Some Notes about Efficiency Balanced Block Designs with Repeated Blocks Bronislaw Ceranka and Malgorzata Graczyk1 Abstract Various problems for a class of efficiency balanced block designs based on balanced incomplete block designs with repeated blocks are considered. 1 Introduction Let us consider a class of block designs in which v treatments are arranged in b blocks according to the incidence matrix N = (nj), nj denotes the number of experimental units in the jth block getting the ith treatment, i = 1, 2,...,v, j = 1, 2,..., b. The ith treatment is replicated r times and the size of j th block is k j, r = [ri r2 ... rv ]', k = [k1 k2 ... kb]'. If r1 = r2 = ... = rv, then the design is called equireplicated. In the course of performing an experiment using a block design, a cost is often attached to each block. Thus it is of interest having the property that each pair of distinct treatments appear in the same, or nearly the same, number of blocks. The set of all distinct blocks in a block design is called support and denoted as b*. The constructions of the designs admitting block designs with repeated blocks are available in the literature (Ceranka and Graczyk, 2007). Balancing is a desirable property of any block design. There are balanced designs of various types. Definition 1 A block design is called variance balanced (VB) if every normalized estimable linear function of treatment effects is estimated with the same variance. Let us consider the matrix C = R - NK-1N', (1.1) where R = diag{r1 r2 ... rv}, K = diag{k1 k2 ... kb}. Kageyama (1974) showed that N is a VB block design if and only if C = p(Iv - 11v iv) , (1.2) 1 Department of Mathematical and Statistical Methods, Poznan University of Life Sciences, Wojska Polskiego 28, 60-637 Poznali, Poland; bronicer@au.poznan.pl, magra@au.poznan.pl where p is the unique nonzero eigenvalue of C with multiplicity v — 1, Iv is the unit matrix of order v, 1v is v x 1 vector all of whose elements are 1. Let us consider the matrix M0 given by Calinski (1971) M0 = R-1NK-1N' — 11v r', (1.3) n b where n is total number of experimental units and n = YIV=1 r = ^j=1 k. Definition 2 A block design is called efficiency balanced (EB) if all treatments contrasts s T (r's = 0) satisfy M0s = ßs, where T = [T1 T2 ... Tv] , T is the total yield for the ith treatment, ß is the unique nonzero eigenvalue of M0 with multiplicity v — 1 and M0 is given in (1.3). Calinski (1971) showed that for such designs, every treatment contrast is estimated with the same efficiency 1 — ß and N is a EB block design if and only if M0 = ß^Iv — n 1vr') . (1.4) Kageyama (1980) proved that for the EB block design N, equation (1.4) is fulfilled if and only if C = (1 — ß) ^R — 1 rr'^ . (1.5) In the paper we present new construction methods of EB block designs with repeated blocks for v treatments and some ways of admitting given design structures to construct new designs for other numbers of treatments. 