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Journal of Statistical Planning and Inference 106 (2002) 135 – 157 www.elsevier.com/locate/jspi Optimality and constructions of incomplete split-block designs Kazuhiro Ozawaa; ∗ , Masakazu Jimbob , Sanpei Kageyamac , Stanis law Mejzad a Department of Nursing, Gifu College of Nursing, 3047-1 Egira Hashima, Gifu 501-6295, Japan b Department of Mathematics, Keio University, Yokohama 223-8522, Japan c Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739-8524, Japan d Department of Mathematical and Statistical Methods, Agricultural University of PoznaÃ , Poland n Received 18 December 1998; accepted 29 June 2000 To the memory of Professor Sumiyasu Yamamoto Abstract A su cient condition for an incomplete split-block design to be universally optimal is given. Optimal properties are examined under two linear models, i.e., with interaction e ects and with- out these e ects. Furthermore, some methods of constructing universally optimal incomplete split-block designs are presented. c 2002 Elsevier Science B.V. All rights reserved. MSC: 05B05; 62K10 Keywords: Split-block design; Universally optimum; Balanced incomplete block design; Balanced incomplete split-block design 1. Introduction Many two-factorial experiments are laid out in the designs with split-units. This type of experiment is especially useful in agricultural (ÿeld) experiments. A split-plot design and a split-block design are two types of designs with split-units that are of great interest from a theoretical point of view as well as practical applications. In many practical situations the experimental material is limited and then only some kind of ∗ Corresponding author. E-mail addresses: ozawa@gifu-cn.ac.jp (K. Ozawa), jimbo@math.keio.ac.jp (M. Jimbo), ksanpei@sed. hiroshima-u.ac.jp (S. Kageyama), smejza@owl.au.poznan.pl (S. Mejza). 0378-3758/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 2 0 9 - 4 136 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Fig. 1. Experiment in blocks. incomplete (non-orthogonal) designs with split-units can be applied. Incomplete split- plot designs have been discussed in the literature, e.g., Bhargava and Shah (1975), Mejza and Mejza (1996), Rees (1969) and Robinson (1970). On the other hand, some problems connected with planning and analysis of experiments carried out in an incom- plete split-block design were considered by Hering and Mejza (1997). In a split-block design, we consider three types of randomizations, that is, within rows, within columns and within blocks. By combining three error terms into one, we take into account linear models with correlations between units. In this paper, optimal properties of the incomplete split-block design are examined under two linear models of observations, i.e., with or without interaction e ects. More- over, some constructions that lead to universal optimum designs are given. Let us consider a two-factorial experiment in which the ÿrst factor A occurs on v1 levels (treatments) A1 ; : : : ; Av1 and the second factor C occurs on v2 levels (treatments) as C1 ; : : : ; Cv2 . In Fig. 1, the split-block design is a set of b blocks each of which is a k1 ×k2 matrix. Assume that the treatments of the factor A occur on the rows and the treatments of the factor C occur on the columns. Let Ai(t; r) denote a treatment Ai of the factor A applied to the rth row of a block Bt , while Cj(t; c) denotes a treatment Cj of the factor C applied to the cth column of a block Bt . Furthermore, the (t; r; c)-plot means the (r; c)th plot in a block Bt . Then Bt is represented by Bt = {Ai(t; 1) ; : : : ; Ai(t; k1 ) | Cj(t; 1) ; : : : ; Cj(t; k2 ) } (see Fig. 2). For convenience, this is simply denoted by Bt = {i(t; 1); : : : ; i(t; k1 ) | j(t; 1); : : : ; j(t; k2 )}: Let V1 = {1; : : : ; v1 } be the set of indices of v1 treatments on A, V2 = {1; : : : ; v2 } be another set of indices of v2 treatments on C and B = {B1 ; : : : ; Bb } be a collection of (k1 + k2 )-subsets, called superblocks, of V1 ∪ V2 . If each superblock is divided into a k1 -subset of V1 and a k2 -subset of V2 , then a binary design (V1 ; V2 ; B) is called an incomplete split-block design. In this paper, the following two linear models for an observation ytrc of the (t; r; c)- plot will be considered. (Model I) ytrc = i(t; r) + j(t; c) + ÿt + trc , (Model II) ytrc = i(t; r) + j(t; c) + ( )i(t; r) j(t; c) + ÿt + trc , K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 137 Fig. 2. Treatments in block Bt . where denotes the error of the (t; r; c)-plot, E( trc trc ) = 0 and 1 if t = t ; r = r and c = c ; 1 if t = t ; r = r and c = c ; 2 Cov( trc ; t r c ) = × 2 if t = t ; r = r and c = c ; (1) 3 if t = t ; r = r and c = c ; 0 otherwise: Furthermore, i is a main e ect of Ai ; j is a main e ect of Cj , ( )ij is an interaction e ect between Ai and Cj , and ÿt is a block e ect of Bt . Without loss of generality, it can be assumed that v1 v2 i = 0; j = 0; i=1 j=1 v1 v2 ( )ij = 0; ( )ij = 0: (2) i=1 j=1 A lexicographical order of observations is used, i.e., a (t; r; c)-plot has the number k1 k2 (t − 1) + k2 (r − 1) + c and the interaction e ect ( )ij has the number (i − 1)v2 + j. Four matrices X1 = (x1 (t; r; c; i)), X2 = (x2 (t; r; c; j)), X12 = (x12 (t; r; c; i; j)) and X3 = (x3 (t; r; c; t )) are deÿned as follows: 1 if a treatment Ai is applied to a (t; r; c)-plot; x1 (t; r; c; i) = 0 otherwise; 1 if a treatment Cj is applied to a (t; r; c)-plot; x2 (t; r; c; j) = 0 otherwise; 1 if treatments Ai and Cj are applied to a (t; r; c)-plot; x12 (t; r; c; i; j) = 0 otherwise; 1 if t = t ; x3 (t; r; c; t ) = (3) 0 otherwise: 138 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Here X1 ; X2 ; X12 and X3 are of sizes bk1 k2 × v1 , bk1 k2 × v2 , bk1 k2 × v1 v2 and bk1 k2 × b, respectively. Models I and II can be rewritten as follows: (Model I) y = X1 + X2 + X3 ÿ + ; E( ) = 0; Cov( ) = , (Model II) y = X1 + X2 + X12 ( ) + X3 ÿ + ; E( ) = 0; Cov( ) = , where y = (y1 ; : : : ; ybk1 k2 ) , = ( 1 ; : : : ; v1 ) , = ( 1 ; : : : ; v2 ) , ( ) = (( )11 ; : : :, ( )v1 v2 ) , ÿ = (ÿ1 ; : : : ; ÿb ) , = ( 1 ; : : : ; bk1 k2 ) (x denotes the transpose of x), and is the covariance matrix of given by (1). Throughout this paper, is assumed to be positive deÿnite. Example 1.1. Suppose that there are three blocks of size 2 × 3 and the ÿrst factor A has three treatments A1 ; A2 ; A3 ; while the second factor C has four treatments C1 ; C2 ; C3 ; C4 . For the design, the matrices X1 , X2 , X3 and X12 are given as follows: 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 X1 = 0 0 1 ; X2 = 1 ; X3 = 0 1 0; 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 139 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 X12 = 0 : 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2. Estimation of main e ects under Model I without interaction e ects We consider some combinatorial property of optimal designs for estimating main e ects under Model I (without interaction e ects). For the design matrices X1 ; X2 and X3 , let X = (X1 : X2 : X3 ) and let Â = ; ÿ where ; are the main e ects and ÿ is the blocks e ects. Then Model I can be represented by y = XÂ + : ˆ The generalized least squares estimator (GLSE) Â of Â is given by a solution of the normal equation −1 −1 X XÂ = X y; 140 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 that is, the GLSEs of ; and ÿ are obtained as solutions of the following equations: X1 −1 X1 ˆ + X 1 −1 X2 ˆ + X 1 −1 ˆ X3 ÿ = X 1 −1 y; X −1 X1 ˆ + X2 −1 X2 ˆ + X 2 −1 ˆ X3 ÿ = X 2 −1 y; (4) 2 ˆ X3 −1 X1 ˆ + X3 −1 X2 ˆ + X 3 −1 X3 ÿ = X 3 −1 y: Here, is expressed as 2 2 = = {(1 − 1 − 2 + 3 )Ibk1 k2 +( 1 − 3 )R +( 2 − 3 )C + 3 RC}; where I! is the identity matrix of order !, R = Ib ⊗ Ik1 ⊗ Jk2 and C = Ib ⊗ Jk1 ⊗ Ik2 . Here ⊗ denotes the Kronecker product and J! denotes an ! × ! matrix with all its elements one. Two matrices I! and J! are simply denoted by I and J in most places. For any 1 6 t 6 b; 2 6 i 6 k1 and 2 6 j 6 k2 ; it follows that (t) (1) (1) ( j) (t) (i) (1) ( j) eb ⊗ ek1 ⊗ (ek2 − ek2 ) + eb ⊗ ek1 ⊗ (−ek2 + ek2 ); (t) (1) (i) (t) (1) ( j) (t) eb ⊗ (−ek1 + ek1 ) ⊗ 1k2 ; eb ⊗ 1k1 ⊗ (−ek2 + ek2 ); eb ⊗ 1k1 ⊗ 1k2 are eigenvectors of each corresponding to eigenvalues Á0 = 1 − 1 − 2 + 3 ¿ 0; Á1 = 1 − 2 + (k2 − 1)( 1 − 3 ) ¿ 0; (5) Á2 = 1 − 1 + (k1 − 1)( 2 − 3 ) ¿ 0; (6) Á3 = 1 + (k2 − 1) 1 + (k1 − 1) 2 + (k1 − 1)(k2 − 1) 3 ¿ 0; (i) where 1m is an m-dimensional all-one column vector and en is an n-dimensional column vector such that the ith element equals 1 while others are all 0. Recall that is positive deÿnite. Some calculation can show that −1 1 −1 1 = 2 = 2 { 0I + 1R + 2C + 3 RC}; where 1 0( 1 − 3) 0( 2 − 3) 0 = ; 1 =− ; 2 =− ; Á0 Á1 Á2 1{ 2 + (k2 − 1) 3 } + 2{ 1 + (k1 − 1) 3 } + 0 3 3 =− : Á3 