Tri-hierarchical incomplete block designs

A tri-hierarchical incomplete block (TriHIB) design is defined as an arrangement of v equi-replicated treatments in three levels of blocking such that each block of the first level contains m1 blocks of second level and that of second level contains m2 blocks of the third level. Every level of a TriHIB design, treated independently, is an incomplete block design (IBD) and ignoring any one level of blocking, results into a bi-hierarchical incomplete block (BiHIB) design. Here, two general methods of construction of TriHIB designs are proposed. In the first method, popular association schemes (viz., triangular, Latin square and rectangular) of partially balanced IBDs have been used for developing three series of designs with equal/unequal block sizes at higher levels of blocking. Second method yields another series of designs, through cyclic development of two initial sequences, with equal block sizes at first two levels and equal/unequal block sizes at the third level of blocking. A list of parameters of TriHIB designs for number of treatments up to 30 is presented, along with efficiency factors. Further, for easy accessibility and wider adaptability of these designs, an R package 'Tri.Hierarchical.IBDs' has been developed for the generation of the proposed designs.