Breeder's Equation

Everything in applied breeding reduces to this line. Select parents that exceed the population mean by $S$, and the next generation's mean moves by $h^2 S$. The heritability is the fraction of your selection that actually transmits; the rest was environment, and environment does not inherit.

Both terms are levers, and they behave differently. Selection differential is under the breeder's control — select more intensely and $S$ rises — but the returns diminish, because taking the top 1% instead of the top 10% raises the standardised differential only from about 1.76 to 2.67 standard deviations while cutting the number of parents by a factor of ten. That reduction is not free: fewer parents means a smaller effective population, faster inbreeding and less variation to select on next time. Heritability, by contrast, can be raised by better trial design rather than by any genetic change at all.

The equation's honest limits are worth stating. It predicts one generation, not many, because selection itself changes the variance it depends upon — repeated selection erodes additive variance and response decays. It assumes the trait is measured on the selected individuals and that they are the actual parents. And it assumes no genotype-by-environment interaction, which is exactly the assumption that fails when a variety selected in one region underperforms in another.

A realised heritability computed from an observed response, $R/S$, that comes out above 1 is a warning rather than a discovery. The usual cause is comparing a selected group grown in one year against a base population grown in another, so a year effect has been read as genetic gain.