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 141 Then, 1 1 Á1 = ; Á2 = : 0 + k2 1 0 + k1 2 Furthermore, it is easy to check that RC = CR = X3 X3 = Ib ⊗ Jk1 ⊗ Jk2 ; (7) RX1 = k2 X1 ; (8) CX2 = k1 X2 ; (9) RX3 = k2 X3 ; CX3 = k1 X3 ; −1 RC = RC; (10) X 3 X 3 = k 1 k2 I hold, where = 0 + k2 1 + k1 2 + k1 k2 3 . Therefore, it holds that −1 X1 X2 = X1 X2 (= K = (kij ); say); (11) −1 X1 X3 = X1 X3 (= k2 M = k2 (mit ); say); −1 X2 X3 = X2 X3 (= k1 N = k1 (njt ); say); −1 X3 X3 = k1 k2 I; where kij is the number of blocks corresponding to both Ai and Cj , mit is the number of rows corresponding to Ai in a block Bt , and njt is the number of columns corresponding to Cj in a block Bt . Thus (4) is expressed as follows: X1 −1 ˆ X1 ˆ + K ˆ + k2 M ÿ = X1 −1 y; ˆ K ˆ + X2 −1 X2 ˆ + k1 N ÿ = X2 −1 y; (12) k2 M ˆ + k1 N ˆ + k1 k2 ÿˆ = X3 −1 y: ˆ Hence ÿ is given by ˆ 1 −1 ÿ= (X y − k2 M ˆ − k1 N ˆ): k 1 k2 3 Then it follows from the ÿrst equation of (12) that −1 k2 −1 1 −1 X1 X1 − MM ˆ + ( K − MN ) ˆ = X1 − MX3 y: (13) k1 k1 142 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Here, by (7) – (9), (10) and (11), the coe cient matrix of ˆ in (13) is further reduced to 1 K − MN = X1 −1 X2 − X −1 X3 X3 −1 X2 k 1 k2 1 = X 1 X2 − X1 RCX2 k1 k2 = X 1 X 2 − X 1 X2 = O: (14) Therefore the normal equation (13) of ˆ is written as follows: k2 1 X1 −1 X1 − MM ˆ = X1 −1 − MX3 −1 y: (15) k1 k1 Now for a design matrix X = (X1 : X2 : X3 ) of a split-block design, the C-matrix is deÿned by k2 C (X ) = X1 −1 X1 − MM : k1 It is easy to see that C (X )1v1 = 0 and rank(C (X )) 6 v1 − 1. Now assume that rank(C (X )) = v1 − 1. Let C − (X ) be a g-inverse of the C-matrix C (X ) such that C − (X )C (X )C − (X ) = C − (X ). By (15), 1 ˆ = C − (X ) X1 −1 − MX3 −1 y: k1 Moreover, it is obvious that Cov( ˆ) = 2 C − (X ) holds. Similarly, assuming that rank(C (X )) = v2 − 1, ˆ is given by 1 ˆ = C − (X ) X2 −1 − NX3 −1 y; k2 where −1 k1 C (X ) = X2 X2 − NN k2 and C − (X ) is a g-inverse of the C-matrix C (X ) such that C − (X )C (X )C − (X ) = C − (X ). Then Cov( ˆ) = 2 C − (X ). 2.1. Criteria of optimality Denote by the set of design matrices X in which all elementary contrasts of main e ects are estimable by GLSE, that is, rank(C (X )) = v1 − 1, and M is the set of C (X ) for any X ∈ . Let M be the convex hull of M and be the set of functions on M such that (i) is convex, (ii) (C) ¿ (gC) for C ∈ M and for any g ¿ 1, and (iii) for any permutation matrix P and C ∈ M , (P CP) = (C) holds. K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 143 The design matrix X ∗ is said to be universally optimum relative to if (C (X ∗ )) = min (C (X )) X∈ holds for all functions ∈ satisfying (i) – (iii). It is known that the universal optimum criterion includes those of A-, D- and E-optimality as its special cases. Proposition 2.1 (Kiefer, 1975). The design matrix X ∗ ∈ is universally optimum relative to if C (X ∗ ) is completely symmetric and tr(C (X ∗ ))=maxX ∈ tr(C (X )); where tr(C) is the trace of C. 2.2. Universally optimum designs for GLSE under Model I For an incomplete split-block design (V1 ; V2 ; B), let B1 = {Bt ∩ V1 | Bt ∈ B} and B2 = {Bt ∩ V2 | Bt ∈ B}: For X = (X1 : X2 : X3 ) of the incomplete split-block design (V1 ; V2 ; B), the incidence matrix M of a binary design (V1 ; B1 ) is given by 1 M= X X3 : k2 1 −2 If MM (=k2 X1 X3 X3 X1 )=(r1 − 1 )I + 1 J holds, where r1 is the number of replications for each treatment in V1 , then a binary design (V1 ; B1 ) is called a balanced incomplete block design, denoted by BIBD(v1 ; k1 ; 1 ), and then X1 X1 = k2 r1 I . Now assume that (V1 ; B1 ) is a BIBD(v1 ; k1 ; 1 ) and let X ∗ = (X1∗ : X2∗ : X3∗ ) be a design matrix of the incomplete split-block design (V1 ; V2 ; B). By (8), it holds that X1∗ −1 X1∗ = X1∗ ( 0 I + 1R + 2C + ∗ 3 RC)X1 k 2 r1 ∗ = I +( 2 + 3 k2 )X1 CX1∗ Á1 and 1 ∗ ∗ ∗ ∗ 1 MM = X X X X = X1∗ CX1∗ : k2 1 3 3 1 2 k2 Therefore, by X1∗ CX1∗ = k2 {(r1 − 1 )I + 1 J } and by the relation of a BIBD, ∗ 1 (v1 − 1) = r1 (k1 − 1), C (X ) can be rewritten as k2 r1 C (X ∗ ) = I +( 2 + ∗ 3 k2 )X1 CX1∗ − X1∗ CX1∗ Á1 k1 k2 r1 = I + k2 2 + 3 k2 − {(r1 − 1 )I + 1J } Á1 k1 k2 v1 1 1 = I− J ; k1 Á 1 v1 144 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 which implies that C (X ∗ ) is completely symmetric. While, for any design matrix X = (X1 : X2 : X3 ), it is easily checked that bk2 (k1 − 1) tr(C (X )) = = constant; Á1 which has the same