The Breeder’s Equation
\[R = h^2 S\]where $S$ is the selection differential (the mean phenotype of selected parents minus the population mean) and $h^2$ is narrow-sense heritability. Equivalently, in terms of selection intensity $i$ (the standardised selection differential):
\[R = i\, h^2\, \sigma_P = i\, h\, \sigma_A\]For a truncation selection scheme where the top fraction $p$ of a normal distribution is selected, $i = \phi(t)/p$ where $t$ is the truncation point and $\phi$ is the standard normal density.
Selection Differential and Intensity
| Proportion selected ($p$) | Selection intensity ($i$) |
|---|---|
| 0.01 | 2.665 |
| 0.05 | 2.063 |
| 0.10 | 1.755 |
| 0.25 | 1.271 |
| 0.50 | 0.798 |
The Selection Index
When multiple traits are measured, the optimal linear index for selection on a single goal trait uses index coefficients $\mathbf{b}$:
\[I = \mathbf{b}^\top \mathbf{x}, \quad \mathbf{b} = \mathbf{P}^{-1}\mathbf{G}\mathbf{a}\]where $\mathbf{P}$ is the phenotypic variance-covariance matrix, $\mathbf{G}$ is the genetic variance-covariance matrix, and $\mathbf{a}$ is a vector of economic weights. The index maximises the correlation between $I$ and the aggregate genotype $H = \mathbf{a}^\top \mathbf{g}$.
Long-Term Response and Limits
Sustained directional selection erodes additive genetic variance: $V_A$ declines as favourable alleles approach fixation. The Robertson–Hill limit estimates the total response from drift and selection in a finite population of size $N_e$:
\[R_{\infty} \approx 2\, N_e\, R_1\]where $R_1$ is the single-generation response. In practice, mutation-selection balance and frequency-dependent effects sustain long-term response beyond this prediction.