Experiments in square lattice with some common treatments added to each block: intrablock analysis

Abstract

A statistical investigation was conducted for the case in which experiments are designed in square lattices, but some common treatments are added to each block. Treatments of the basic or initial design were designated as "regular treatments" and the other ones as "common treatments". The main objective was to develop a general method of analysis for this specific kind of experiment. The following parameters were considered as pertinent to the initial design: k (number of plots per block), v = k² (number of regular treatments), b (number of blocks), i (number of orthogonal replications), n (number of repetition of orthogonal replications) and r = ni (number of replication of each treatment). The inclusion of c common treatments to each block, resulted in augmented design with the following parameters: v' = v + c (total number of treatments), b (number of blocks), k' = k + c (number of plots per block), r' (number of replications of each treatment) and λ_uu (number of blocks in which treatments u and u' occur together). The following mathematical model was considered: y_uh = m + t_u + b_h + e_uh, where y_uh denotes the observation of the u^th treatment in the h^th block; m denotes the effect of the general mean; t_u denotes the u^th treatment effect (u = 1,2,...., v'); b_h denotes the h^th block effect (h = 1,2, ...., b); e_uh is a random error component where e_uh ᴖ (0, σ²). Formulas for sums of squares in the analysis of variance, the adjusted treatment means and the variances of treatment differences, were obtained.