Two classes of group divisible partial diallel crosses

Abstract

Various forms of the diallel cross have been used in plant and animal breeding, but this form of investigation requires a large number of lines or individuals which very often leads to rather unmanageable experiments. For this reason the concept of the partial diallel cross (PDC) has been introduced (Kempthorne, 1957; Gilbert, 1958; Hinkelmann & Stern, 1960; Kempthorne & Curnow, 1961), in which only a sample of all possible crosses of the complete diallel cross (CDC) is carried out. Gilbert (1958) has mentioned the analogy between PDC's and incomplete block designs with blocks of size two. Since a CDC corresponds to a balanced incomplete block design (Kempthorne & Curnow, 1961) it was to be expected that partially balanced incomplete block designs (PBIB) would be related to certain types of PDC's. Curnow (1963) and Fyfe & Gilbert (1963) have given some PDC's derived from PBIB's with two associate classes, and Fyfe & Gilbert (1963) further introduce PDC's derived from a PBIB with three associate classes (their factorial designs). In this paper we shall consider the correspondence between PDC's and PBIB's with m associate classes and give a general method of analysing these designs (?2). In search for some more flexible designs than those derived from PBIB's with two associate classes we came across two rather general classes. One class is derived from a generalization of group divisible PBIB's with two associate classes (Roy, 1953-54). The other class is believed to be new, representing an extension of a design with three associate classes given by Vartak (1959). The definitions of these PBIB's are given in ? 3. The formation of PDC's from these PBIB's is discussed briefly in ? 4. An example is presented in Appendix A. In ?? 5 and 6 the analysis of these PDC's is given. For some selected values of N, the number of lines involved in a diallel cross experiment, we list the best designs out of these two classes in Appendix B.