two positions. The permutation π taking (x 1,x 2,x 3,x 4,x 5 ) to (x 2,x 3,x 4,x 5,x 1 ) is an automorphism of Ω. Using GAP (2014), we foun a very large number of Hamiltonian ecompositions of Ω which are invariant uner π (as in Example 6, it is sufficient to fin a single Hamiltonian cycle which has no eges in common with any of its images uner powers of π). For any one of these ecompositions, using each cycle in both irections gives the require CWBD. One is shown in Figure 2(b), where vertex (x 1,x 2,x 3,x 4,x 5 ) is ientifie as the integer 8x 1 +4x 2 +2x 3 + x For t>16, Ruvalis (1971) showe that the smallest value of t for which there exists a graph with the common-neighbour property is t = Designs of Type III For a esign of Type III, we consier A to be the ajacency matrix of a irecte graph Ξ. Now λ 1 an conition (3.3) is satisfie. However, neither A A nor A + A is completely symmetric, so at most one value of λ is possible for any given irecte graph Ξ. As in Section 5.1, we buil larger matrices from smaller ones. Let A 1 be the ajacency matrix of a oubly regular tournament Γ on r vertices, where r =4q+3. Let t = mr, where m 2, an put A 2 = J m (I r +A 1 ) I t. Then A 2 +A 2 = J m (J r +I r ) 2I t an A 2 A 2 =(mq+m 1)J m (J r +I r )+I t. Thus A 2 satisfies conition (3.3) with λ = m(q + 1), k =2m(q + 1) 1, an n = m 2 (4q + 3)(q + 1) m(3q + 2).

19 CIRCULAR WEAKLY BALANCED DESIGNS (a) (b) Figure 2. Two CWBDs for 16 treatments on n subjects in 16 perios which are uniform on the subjects: (a) n = 6; (b) n = 10. Example 8. When q = 0 we may let A 1 be the ajacency matrix of the oubly regular tournament efine by S in GF(3). When m = 2 then t = 6, n = 8, an A 2 is A for the esign in Example 4.4 of Filipiak an Markiewicz (2012) with its treatments written in the orer 1, 3, 5, 6, 2, 4. Babai an Cameron (2000) give a oubling construction for what they call an S-igraph. Let A 1 be the ajacency matrix of a oubly regular tournament Γ on r vertices, where r =4q + 3. Put A 2 = 0 1 r 0 0 r 0 r A 1 1 r A r 0 1 r 1 r A 1 0 r A 1 an I 8q =(J 2 I 2 ) I 4q. Then the S-igraph Ξ has ajacency matrix A 2. Now, A 2 + A 2 = J 8q I 8q I 8q an A 2 A 2 = (4q + 3)I 8q + (2q + 1)(J 8q I 8q I 8q ). Thus A 2 satisfies conition (3.3) with t = 8(q + 1), k =4q + 3, λ = 2(q + 1), an n = 16q q Example 9. If q = 0 an an A 1 is as in Example 8, then this oubling construction gives a matrix A 2 which, after relabelling of the treatments, is the matrix A for the esign in Example 4.3 of Filipiak an Markiewicz (2012). If t/ {4, 6}, then there is a CBD for t treatments with n = t 1 an λ = 1, see Tillson (1980). Examples for t = 3, t = 5, an 7 t 16 are given by

20 20 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Azaïs, Bailey an Mono (1993). When t = 4 or t = 6 then there is a CBD with n = 2(t 1), an there is no CWBD for t = 4 with n 5. Thus Type III esigns o not give a CWBD with fewer subjects than a CBD unless t = 6. However, in a situation like Example 9, the CWBD with 10 subjects gives lower variances of all treatment estimators than the CBD with 7 subjects, so there may still be some interest in constructing such esigns. The methos in this section give two possible ways of constructing the matrix A. A computer search shoul quickly fin whether the corresponing igraph Ξ has a Hamiltonian ecomposition. If so, this can be juxtapose with λ 1 copies of the relevant CBD to obtain a CWBD. Acknowlegements This paper was starte in the Isaac Newton Institute for Mathematical Sciences in Cambrige, UK, uring the 2011 programme on the Design an Analysis of Experiments. This research was partially supporte by the National Science Center Grant DEC-2011/01/B/ST1/01413 (K. Filipiak an A. Markiewicz) an by the Collaborative Research Center Statistical moeling of nonlinear ynamic processes (SFB 823, Teilprojekt C2) of the German Research Founation (J. Kunert). Part of the work was one while R. A. Bailey an P. J. Cameron hel Hoo Fellowships at the University of Aucklan in References Azaïs, J.-M., Bailey, R. A. an Mono, H. (1993). A catalogue of efficient neighbour-esigns with borer plots. Biometrics 49, Babai, L. an Cameron, P. J. (2000). Automorphisms an enumeration of switching classes of tournaments. Electron. J. Combin. 7, article #R38 (25pp.) Bailey, R. A. (2008). Design of Comparative Experiments. Cambrige University Press, Cambrige. Bailey, R. A. an Druilhet, P. (2004). Optimality of neighbour balance esigns for total effects. Ann. Statist. 32, Baumert, L. D. (1971). Cyclic Difference Sets. Springer-Verlag, Berlin. Cameron, P. J. (1994). Combinatorics: Topics, Techniques, Algorithms. Cambrige University Press, Cambrige. Cheng, C.-S. an Wu, C.-F. (1980). Balance repeate measurements esigns. Ann. Statist. 8, Craigen, R. (1996). Chapter IV.24 Haamar matrices an esigns. In The CRC Hanbook of Combinatorial Designs (Eite by Charles J. Colbourn an Jeffrey H. Dinitz), CRC Press, Boca Raton. Druilhet, P. (1999). Optimality of circular neighbor balance esigns. J. Statist. Plann. Inference 81,

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22 22 R. A. BAILEY, P. J. CAMERON, K. FILIPIAK, J. KUNERT AND A. MARKIEWICZ Rei, K. B. an Brown, E. (1972). Doubly regular tournaments are equivalent to skew Haamar matrices. J. Combin. Theory Ser. A 12, Ruvalis, A. (1971). (v,k,λ)-graphs an polarities of (v,k,λ)-esigns. Math. Z. 120, Seiel, J. J. (1968). Strongly regular graphs with ( 1, 1, 0) ajacency matrix having eigenvalue 3. Linear Algebra Appl. 1, Stufken, J. (1991). Some families of optimal an efficient repeate measurements esigns. J. Statist. Plann. Inference 27, The GAP Group (2014). GAP Groups, Algorithms, an Programming, Version (http: // Tillson, T. W. (1980). A Hamiltonian ecomposition of K 2m, 2m 8. J. Combin. Theory Ser. B 29, Wilkinson, G. N., Eckert, R., Hancock, T. W. an Mayo, O. (1983). Nearest neighbour (NN) analysis of fiel experiments. J. R. Stat. Soc. B 45, School of Mathematics an Statistics, University of St Anrews, UK School of Mathematical Sciences, Queen Mary, University of Lonon, UK. rab24@st-anrews.ac.uk School of Mathematics an Statistics, University of St Anrews, UK. School of Mathematical Sciences, Queen Mary, University of Lonon, UK. pjc20@st-anrews.ac.uk Institute of Mathematics, Poznań University of Technology, Polan. katarzyna.filipiak@put.poznan.pl Department of Statistics, TU Dortmun University, Germany. joachim.kunert@tu-ortmun.e Department of Mathematical an Statistical Methos, Poznań University of Life Sciences, Polan. amark@up.poznan.pl (Receive February 2015; accepte January 2016)