The Theory and Analysis of Diallel Crosses.

T H E diallel cross method of investigating the genetical properties of a group of homozygous lines has recently received much attention. HULL (1945) has considered some aspects of the method. A short summary of a more general approach by JINKS and HAYMAN (1953) and its application to several published sets of maize data has also appeared. In another paper JINKS (1954) has described experiments on inbred lines of Nicotina rustica, and has given an account of some of the associated statistics together with a discussion of the results. In this paper we apply a genetic algebra to the theory of the diallel cross, not only to re-establish the formulae of JINKS, but also to investigate more complex genetical systems. We will show how to measure additive and dominance variation, how to describe the relative dominance properties of the parental lines and how to detect non-allelic genic interaction. A ivorked example illustrates the theory. The following definitions will be used. A diallel cross is the set of all possible matings between several genotypes. The genotypes may be defined as individuals, clones, homozygous lines, etc., and, if there are n of them, there are n2 mating combinations, counting reciprocals separately. A diallel table is an arrangement in a square of n2 measurements corresponding one-to-one to the mating combinations of a diallel cross, each row and column of the square corresponding to offspring with a common parental genotype. This general definition is necessary because a diallel table need not be restricted to containing measurements on the progeny of a diallel cross, but may be used for later generations obtained by selfing these progeny or backcrossing them to their parents. We shall investigate the diallel cross consisting of the progeny of n selfed homozygous lines and their n2 n crosses.