Blocking in incomplete split plot designs

Abstract

SUMMARY An incomplete split plot design with whole plots arranged in a completely randomized design was proposed by Robinson (1967). In this note, designs in which the whole plots are arranged in blocks, are considered. A method of construction and estimates of treatment effects are given. Robinson (1967) discussed certain incomplete split plot designs for factorial experiments with a factor A1 on whole plots and a factor A2 on subplots. In these designs the number of subplots per whole plot is less than the number of levels of A2. For each level of A1, the allocation of levels of A2 is balanced with respect to whole plots in the sense that, if whole plots are thought of as blocks, the allocation produces a balanced incomplete block design. The same balanced incomplete block design is used for each level of A1. Robinson (1967) did not consider grouping whole plots into blocks. However, the incomplete split plot design for the treatment combinations of two factors A1 and A2 at v1 and v2 levels, respectively, consists of whole plots made up of k2 < v2 subplots, with each level of the factor A1 applied to b2 whole plots and the v2 levels of the factor A2 arranged in a balanced incomplete block design within each level of A1 by considering the whole plots as blocks. On any level of A1, each level of A2 occurs r2 times and each pair of levels of A2 occurs in the same whole plot A' times. The design may be considered to be repeated r, times. If the whole plots are arranged in blocks, the between whole plot error may be reduced. A procedure, based on designs for confounding in asymmetrical factorial experiments as described by Nair & Rao (1948), is suggested for generating these designs. Neglecting the whole plots, we see that the block designs derived by this procedure are partially balanced incomplete block designs with three associate classes in a rectangular association scheme and they give a generalization of the direct product designs of Vartak (1955, 1959). Rees (1969) considered incomplete split plot designs where the blocks were direct product designs.