2 Construction for v1 < v treatments In order to construct a EB block design, we consider the VB block design, N with repeated blocks with parameters v, b, r, k, b* and the v1 x v matrix E = (elt), where for each t, elt = 1 for a certain I, el't = 0, I = l', l,l' = 1, 2, ...,v1, t = 1, 2,...,v, i.e., E' 1 = 1 Now, we form the matrix N1 = EN. (2.1) From the design matrix N for v treatments, we construct the design matrix N1 for v1 treatments, v1 < v. Hence we have N11b = EN1b = rE1b = r1, N^1vi = N'E' 1vi = N' 1v = k = k1, n = n1. Theorem 1 If N is the equireplicated VB block design with repeated blocks, then Ni given in the form (2.1) is a EB block design with repeated blocks and with the parameters v1, b1 = b, r1, k1 = k, b*1 = b*. Proof. In any equireplicated design n = rv. From (1.1) and (1.2), for the VB block design N, we have C1 = ECE' = E (p (Iv - 11v 1V)) E' PE (rIv - r2MvJ E PE K - n rr' ) E' r ^R1 ni r1r1 where p = r:=P. Hence, the Theorem is proven. Theorem 2 If N is the EB block design with repeated blocks, then N1 given in the form (2.1) is a EB block design with repeated blocks. Proof. From (1.5) for the EB block design N, we have C1 = ECE' = (1 - p)E (R - nrr') E' = (1 - p) (R1 - ^r1r1 ) . Hence the Theorem is proven. For example, let us consider the VB block design with repeated blocks and parameters v = 9, b = 25, r = 9, k = [3 ■ 124 9]', b* = 13 (Example 2.7. Ceranka and Graczyk, 2007), N 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 and we consider the matrix E 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 000000100 000000010 000000001 Thus the matrix N1 in the form (2.1) is given as 2111111000002111111000002 0110110211000110110211002 1001001111201001001111202 0100001001010100001001011 0010100000110010100000111 0001010010010001010010011 Ni Here, N1 is the matrix of a EB block design with repeated blocks for v1 = 6 objects. In this design, bi = b = 25, n = [18 ■ 1'3 9 ■ l'3]', ki = k = [3 ■ I24 9]' and b* = b* = 13. For the design N1, we have 14 —4 —4 —2 —2 —2 —4 14 —4 —2 —2 —2 7 —4 —4 14 —2 —2 —2 9 —2 —2 —2 8 —1 —1 —2 —2 —2 —1 8 —1 —2 —2 —2 —1 —1 8 and 1 — ß = 9. Thus it is easy to see that C1 is given in the form (1.5). On the other hand, Mo / 7 —2 —2 —1 —1 —2 7 —2 —1 —1 2 1 —2 —2 7 —1 —1 9 9 —2 —2 —2 8 —1 —2 —2 —2 —1 8 V —2 —2 —2 —1 —1 and its eigenvalues are 0 and ß = 9 with the multiplicity 5. Hence, N1 is EB. 3 Construction for v treatments We consider the set of £ EB block designs with repeated blocks Ns with the parameters v, bs, rs, ks, b*, s = 1, 2, ...,£. We form the matrix N = [N1 N2 .... N]. (3.1) Theorem 3 If Ns is a EB block design with repeated blocks, s = 1, 2, ...,£, for which vectors of treatment replications are mutually proportional, then N in the form (3.1) is a EB block design with repeated blocks with the parameters v, b = S=1 bs, r = ES=1 rs, k =[k1 k2 ... kç]' , b* where as and aw are some constants. For example, let us consider the EB block designs with repeated blocks and with the parameters v = 4, bi = 10, ri = [8 4 ■ l'3] , ki = 2, b\ = 7 and v = 4, b2 = 10, r2 = [12 6 ■ l'3] , k2 = 3, b*2 = 7 given by the incidence matrices Ni N 2= 2 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 2 2 2 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 0 and Based on the matrices Ni and N2, we form the design matrix N in the form (3.1) as N 21111110000111111222 01001001101011011001 00100101011101101010 00010010111110110100 Here, N is the incidence matrix of the EB block design with the parameters v = 4, b = 20, r = [20 10 ■ l'3]' , k = [2 ■ l'10 3 ■ l'10]', b* = 14. For this design, we have ( C=4 V 6 -2 -2 -2 "I \ -2 4 -1 -1 -2 -1 4 -1 -2 -1 -1 4 and 1 — ß = 4. Thus it is easy to see that C is given in (1.5). On the other hand, Mo ( 3 -1 -1 -2 4 -1 -2 -1 4 211 -1 -1 -1 4 \ and its eigenvalues are 0 and ß = 4 with the multiplicity 3. Hence, N is EB. 4 Construction for