Although linkage disequilibrium, epistasis and inbreeding are common phenomena in genetic systems that control quantitative traits, theory development and analysis are very complex, especially when they are considered together. The objective of this study is to offer additional quantitative genetics theory to define and analyze, in relation to non-inbred cross pollinating populations, components of genotypic variance, heritabilities and predicted gains, assuming linkage disequilibrium and absence of epistasis. The genotypic variance and its components, additive and due to dominance genetic variances, are invariant over the generations only in regard to completely linked genes and to those in equilibrium. When the population is structured in half-sib families, the additive variance in the parents' generation and the genotypic variance in the population can be estimated. When the population is structured in full-sib families, none of the components of genotypic variance can be estimated. The narrow sense heritability level at plant level can be estimated from the parent-offspring or mid parent-offspring regression. When there is dominance, the narrow sense heritability estimate in the in F2 is biased due to linkage disequilibrium when estimated by the Warner method, but not when estimated by means of the plant F2-family F3 regression. The bias is proportional to the number of pairs of linked genes, without independent assortment, and to the degree of dominance, and tends to be positive when genes in the coupling phase predominate or negative and of higher value when genes in the repulsion phase predominate. Linkage disequilibrium is also cause of bias in estimates of the narrow sense heritabilities at full-sib family mean and at plant within half-sib and full-sib families levels. Generally, the magnitude of the bias is proportional to the number of pairs of genes in disequilibrium and to the frequency of recombining gametes.
components of genotypic variance; heritabilities; half-sib families; full-sib families