value as tr(C (X ∗ )) = k2 1 v1 (v1 − 1)=(k1 Á1 ), since 1 v1 (v1 − 1) = bk1 (k1 − 1) holds. Thus, by Proposition 2.1, X ∗ is universally optimum relative to . The result with respect to can be obtained similarly. Thus Proposition 2.1 has established the following: Theorem 2.1. (i) Suppose ={X | rank(C (X ))=v1 −1} and (V1 ; V2 ; B) is an incom- plete split-block design with a design matrix X ∗ ∈ . If (V1 ; B1 ) is a BIBD(v1 ; k1 ; 1 ); then X ∗ is universally optimum relative to . (ii) Suppose = {X | rank(C (X )) = v2 − 1} and (V1 ; V2 ; B) is an incomplete split-block design with a design matrix X ∗ ∈ . If (V2 ; B2 ) is a BIBD(v2 ; k2 ; 2 ), then X ∗ is universally optimum relative to . In the considerations above the BIB structure of the association matrix MM (NN ) was used. It is also possible to consider a wider class of block designs satisfying the property that MM (or NN ) = (r − )I + J . As an example of such a class let us consider a completely (orthogonal) randomized block design, i.e., in which r = . Then immediately Theorem 2.1 yields the following: Corollary 2.1. (i) Suppose (V1 ; V2 ; B) is an incomplete split-block design with a design matrix X ∗ ∈ . If (V1 ; B1 ) is a completely randomized block design for row treatments; then X ∗ is universally optimum relative to . (ii) Suppose (V1 ; V2 ; B) is an incomplete split-block design with a design matrix X ∗ ∈ . If (V2 ; B2 ) is a completely randomized block design for column treatments, then X ∗ is universally optimum relative to . The above corollary is rather obvious but to have consideration complete it is worth stating, since such designs are often used in a practical setting (see Hering and Mejza, 1997). 3. Estimation of interaction e ects under Model II A combinatorial property of optimal designs will be considered for estimating inter- action e ects under Model II (with interaction e ects). Assume that (V1 ; B1 ) and (V2 ; B2 ) are BIBD(v1 ; k1 ; 1 ) and BIBD(v2 ; k2 ; 2 ), respec- tively. The normal equation under Model II is given as follows: ˆ X1 −1 X1 ˆ + X1 −1 X2 ˆ + X1 −1 X12 ( ) + X1 −1 X3 ÿ = X1 −1 y; X −1 X ˆ + X −1 X ˆ + X −1 X ( ) + X −1 X ÿ = X −1 y; ˆ 2 1 2 2 2 12 2 3 2 X −1 X1 ˆ + X −1 X2 ˆ + X −1 X12 ( ) + X −1 X3 ÿ = X −1 y; 12 ˆ 12 12 12 12 ˆ X3 −1 X1 ˆ + X3 −1 X2 ˆ + X3 −1 X12 ( ) + X3 −1 X3 ÿ = X3 −1 y: K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 145 In a manner similar to the argument under Model I, this equation can be repre- sented by ˆ X1 −1 X1 ˆ + K ˆ + X1 −1 X12 ( ) + k2 M ÿ = X1 −1 y; K ˆ + X −1 X ˆ + X −1 X ( ) + k N ÿ = X −1 y; ˆ 2 2 2 12 1 2 (16) X −1 X1 ˆ + X −1 X2 ˆ + X −1 X12 ( ) + X −1 X3 ÿ = X −1 y; 12 ˆ 12 12 12 12 ˆ k2 M ˆ + k1 N ˆ + X3 −1 X12 ( ) + k1 k2 ÿ = X3 −1 y: ˆ Then ÿ is obtained by ˆ 1 −1 −1 ÿ= {X y − k2 M ˆ − k1 N ˆ − X3 X12 ( )}: (17) k 1 k2 3 Hence the ÿrst equation of (16) is represented by −1 k2 X1 X1 − MM ˆ + ( K − MN ) ˆ k1 −1 1 −1 −1 1 −1 + X1 X12 − MX3 X12 ( ) = X1 − MX3 y; k1 k1 which, by (14), implies −1 k2 −1 1 −1 X1 X1 − MM ˆ + X1 X12 − MX3 X12 ( ) k1 k1 −1 1 −1 = X1 − MX3 y: (18) k1 Similarly, by (17), the second equation of (16) is shown to be −1 k1 −1 1 −1 X2 X2 − NN ˆ + X2 X12 − NX3 X12 ( ) k2 k2 −1 1 −1 = X2 − NX3 y: (19) k2 Since (V1 ; B1 ) is a BIBD(v1 ; k1 ; 1 ), the coe cient matrix of ˆ in (18) is reduced to −1 k2 k 2 v1 1 ( 0 + k 2 1 ) 1 k 2 v1 1 1 X1 X1 − MM = I− J = I− J k1 k1 v1 k1 Á1 v1 which, through (18), implies k1 Á1 −1 1 −1 ˆ= X1 − MX3 y k2 1 v 1 k1 −1 1 −1 − X1 X12 − MX3 X12 ( ) : (20) k1 146 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Similarly, (19) leads to k 2 Á2 −1 1 −1 ˆ= X2 − NX3 y k 1 2 v2 k2 −1 1 −1 − X2 X12 − NX3 X12 ( ) : (21) k2 For a design matrix X = (X1 : X2 : X12 : X3 ) of a split-block design such that (Vi ; Bi ) are BIBDs, using (16) and (17) it follows from (20) and (21) that C( ) (X )( ) = F(X )y; where the C-matrix C( ) (X ) is given by −1 1 −1 −1 C( ) (X ) = X12 X12 − X X3 X 3 X12 k1 k2 12 k1 Á1 −1 1 −1 − X12 X1 − X X3 M k 2 1 v1 k1 12 −1 1 −1 × X1 X12 − MX3 X12 k1 k2 Á2 −1 1 −1 − X12 X2 − X X3 N k 1 2 v2 k2 12 −1 1 −1 × X2 X12 − NX3 X12 (22) k2 and −1 1 −1 −1 F(X ) = X12 − X X3 X 3 k1 k2 12 k1 Á1 −1 1 −1 −1 1 −1 − X12 X1 − X X3 M X1 − MX3 k 2 1 v1 k1 12 k1 k2 Á2 −1 1 −1 −1 1 −1 − X12 X2 − X X3 N X2 − NX3 : k 1 2 v2 k2 12 k2 It is easily checked that ( j) (i) C( ) (X )(1v1 ⊗ ev2 ) = 0; C( ) (X )(ev1 ⊗ 1v 2 ) = 0 for any i and j, and rank(C( ) (X )) 6 (v1 − 1)(v2 − 1). Hence by (2), if rank(C( ) (X )) = (v1 − 1)(v2 − 1), then an estimator of the elementary contrasts of the interaction e ects ( ) can be obtained. 