Balanced N-ary designs with equal or unequal block size & equal or unequal replications

Abstract

Tocher (1952) introduced n-ary designs as generalization of balanced incomplete block designs. But the properties of the parameters of the design have not been discussed so far. We have shown that some important properties of the balanced incomplete block binary design are also true in the case of balanced n-ary symmetrical proper equireplicate designs. That is if h =∑jnij2 , λ=∑jnijnpj; in a proper equireplicate balanced design then (i) h > λ (ii) b ≥ v (iii) rk = h+(v-1) λ Among the methods block section, block intersection, complementation and inversion considered by us for the construction of designs the method of complementation is only found fruitful for the construction of proper equireplicate balanced designs. There are situations like comparison of new varieties of seeds of which are in short supply where equal replication of treatments is not possible. There may also be contexts in which the available few animals cannot be used completely for the experiment using conventional designs. For such circumstances we have proposed a systematic method of construction of balanced n-ary designs with equal or unequal replications and equal or unequal block sizes. The method of Kronecker product has been formally introduced to the literature for the construction of proper equireplicate balanced n-ary designs and the methods is contained in the following results. If N1 and N2 are two BIB designs with parameters v, b1, r1, k1, λ1 and v, b2, r2, k2, λ2 respectively, for positive integral values of a1 and a2, a1E(1,b2)xN1+a2N2xE(1,b1) is in general a proper equireplicate n-ary design provided a1+a2+1= n. If N1 and N2 are two balanced proper equireplicate n1-ary and n2-ary designs in v treatments with b1,b2 blocks respectively, for positive integers a1 and a2, a1E(1,b2)xN1+a2N2xE(1, b1)is a n-ary balanced equireplicate proper design with b1b2 blocks where n=a1 (n1-1)+a2(n2-1)+1.