Heritability at family mean level

Based on a polygenic system of a diploid species, without epistasis, and a population in Hardy-Weinberg equilibrium, without inbreeding and under linkage equilibrium, it can be shown that: (1) the narrow sense heritability at half-sib family level is equal to the square of the correlation coefficient between family mean and the additive genetic value of its common parent; (2) the narrow sense heritability at full-sib family level is equal to the square of the correlation coefficient between family mean and the mean of the additive genetic values of its parents; (3) the narrow sense heritability at Sn family level is exactly equal to the square of the correlation coefficient between family mean and the additive genetic value of its parent only in absence of dominance or when allele frequencies are equal; and (4) the broad sense heritability at full-sib or Sn family level can be used to analyze selection efficiency, since the progeny genotypic mean is, in general, a good indicator of parents, or Sn-1 plant superiority with respect to the frequency of favorable genes.