Classification and Analysis of Partially Balanced Incomplete Block Designs with Two Associate Classes

INCOMPLETE block designs are now in fairly general use, especially the I balanced incomplete block and the lattice designs. One of the authors in collaboration with Nair 1 introduced in 1939 a wider class of designs, viz. partially balanced incomplete block designs which included as a special case the balanced incomplete block designs and the square lattices. A slight generalization by Nair and Rao 2, resulted in the inclusion of cubic and other higher dimensional lattices as special cases. Recently other special cases of partially balanced designs are beginning to be used, for example, the rectangular lattices of Harshbarger 3, 4, 51 and the linked block designs of Youden [6. It is the object of this paper to review the more important properties of partially balanced designs with two associate classes and to give both the intrablock and the combined intraand inter-block analysis in a simplified and compact form, together with an illustrative numerical example. A special feature is the division of these designs into a small number of distinct types, for each of which the association scheme can be explicitly exhibited. This simplifies the numerical computations as well as the interpretation of the results. A number of useful designs belonging to each type have been given for illustrative purposes, but no complete tabulation has been attempted. Complete tables (up to ten replications) will be published by the Institute of Statistics, University of North Carolina.