Methods of calculating the inbreeding coefficient In a finite population undergoing recurrent selection (self-select-intercross in succeeding generations) were investigated. It was noted that, in a population under selection, the inbreeding coefficient does not provide the experimenter with a measure of expected degree of variability; instead an index of total heterozygosity is required, and such an Index was derived.

Formulas necessary to calculate both the inbreeding coefficients and the heterozygosity indexes were derived for the cases: one-locus, two-allele, random selection; k independent loci and random selection; one-locus, two-allele and effective directional selection; and k linked loci with effective directional selection. These formulas Involved defining a generalized inbreeding coefficient and a generalized index of homozygosity (or heterozygosity) in terms of vectors whose components reflected the various possible patterns of genes identical by descent at a given stage of the recurrent selection breeding program. Formulas were derived whereby the mean and the variance of the total number of loci homozygous (or heterozygous) by descent or in state may be obtained.

The progress of the panmictic index and/or the index of total heterozygosity through at least twenty-five cycles of recurrent selection was observed in computer-simulated populations ranging in sizes from ten through one hundred, assuming varying recombination probabilities both in the one-locus and in the two linked-loci case and assuming both minimum and maximum inbreeding selection patterns. Tables resulting from these simulated studies could be used to estimate minimum and maximum inbreeding coefficients and/or minimum and maximum heterozygosity indexes in experimental populations for which the initial conditions approximate those assumed in the simulated populations.

It was observed that the coefficient of relationship in the source population was extremely important in tracing the progress of the degree of Inbreeding and/or total homozygosity, that linkage played a major role in promoting heterozygosity in a recurrent selection system, and that careful intercrossing rather than random mating in alternate generations of the recurrent selection cycle was important in promoting maximum heterozygosity in the selected population. In the simulated populations the effect of small population sizes was observed and, in general, indications were that unless more than five complete recurrent cycles are contemplated, increasing population size results In only relatively minor increases in panmixia, especially when linked loci are involved in the selected trait and when care Is taken to avoid a maximum inbreeding selection pattern.