v + 1 treatments 1, 2, be the incidence matrix of the balanced incomplete block design with bn, rn, kn, An, b*. Now, we form the matrix N Let Nn, n repeated blocks with the parameters v, as N Ni 0 1t d1bi 0 1t N2 0 1, oL 01' (4.1) Theorem 4 A block design with an incidence matrix N of the form (4.1) is a EB block design with repeated blocks with the parameters v + 1, b = tb1 + ub2, r = [(tri + ur-2) 1v dtbi]', k = [(ki + d) !tbl k21'b2]', b* = tb* + ub* if and only if the constants t, u and d satisfy the equalities t (ri - Ai) k2 = u (vdA2 - ki(r2 - A2)) (4.2) and d>ki(r2 ~ A2). (4.3) vA2 Proof. For the block design with the matrix N given in (4.1), we have C t ri - ri-Ai k1+d + u r2 - r2 -A2 k2 dtri -k1+d 1 tAi + uA2 ki+d + k2 1 l' dtri -1v 1v ki+d 1 dtbiki ki+d (4.4) I v v Now, we compare the matrix C of the form (1.5) and (2.6). We have tA1 + uA2 (tr1 + ur2)2 n and k1 + d k2 dtr1 (tr1 + ur2) dtb1 k1 + d n If the conditions (4.2) and (4.3) hold, then the matrix C is of the form (1.5), where 1 - ^ = Tkt^^+S • ^ the Theorem is proven. In particular case when t = u = 1, we have Corollary 2. A block design with the incidence matrix N of the form N = (4.5) N1 N2 dlbi 0b2 is a EB block design with repeated blocks with the parameters v + 1, b = b1 + b2, r = [(r1 + r2)lV dbj' , k = [(k1 + d)1^ k2l'b2]' ,b* = b* + b*2 ifandonlyif (r1 — A1)k2 = vA2d — k1(r2 — A2) and d > h(r2 - A2) vA2 As an example, let us consider the balanced incomplete block design with the parameters v = 7, b1 = 28, r1 = 12, k1 = 3, A1 = 4, b* = 11 given by the incidence matrix N1 111111000000111100000000110 111000111000000011110000101 111000000111000000001111011 000000111111111100000000011 0001110001110000111100000101 0001111110000000000011111001 0000000000001111111111110000 and v = 7, b2 = 21, r2 = 9, k2 = 3, A2 = 3, b* = 7 given by the incidence matrix N2 111000011100001110000 100110010011001001100 010101001010100101010 100001110000111000011 001100100110010011001 001011000101100010110 010010101001010100101 Hence d = 2, and N in (4.1) given as N 111111000000111100000000110111000011100001110000 111000111000000011110000101100110010011001001100 111000000111000000001111011010101001010100101010 000000111111111100000000011100001110000111000011 0001110001110000111100000101001100100110010011001 0001111110000000000011111001001011000101100010110 0000000000001111111111110000010010101001010100101 is the incidence matrix of the EB block design with the parameters v 21, k = 3, b* = 18. For the design N, we have ( C 7 9 6 -1 6 1 -1 -1 -1 -1 1 -1 -1 -1 -1 6 -1 -1 -1 -1 1 6 -1 -1 -1 1 -1 6 -1 -1 1 -1 -1 6 -1 1 1 11 6 \ and 1 - ß = 9. 7, b = 49, Thus it is easy to see that C is given in the form (1.5). On the other hand, ( M, o = V its eigenvalues are 0, and ß = § wi 1 -1 -1 -1 -1 1 -1 -1 -1 -1 6 -1 -1 -1 -1 1 6 -1 -1 -1 1 -1 6 -1 -1 1 -1 -1 6 -1 1 1 -1 -1 6 6 h the multiplicity 6. Hence, N is EB r 3 References [1] Calinski, T. (1971): On some desirable patterns in block designs. Biometrics, 27, 275-292. [2] Ceranka, B. and Graczyk, M. (2007): Variance balanced block designs with repeated blocks. Applied Mathematical Sciences, Hikari Ltd., 1, 2727-2734. [3] Kageyama, S. (1974): Reduction of associate classes for block designs and releated combinatorial arrangements. Hiroshima Math. J. , 4, 527-618. [4] Kageyama, S. (1974): On properties of efficiency - balanced designs. Commun. Statist. - Theor. Meth. , A 9, 597-616.