Intra-Block Analysis for Factorials in Two-Associate Class Group Divisible Designs

Abstract

Group divisible incomplete block designs form an important class of incomplete block designs useful in a wide variety of experimental situations. Their properties, construction, and analysis have been thoroughly discussed in statistical literature, and we cite only several recent references [1], [2], and [3] to work of Bose and his co-workers dealing with partially balanced designs with two associate classes with which we shall be concerned. The utility of incomplete block designs would be increased with means of incorporating factorial treatment combinations in them. The use of factorials is widespread and stimulated by the concepts of confounding, partial confounding, and fractional replication. A mathematical summary on factorials is given by Kempthorne [4]. Kramer and Bradley [5] considered factorials in near-balance incomplete block designs, and here we generalize to the wider class of group divisible designs with two associate classes. Harshbarger [6] used a $2^3$ factorial in a Latinized rectangular lattice and this seems to be the first use of a factorial in a partially balanced incomplete block design. We obtain the intra-block analysis of variance for two-associate class group divisible designs with the adjusted treatment sum of squares in a modified form that more clearly indicates the structure of that quantity. Factorial treatment combinations are then identified with basic treatments through the association scheme of a design. This identification is effected in such a way that the factors are divided into two groups. For example, the design for 18 treatments (see [2], Design S60), divisible into six groups of three, in blocks of six, treatments replicated five times, can be adapted to a $6 \times 3$ factorial scheme; by regarding the six groups as made up of a $2 \times 3$ classification, the same design can be used for a $2 \times 3^2$ factorial scheme. Single-degree-of-freedom comparisons are obtainable in much the usual way and use of fractional replication, essentially within the groups of factors, is possible. The analyses for factorials depend on the estimators of basic treatment effects. We are not concerned with the construction of two-associate class group divisible designs and all known such designs for which $r \leqq 10, 3 \leqq k \leqq 10$, where $r$ is the number of replications and $k$ is the number of plots per block, are given in [7].