Balanced designs for biological experiments in blocks of natural sizes

Abstract

As a preliminary result we have established Fisher’s inequality associated with a BIB design and generalized it to balanced binary designs with unequal replications and unequal block sizes to balanced n-ary equireplicate designs and also to BIB designs in which one treatment alone is allowed to repeat more than once in a block. Further it is shown that a balanced proper binary design is equireplicate. From existing BIB designs we have constructed balanced binary and ternary designs. A novel method of construction is as follows: Let there be a BIB design with parameters v, b, r, k, λ. From each block form k blocks each of size k+1 with block content as all treatments of the block with one distinct treatment repeated in a block. The resulting design will be a balanced ternary design with parameters v1=v, b1=kb, r1=r(k+1), λ1= λ(k+2). Kroneckor product is applied for the construction of balanced ternary designs by collapsing blocks of a BIB design. We have proved using Kroneckor product, that existence of a resolvable BIB design implies the existence of a proper balanced ternary design and this is an improvement over the results due to Dey (1970). Further it is shown that method of Kroneckor product used for the construction of balanced ternary designs can also be used for the construction of partially balanced ternary designs. Methods have been devised for the construction of balanced ternary designs making use of Finite geometrices and Galois field.