Two-Way Nested Classification

Abstract

In the preceding two chapters, we have considered experimental situations where the levels of two factor s are crossed. In this and the following chapter we consider experiments where the levels of one ofthe factors are nested within the levels of the other factor. The data for a two-way nested classification are similar to that of a single factor classification except that now replications are grouped into different sets arising from the levels of the nested factor for a given level of the main factor. Suppose the main factor A has a levels and the nested factor B has ab levels which are grouped into a sets of b levels each , and n observations are made at each level of the factor B giving a total of abn observations. The nested or hierarchical designs of this type are very important in many industrial and genetic investigations. For example, suppose an experiment is designed to investigate the variability of a certain material by randomly selecting a batches , b sample s are made from each batch, and finally n analyses are performed on each sample . The purpo se of the investig ation may be to make inferences about the relative contribution of each source of variation to the total variance or to make inferences about the variance components individually. For another example, suppose in a breeding experiment a random sample of a sires is taken, each sire is mated to a sample of b dams, and finally n offspring are produced from each sire-dam mating. Again, the purpose of the investigation may be to study the relative magnitude of the variance components or to make inferences about them individually. MATHEMATICAL MODElThe random effects model for a two-way nested classification can be written as Yijk = f.-L + a ; + {3j(i) + eijk.