3.1. Criteria of optimality The notion of universal optimality for the estimation of interaction e ects will be introduced. Let Si be the set of all permutation matrices of order vi for i = 1; 2. Further let P = {P1 ⊗ P2 | P1 ∈ S1 ; P2 ∈ S2 }. K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 147 Deÿnition 3.1. Suppose that ˜ ( ) is the set of design matrices X in which all elemen- tary contrasts of interaction e ects are estimable by GLSE. Let ( ) be any subset of ˜ ( ) and M( ) be the set of C( ) (X ) for any X ∈ ( ) . Moreover; let M( ) be the convex hull of M( ) and be the set of functions on M( ) such that (i) is convex; (ii) (C) ¿ (gC) for C ∈ M( ) and for any g ¿ 1; and (iii) for any permutation matrix P ∈ P and C ∈ M( ) ; (P CP) = (C) holds. The design matrix X ∗ is said to be universally optimum relative to ( ) if ∗ (C( ) (X )) = min (C( ) (X )) X∈ ( ) holds for all functions ∈ satisfying (i) – (iii). Theorem 3.1. The design matrix X ∗ ∈ ( ) is universally optimum relative to ( ) if the following two conditions hold. (i) C( ) (X ∗ ) is a v1 v2 × v1 v2 matrix which consists of v1 sub-matrices C(ij ) (X ∗ ) 2 of order v2 (i; j = 1; : : : ; v1 ); i.e.; C( ) (X ∗ ) = (C(ij ) (X ∗ )); pI + qJ if i = j; C(ij ) (X ∗ ) = rI + sJ if i = j; where p = v2 (v1 − 1)s; q = −(v1 − 1)s; r = −v2 s and s is a positive integer. (ii) tr(C( ) (X ∗ )) = maxX ∈ ( ) tr(C( ) (X )). Proof. Suppose that X ∗ satisÿes conditions (i) and (ii); and that X ∗∗ is universally optimum. It is obvious that ∗∗ ∗ (C( ) (X )) 6 (C( ) (X )): (23) Let ∗∗ 1 ∗∗ C( ) (X )= (P1 ⊗ P2 ) C( ) (X )(P1 ⊗ P2 ); v1 !v2 ! P1 ∈S1 P2 ∈S2 then C ( ) (X ∗∗ ) satisÿes (i). Hence; C ( ) (X ∗∗ ) is a scalar multiple of C( ) (X ∗ ); that is; C( ) (X ∗ ) = gC ( ) (X ∗∗ ) for some g. By (ii) and the deÿnition of C ( ) (X ∗∗ ); ∗ ∗∗ ∗∗ 1 ∗ tr(C( ) (X )) ¿ tr(C( ) (X )) = tr(C ( ) (X )) = tr C( ) (X ) g holds; which implies that g ¿ 1. Thus; by Deÿnition 3.1(ii); ∗∗ ∗ (C ( ) (X )) ¿ (C( ) (X )) (24) 148 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 holds. Furthermore; by (i) and (iii) of Deÿnition 3.1; we have ∗∗ 1 ∗∗ (C ( ) (X )) 6 ((P1 ⊗ P2 ) C( ) (X )(P1 ⊗ P2 )) v1 !v2 ! P1 ∈S1 P2 ∈S2 ∗∗ = (C( ) (X )): Since X ∗∗ is universally optimum; it must hold that ∗∗ ∗∗ (C( ) (X )) = (C ( ) (X )): (25) Therefore; by (23) – (25); it follows that ∗∗ ∗ (C( ) (X )) = (C( ) (X )) and for estimating ; the design matrix X ∗ ∈ ( ) is universally optimum. 3.2. Combinatorial properties of optimal designs for estimating interaction e ects Let (V1 ; V2 ; B) be an incomplete split-block design and further let (i) (i; j) be the number of superblocks including treatments Ai and Cj , (ii) (i; i ; j) be the number of superblocks including treatments Ai , Ai and Cj , (iii) (i; j; j ) be the number of superblocks including treatments Ai , Cj and Cj , (iv) (i; i ; j; j ) be the number of superblocks including treatments Ai , Ai , Cj and Cj . It is easy to see that 1 (i; j; j ) = (i; i ; j; j ); (26) k1 − 1 i =i 1 (i; i ; j) = (i; i ; j; j ); (27) k2 − 1 j =j 1 1 (i; j) = (i; i ; j) = (i; j; j ): (28) k1 − 1 k2 − 1 i =i j =j Now, to give the main theorem of this section, three lemmas on the C-matrix are given. Lemma 3.1. Let 0 = 0 + 1 + 2 + 3 ; 1 = 1 + 3 and 2 = 2 + 3. Then the terms in the expanded form of (22) are represented as follows: −1 (i) tr(X12 X12 ) = 0 k1 k2 b. −1 (ii) tr(X12 X3 X3 −1 X12 ) = 2 k1 k2 b. K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 149 v1 v2 v1 −1 −1 (iii) tr(X12 X1 X1 X12 ) = [ ii (i; j){ 0 + (k2 − 1) 1 } i=1 j=1 i =1 + (1 − ii ) (i; i ; j){ 2 + (k2 − 1) 3 }]2 ; where ii is the Kronecker delta. v1 v2 v2 −1 −1 (iv) tr(X12 X2 X 2 X12 ) = [ jj (i; j){ 0 + (k1 − 1) 2 } i=1 j=1 j =1 + (1 − jj ) (i; j; j ){ 1 + (k1 − 1) 3 }]2 : (v) tr(( k2 )−1 X12 −1 X3 X 3 −1 X1 X 1 −1 X12 ) v1 v2 v1 = [ ii (i; j)2 { 0 + (k2 − 1) 1 } i=1 j=1 i =1 + (1 − ii ) (i; i ; j)2 { 2 + (k2 − 1) 3 }]: (vi) tr(( k1 )−1 X12 −1 X3 X 3 −1 X2 X 2 −1 X12 ) v1 v2 v2 = [ jj (i; j)2 { 0 + (k1 − 1) 2 } i=1 j=1 j =1 + (1 − jj ) (i; j; j )2 { 1 + (k1 − 1) 3 }]: (vii) tr(( k2 )−2 X12 −1 X3 X 3 −1 X1 X 1 −1 X3 X 3 −1 X12 ) v1 v2 v1 2 = { ii (i; j) + (1 − ii ) (i; i ; j)}2 : i=1 j=1 i =1 (viii) tr(( k1 )−2 X12 −1 X3 X3 −1 X2 X 2 −1 X3 X 3 −1 X12 ) v1 v2 v2 2 = { jj (i; j) + (1 − jj ) (i; j; j )}2 : i=1 j=1 j =1 The reader can ÿnd the proof of Lemma 3.1(i) – (viii) in the appendix at the end of the present paper. Hereafter, let ( ) ⊂ ˜ ( ) be the set of design matrices of incomplete split-block designs (V1 ; V2 ; B) such that (Vi ; Bi ) are BIBDs. Then, by virtue of this lemma, the following can be obtained. 