Generalized Genetic Variance and Covariance Formulae for Self-Fertilized Crops Assuming Linkage.

UALITATIVE characters can, by definition, be readily separated into pheno-Q typic classes having a correspondence with genotypic classes. However, in assessing the continuous variability of quantitative characters, class boundaries must be arbitrary, bearing no ascertainable correspondence with genotypic classes. This obscuration of class boundaries of quantitative characters stems from small effects of single factors relative to total variation which must be ascribable either to (1) a considerable amount of non-genetic variation with few genes segregating or (2) a large or small amount of non-genetic variation with large numbers of genes segregating. The description of a quantitative character based upon these arbitrary class frequencies lacks the desired characteristic of genetic inference that enumeration enjoys in the case of qualitative traits. However, means, variances and covariances offer a familiar way of population description and are of value in making genetic inferences in the case of quantitative characters. In quantitative characters where the number of genes is usually assumed to be of a greater magnitude than the number of chromosomes, the effect of linkage may become important in drawing genetic inferences from these variances and covari-ances, especially in crops where genotypic frequencies are not in equilibrium, such as in material derived from a recent cross of inbred lines. Specifically, ROBERTSON (1952) states that in the Fz of a cross between two inbred lines, there will be a correlation between genes at adjacent loci and any analysis will tell us about properties of blocks of genes rather than of individual genes. Several authors have considered various aspects of linkage effects on quantitative characters, although consideration has been limited chiefly to random mating populations. An analysis of quantitative characters was carried out in a classic paper by FISHER (1918) who partitioned continuous variability utilizing principles of Mendelian inheritance. FISHER indicates that in a random mating population additive genetic variance and dominance variance can be defined so their contributions to correlations among relatives are the same as without linkage. WRIGHT (1921) found that the relative increase in homozygosis in successive generations of breeding is independent of dominance, the number of factors , or linkage in absence of selection and assortative mating. WRIGHT (1933) believed that only in exceptional cases should a correlation between characters in a