The Breeder's Equation: Predicting Response to Selection (R = h²S)

What the Breeder's Equation Is and Where It Applies

The breeder's equation, R = h²S, is the central predictive rule of quantitative genetics—the study of traits controlled by many genes plus environment rather than single Mendelian loci. It applies to continuous, polygenic traits: height, body weight, yield, milk volume, egg number, disease resistance, flowering time.

Each symbol has a precise definition:

• R (response to selection) — the difference between the offspring generation's mean and the parental generation's mean. It is what you get.
• S (selection differential) — the difference between the mean of the selected parents and the mean of the whole population before breeding. It is what you impose.
• h² (narrow-sense heritability) — the fraction of total phenotypic variance (V_P) due to additive genetic variance (V_A): h² = V_A / V_P. It ranges from 0 to 1.

The equation lives at the population level, not inside any cell. It was formalized by Jay L. Lush in his 1937 Animal Breeding Plans, building on R. A. Fisher's 1918 partition of variance and Sewall Wright's path analysis.

The Mechanism, Step by Step

The logic runs in a tight sequence each generation:

• 1. Measure the population. Record the trait in every candidate and compute the mean (μ) and phenotypic variance V_P.
• 2. Select parents. Choose the individuals to breed—say the top 20%. Their mean minus μ is the selection differential S.
• 3. Only additive value transmits. An individual's phenotype = additive genetic value + dominance + epistasis + environment. Only the additive part is reliably passed to offspring, because parents transmit single alleles, not their diploid genotype. This is why h² (not broad-sense H²) is the multiplier.
• 4. Regress offspring on parents. Offspring mean deviates from μ by h² × S. Geometrically, h² is the slope of the offspring-on-midparent regression line.
• 5. Realize the response. The offspring-generation mean shifts by R = h²S.

An equivalent, breeder-friendly form separates intensity from scale: R = h² · i · σ_P, where i is the standardized selection intensity (S in standard-deviation units) and σ_P is the phenotypic standard deviation. Selecting the top 20% gives i ≈ 1.40; the top 5% gives i ≈ 2.06.

Key Quantities and a Concrete Example

The variance components underneath h² are the real machinery. Total phenotypic variance partitions as V_P = V_A + V_D + V_I + V_E (additive, dominance, epistatic/interaction, and environmental). Narrow-sense heritability uses only additive variance, h² = V_A / V_P, whereas broad-sense heritability H² = V_G / V_P includes all genetic variance.

Worked example — dairy cattle. Milk yield has h² ≈ 0.30. If elite bulls are chosen whose daughters average 2000 kg above the herd mean (S = 2000 kg), the predicted response is R = 0.30 × 2000 = 600 kg per generation. Over several generations this compounds, which is exactly how Holstein yields climbed from roughly 5000 kg to over 10,000 kg per lactation in the 20th century.

Canonical long-term case — the Illinois maize experiment. Begun in 1896 by Cyril Hopkins, it selected on kernel oil and protein. With h² for oil around 0.3–0.5 and steady selection, the high-oil line rose from about 4.7% to over 20% oil across ~100 generations—an ongoing demonstration that R = h²S predicts real, cumulative gains until additive variance is depleted.

How Heritability and Response Are Measured

Because h² is a population statistic, not a property of an individual, it must be estimated—and every estimate is specific to one population in one environment.

• Parent–offspring regression. The slope of offspring value on midparent value directly estimates h². This is the most intuitive method.
• Sib analysis / ANOVA. Comparing variance among vs. within full-sib and half-sib families partitions V_A from V_E. Half-sib designs are the workhorse of animal breeding.
• Twin and family designs (humans). Contrasting monozygotic and dizygotic twin correlations estimates heritability of traits like height (h² ≈ 0.8).
• Genomic estimation (GREML / GCTA). Modern SNP-based methods estimate additive variance directly from genome-wide markers.
• Realized heritability. Run a selection experiment, then compute h²_realized = R / S after the fact. Comparing predicted vs. realized h² is the definitive test of the equation.

The response is regulated by anything that changes V_A: inbreeding erodes it, new mutation replenishes it (~10⁻³ σ_A² per generation), and environmental noise inflates V_P and thus shrinks h².

How It Differs From Related Rules

The breeder's equation is the simplest member of a family of selection-response models; knowing its boundaries is essential.

• vs. broad-sense heritability (H²). Using H² overpredicts response, because dominance and epistatic value do not transmit intact through gametes. Only additive variance responds to selection—hence h², not H².
• vs. the multivariate Lande equation. Traits are correlated. The Lande–Arnold equation Δz̄ = G·β generalizes R = h²S to many traits at once, replacing h² with the genetic variance–covariance matrix G and S with the selection gradient β. It predicts correlated responses—selecting on one trait drags others along.
• vs. the secondary theorem / Robertson's rule. The change in mean fitness equals the additive genetic variance in fitness (Fisher's fundamental theorem is the special case).
• vs. Hardy–Weinberg. Hardy–Weinberg describes allele frequencies at a single locus under no selection; the breeder's equation describes the mean of a polygenic trait under selection.

Significance, Limits, and Open Questions

The breeder's equation underpins the entire modern food supply: crop yield gains, livestock improvement, and aquaculture programs are all planned with it, and it is the conceptual backbone of evolutionary quantitative genetics for predicting microevolution in the wild.

But it is a short-term tool with well-known failure modes:

• Selection limits (plateaus). As favorable alleles fix, V_A—and therefore h² and R—decline toward zero. Every closed selection program eventually plateaus.
• The wild-population problem. Applied to natural populations, R = h²S routinely fails to predict evolution because the environmental covariance between a trait and fitness inflates S. The Robertson–Price identity and the secondary theorem give more reliable predictions in the wild.
• Missing heritability. For human traits like height, genome-wide SNPs long recovered far less additive variance than pedigree h² implied—an active problem now partly resolved by rare variants and whole-genome sequencing.

Open questions include how non-additive variance and gene-by-environment interaction can be harnessed, and how genomic selection (predicting breeding value directly from markers) reshapes the classic equation.

Worked examples of R = h²S across real traits and selection scenarios

Trait / system	Heritability h²	Selection differential S	Predicted response R = h²S
Human adult height	≈ 0.80	Parents +10 cm above mean	+8.0 cm in offspring mean
Dairy cow milk yield	≈ 0.30	Bulls' daughters +2000 kg	+600 kg/lactation
Maize kernel oil (Illinois)	≈ 0.30–0.50	Top ears each cycle	≈ +0.10–0.30% per generation, cumulative to >20%
Drosophila bristle number	≈ 0.50	Select high-bristle flies +4	+2 bristles per generation
Broiler chicken body weight	≈ 0.35	Heaviest breeders +300 g	+105 g per generation
Fitness (natural selection)	≈ 0.10–0.20	Weak, fluctuating	Small, often near zero net change