150 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Lemma 3.2. For a design matrix X ∈ ( ); tr(C( ) (X )) = (k1 k2 0 − )b v1 v2 Á1 [ − k1 { − 0 + (k2 − 1) 1 }]2 (i; j)2 k1 k 2 v 1 1 i=1 j=1 v1 v2 + [ − k1 { 2 + (k2 − 1) 3 }]2 (i; i ; j)2 } i=1 j=1 i =i v1 v2 Á2 [ − k2 { 2 − 0 + (k1 − 1) 2 }] (i; j)2 k1 k 2 v 2 2 i=1 j=1 v1 v2 + [ − k2 { 1 + (k1 − 1) 3 }]2 (i; j; j )2 } i=1 j=1 j =j holds. Proof. By Lemma 3.1(i) – (viii); the assertion is proved. Furthermore, the trace of the C-matrix can be bounded from the above as follows: Lemma 3.3. Let X ∗ ∈ ( ) be a design matrix for an incomplete split-block design such that (i; i ; j) and (i; j; j ) are constants not depending on the choice of i; i ; j and j . Then (i; j) is also a constant. Moreover; ∗ tr(C( ) (X )) 6 tr(C( ) (X )) 2 ∗ 11 Q = (k1 k2 0 − )b − k1 k2 1 2 (v1 − 1)(v2 − 1) holds for any X ∈ ( ); where 11 = (i; j) and Q∗ = 2 v2 Á1 (v2 − 1)[(v1 − 1)[ − k1 { 0 + (k2 − 1) 1 }]2 + (k1 − 1)2 [ − k1 { 2 + (k2 − 1) 3 }]2 ] + 1 v1 Á2 (v1 − 1)[(v2 − 1)[ − k2 { 0 + (k1 − 1) 2 }]2 + (k2 − 1)2 [ − k2 { 1 + (k1 − 1) 3 }]2 ]: Proof. Firstly; it is obvious that (i; j) is constant if either (i; i ; j) is constant for any i or (i; j; j ) is constant for any j . By (28) with the Schwarz inequality; it is K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 151 shown that 2 1 (k1 − 1)2 (i; i ; j)2 ¿ (i; i ; j) = (i; j)2 v1 − 1 v1 − 1 i =i i =i and 2 1 (k2 − 1)2 (i; j; j )2 ¿ (i; j; j ) = (i; j)2 ; v2 − 1 v2 − 1 j =j j =j where both the equalities hold if and only if k1 − 1 k2 − 1 (i; i ; j) = (i; j) and (i; j; j ) = (i; j) v1 − 1 v2 − 1 hold; not depending on the choice of i and j . By (5); (6) and Lemma 3.2; it is shown that v1 v2 Q∗ tr(C( ) (X )) 6 ( 0 k1 k2 − )b − (i; j)2 : k 1 k 2 v1 v2 1 2 (v1 − 1)(v2 − 1) i=1 j=1 Furthermore; 2 v1 v2 v1 v2 v1 v2 (i; j)2 ¿ (i; j) = (bk1 k2 )2 i=1 j=1 i=1 j=1 holds; where the equality holds if and only if (i; j) = constant(= 11 ); not depending on the choice of i and j. Thus the proof is completed. Now, the main theorem of this section will be given. Theorem 3.2. Let X ∗∗ ∈ ( ) be a design matrix for an incomplete split-block design (V1 ; V2 ; B) such that (i; i ; j; j ) is a constant not depending on the choice of i; i ; j and j . Then (i; j); (i; i ; j); (i; j; j ) are also constants. Hence X ∗∗ is universally optimum relative to the set ( ) of design matrices for incomplete split-block designs such that (Vi ; Bi ) are BIBDs. Proof. The C-matrix C( ) (X ) of (22) is obtained as a weighted sum of the terms in Lemma 3.1(i) – (viii) and the trace of C( ) (X ) is obtained as in Lemma 3.2. If (i; i ; j; j ) is constant not depending on i; i ; j and j ; then for X ∗∗ ; (i; j); (i; i ; j) and (i; j; j ) are constants; because of (26) – (28). Therefore; it holds that the trace 152 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 of C( ) (X ∗∗ ) attains the maximum of Lemma 3.3. Furthermore; by examining each element of C( ) (X ∗∗ ); it can be shown that it has the form of Theorem 3.1(i); where 2 (k1 − 1)(k1 v1 − 3v1 − 2k1 + 4) (k2 − 1)(k2 v2 − 3v2 − 2k2 + 4) 11 s=− + v1 Á1 1 (v1 − 1)2 v2 Á2 2 (v2 − 1)2 k 1 k2 0 + k2 1 + k1 2 − 22 k1 k 2 and 22 = (i; i ; j; j ). Thus; the proof is completed by Theorem 3.1. 2 When = I , Lemmas 3.2, 3.3 and Theorem 3.2 can be reduced to the following. Corollary 3.1. Under the same assumption as Lemmas 3.2; 3.3 and Theorem 3.2; when 1 = 2 = 3 = 0; the following hold: (i) tr(C( ) (X )) = (k1 k2 − 1)b 1 v1 v2 v1 v2 − (k − 1)2 (i; j)2 + (i; i ; j)2 k 1 k2 v 1 1 1 i=1 j=1 i=1 j=1 i =i 1 v1 v2 v1 v2 − (k − 1)2 (i; j)2 + (i; j; j )2 ; k 1 k2 v 2 2 2 i=1 j=1 i=1 j=1 j =j ∗ (ii) tr(C( ) (X )) 6 tr(C( ) (X )) v1 v2 {v1 k2 (k1 − 1) + v2 k1 (k2 − 1)} 2 = (k1 k2 − 1)b − 2 2 11 ; k1 k2 b (iii) X ∗∗ is universally optimum relative to the set ( ) of incomplete split-block designs such that (Vi ; Bi ) are BIBDs. 4. A construction of a balanced incomplete split-block design and its combinatorial property We now consider constructions of universally optimum designs under Model II. When (i; i ; j; j ) is a constant, an incomplete split-block design (V1 ; V2 ; B) is called a balanced incomplete split-block design, denoted by BISBD(v1 ; k1 ; v2 ; k2 ; 22 ). In this design, for any distinct four treatments Ai ; Ai ∈ V1 and Cj ; Cj ∈ V2 , there are ex- actly 22 superblocks which contain the four treatments simultaneously. For 0 6 m 6 2 and 0 6 n 6 2, let mn be the number of the blocks containing any m-subset of V1 and n-subset of V2 . In the BISBD, (i; i ; j; j ) = 22 , (i; j; j ) = 12 , (i; i ; j) = 21 , (i; j) = 11 , are all constants, and there are the following relationships among K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 153 parameters: v1 − 1 v2 − 1 v2 − 1 v1 − 1 12 = 22 ; 21 = 22 ; 11 = 12 = 21 ; k1 − 1 k2 − 1 k2 − 1 k1 − 1 v2 v1 v1 − 1 v2 − 1 20 = 21 ; 02 = 12 ; 10 = 20 ; 01 = 02 ; k2 k1 k1 − 1 k2 − 1 v2 v1 v1 v2 10 = 11 ; 01 = 11 and 00 = 10 = 01 = b: k2 k1 k1 k2 It is obvious that (V1 ; B1 ) is a BIBD(v1 ; k1 ; 20 ), while (V2 ; B2 ) is a BIBD(v2 ; k2 ; 02 ). Here a construction of BISBDs is provided. Theorem 4.1. The existence of a BIBD(v1 ; k1 ; 1) and a BIBD(v2 ; k2 ; 2) implies the existence of a BISBD(v1 ; k1 ; v2 ; k2 ; 1 2 ). Proof. When (Vi ; Bi ) is a BIBD(vi ; ki ; i ); i =1; 2; let B={(B1 |B2 ) | B1 ∈ B1 ; B2 ∈ B2 }. Then (V1 ; V2 ; B) is the required design. The BISBD constructed in the proof of Theorem 4.1 is here called a direct product of two BIBDs. Lemma 4.1. If there exists a BISBD(v1 ; k1 ; v2 ; k2 ; 22 ); then there are a BIBD (v1 ; k1 ; 22 ) and a BIBD(v2 ; k2 ; 22 ). In this case b1 b2 =b = 22 holds; where b is the number of superblocks of the BISBD(v1 ; k1 ; v2 ; k2 ; 22 ); and b1 and b2 are the num- bers of blocks of the BIBD(v1 ; k1 ; 22 ) and the BIBD(v2 ; k2 ; 22 ); respectively. Proof. For any x; y ∈ V1 ; let Bxy = {B ∩ V2 | x; y ∈ B; B ∈ B}. Then Bxy forms a BIBD(v2 ; k2 ; 22 ). Note that |Bxy | = 20 = b2 . Similarly; for any a; b ∈ V2 ; let Bab = {B ∩ V1 | a; b ∈ B; B ∈ B}. Then Bab forms a BIBD(v1 ; k1 ; 22 ) and hence |Bab | = 02 = b1 holds. Moreover; by the combinatorial properties of BISBDs 22 = 02 20 = 00 = b1 b2 =b holds. Theorem 4.2. If there exists a BISBD(v1 ; k1 ; v2 ; k2 ; 1); then a BISBD having the same parameters as this design is constructed as a direct product of a BIBD(v1 ; k1 ; 1) and a BIBD(v2 ; k2 ; 1). Proof. By Lemma 4.1; the assumption shows that two BIBD(v1 ; k1 ; 1) and BIBD (v2 ; k2 ; 1) exist. Hence; by Theorem 4.1; a BISBD(v1 ; k1 ; v2 ; k2 ; 1) can be constructed. Therefore the BISBD(v1 ; k1 ; v2 ; k2 ; 1) can be constructed as a direct product of a BIBD (v1 ; k1 ; 1) and a BIBD(v2 ; k2 ; 1). Theorem 4.2 means that if 22 =1, there cannot exist a BISBD with lesser number of superblocks than that given as a direct product of two BIB designs. But when 22 ¿ 1, the authors do not know whether there is a set of such parameters that there does not exist any BISBD of direct product type but that there exists a BISBD of non-direct product type. 154 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 Let i, i = 1; 2, be the smallest positive integer such that i (vi − 1) = ri (ki − 1) and v i ri = b i k i are satisÿed for given vi and ki . If there are a BIBD(v1 ; k1 ; 1 ) and a BIBD(v2 ; k2 ; 2 ), then Theorem 4.1 shows the existence of a BISBD(v1 ; k1 ; v2 ; k2 ; 1 2 ). The present question is whether we can construct a BISBD(v1 ; k1 ; v2 ; k2 ; 22 ) such that 22 ¡ 1 2 . For this question, we checked the possibility of 22 ¡ 1 2 within the scope of param- eters 3 6 v1 ; v2 6 1000, 2 6 k1 ; k2 6 50 and for 1 6 1 ; 2 6 20 by using computer. Then it reveals that there are no such parameters within these ranges. Thus it may be hard to ÿnd a BISBD of non-direct product type. Acknowledgements The authors are thankful to the referees for their constructive comments on this paper. Appendix We give the proof of Lemma 3.1(i) – (viii) in Section 3.2. Proof of Lemma 3.1. (i) Let (X12 −1 X12 )[i; j; i ; j ] be the ((i; j); (i ; j ))th entry of the matrix X12 −1 X12 . For x12 (t; r; c; i; j) deÿned by (3); let 1 if x12 (t; r; c; i; j) = 1 for some r; x12 (t; ∗; c; i; j) = 0 otherwise; 1 if x12 (t; r; c; i; j) = 1 for some c; x12 (t; r; ∗; i; j) = 0 otherwise; 1 if x12 (t; r; c; i; j) = 1 for some r and c; x12 (t; ∗; ∗; i; j) = 0 otherwise: Then the (i; j; t; r; c)-elements of X12 R; X12 C and X12 RC are written by x12 (t; r; ∗; i; j); x12 (t; ∗; c; i; j) and x12 (t; ∗; ∗; i; j). It follows from (3) that −1 (X12 X12 )[i; j; i ; j ] = 0 X12 X12 + 1 X12 RX12 + 2 X12 CX12 + 3 X12 RCX12 b k1 k2 = { 0 x12 (t; r; c; i; j) + 1 x12 (t; r; ∗; i; j) t=1 r=1 c=1 + 2 x12 (t; ∗; c; i; j) + 3 x12 (t; ∗; ∗; i; j)}x12 (t; r; c; i ;j ) = ii jj 0 (i; j) + ii (1 − jj ) 1 (i; j; j ) + (1 − ii ) jj 2 (i; i ; j) + (1 − ii )(1 − jj ) 3 (i; i ; j; j ); K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 155 where (i; j); (i; i ; j); (i; j; j ) and (i; i ; j; j ) are deÿned in Section 3.2. Hence it is seen that v1 v2 −1 tr(X12 X12 ) = 0 (i; j) = 0 k1 k2 b; i=1 j=1 which completes the proof. (ii) Let (X12 −1 X3 )[i; j; t] be the ((i; j); t)th entry of the matrix X12 −1 X3 . By (3), b k1 k2 −1 (X12 X3 )[i; j; t] = { 0 x12 (t ; r; c; i; j) + 0 x12 (t ; r; ∗; i; j) t =1 r=1 c=1 + 0 x12 (t ; ∗; c; i; j) + 0 x12 (t ; ∗; ∗; i; j)}x3 (t ; r; c; t) = 12 (i; j; t); holds, where 1 if Ai and Cj are applied to a block Bt ; 12 (i; j; t) = 0 otherwise: Furthermore, it follows that b −1 −1 2 (X12 X3 X 3 X12 )[i; j; i ; j ] = 12 (i; j; t) 12 (i ; j ; t) t=1 2 = { ii jj (i; j) + ii (1 − jj ) (i; j; j ) + (1 − ii ) jj (i; i ; j) + (1 − ii )(1 − jj ) (i; i ; j; j )}; which completes the proof. (iii) It follows from (3) that b k1 k2 −1 (X12 X1 )[i; j; i ] = { 0 x12 (t; r; c; i; j) + 1 x12 (t; r; ∗; i; j) t=1 r=1 c=1 + 2 x12 (t; ∗; c; i; j) + 3 x12 (t; ∗; ∗; i; j)}x1 (t; r; c; i ) = ii (i; j){ 0 + (k2 − 1) 1 } + (1 − ii ) (i; i ; j){ 2 + (k2 − 1) 3 } and −1 −1 (X12 X1 X 1 X12 )[i; j; i ; j ] v1 = [ ii (i; j){ 0 + (k2 − 1) 1 } + (1 − ii ) (i; i ; j){ 2 + (k2 − 1) 3 }] i =1 ×[ i i (i ; j ){ 0 + (k2 − 1) 1 } + (1 − i i ) (i ; i ; j ){ 2 + (k2 − 1) 3 }]: Hence, the assertion is proved. 156 K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 (iv) By (3), b k1 k2 −1 (X12 X2 )[i; j; j ] = { 0 x12 (t; r; c; i; j) + 1 x12 (t; r; ∗; i; j) t=1 r=1 c=1 + 2 x12 (t; ∗; c; i; j) + 3 x12 (t; ∗; ∗; i; j)}x2 (t; r; c; j ) = jj (i; j){ 0 + (k1 − 1) 2 } + (1 − jj ) (i; j; j ){ 1 + (k1 − 1) 3 } and −1 −1 (X12 X2 X 2 X12 )[i; j; i ; j ] v2 = [ jj (i; j){ 0 + (k1 − 1) 2 } + (1 − jj ) (i; j; j ){ 1 + (k1 − 1) 3 }] j =1 ×[ j j (i ; j ){ 0 + (k1 − 1) 2 } + (1 − j j ) (i ; j ; j ){ 1 + (k1 − 1) 3 }] hold. Hence, the proof is completed. (v) We deÿne that 1 if x1 (t; r; c; i) = 1 for some r and c; x1 (t; ∗; ∗; i) = 0 otherwise: −1 It is easy to check that (X3 X1 )[t; i] = k2 x1 (t; ∗; ∗; i). Then b −1 −1 (X12 X3 X 3 X1 )[i; j; i ] = 12 (i; j; t ) k2 x1 (t ; ∗; ∗; i ) t =1 2 = k2 { ii (i; j) + (1 − ii ) (i; i ; j)}: (A.1) Hence it can be shown that −1 −1 −1 (X12 X3 X 3 X1 X1 X12 )[i; j; i ; j ] v1 = 2 k2 { ii (i; j) + (1 − ii ) (i; i ; j)} i =1 ×[ i i (i ; j ){ 0 + (k2 − 1) 1 } + (1 − i i ) (i ; i ; j ){ 2 + (k2 − 1) 3 }]; which proves the required result. (vi) We deÿne that 1 if x2 (t; r; c; j) = 1 for some r and c; x2 (t; ∗; ∗; j) = 0 otherwise: K. Ozawa et al. / Journal of Statistical Planning and Inference 106 (2002) 135 – 157 157 It is easy to check that b −1 −1 (X12 X3 X 3 X2 )[i; j; j ] = 12 (i; j; t ) k1 x2 (t ; ∗; ∗; j ) t =1 2 = k1 { jj (i; j) + (1 − jj ) (i; j; j )}: (A.2) Then it follows that −1 −1 −1 (X12 X3 X 3 X2 X2 X12 )[i; j; i ; j ] v2 = 2 k1 { jj (i; j) + (1 − jj ) (i; j; j )} j =1 ×[ j j (i ; j ){ 0 + (k1 − 1) 2 } + (1 − j j ) (i ; j ; j ){ 1 + (k1 − 1) 3 }]: Hence the result is proved. (vii) By (A.1), −1 −1 −1 −1 (X12 X3 X 3 X1 X1 X3 X 3 X12 )[i; j; i ; j ] v1 2 = 4 k2 { ii (i; j) + (1 − ii ) (i; i ; j)} i =1 ×{ i i (i ; j ) + (1 − i i ) (i ; i ; j )} which completes the proof. (viii) By (A.2), −1 −1 −1 −1 (X12 X3 X 3 X2 X2 X3 X 3 X12 )[i; j; i ; j ] v2 2 = 4 k1 { jj (i; j) + (1 − jj ) (i; j; j )} j =1 ×{ j j (i ; j ) + (1 − j j ) (i ; j ; j )}: Hence the required result is readily proved. References Bhargava, R.P., Shah, K.R., 1975. Analysis of some mixed models for block and split-plot designs. Ann. Inst. Statist. Math. 27, 365–375. Hering, F., Mejza, S., 1997. Incomplete split-block designs. Biometrical J. 39, 227–238. Kiefer, J., 1975. Construction and optimality of generalized Youden designs II. In: Srivastava, J.N. (Ed.), A Survey of Statistical Designs and Linear Models. North-Holland, Amsterdam, pp. 333–353. Mejza, I., Mejza, S., 1996. Incomplete split-plot designs generated by GDP-BIBD(2). Calcutta Statist. Assoc. Bull. 46, 117–127. Rees, H.D., 1969. 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