Narrow-Sense vs Broad-Sense Heritability: Additive Variance Partitioned

What Heritability Actually Measures

Heritability is a property of a population in an environment, not of an individual or a trait in the abstract. It answers one narrow question: of the variation we see in a trait among individuals, how much tracks genetic differences between them? The total phenotypic variance is partitioned as V_P = V_G + V_E + V_GxE, where V_G is genetic variance, V_E environmental variance, and V_GxE the gene-environment interaction.

Genetic variance itself splits further: V_G = V_A + V_D + V_I. V_A is additive variance (the sum of average allele effects that parents transmit), V_D is dominance variance (interactions between the two alleles at one locus), and V_I is epistatic/interaction variance (interactions between loci).

• Broad-sense heritability H² = V_G / V_P — the whole genetic slice.
• Narrow-sense heritability h² = V_A / V_P — only the additive, transmissible slice.

Because only additive effects reliably pass parent-to-offspring, h² is the number that predicts response to selection, while H² captures all genetic determination including non-transmissible dominance and epistasis.

How the Variance Is Partitioned, Step by Step

Consider a single biallelic locus with alleles A and a at frequencies p and q. Assign genotypic values +a (AA), d (Aa), and −a (aa), where d encodes dominance. The average effect of an allele substitution is α = a + d(q − p). From this Fisher derived the components:

• V_A = 2pq·α² — variance in the additive (breeding) values that offspring inherit.
• V_D = (2pq·d)² — variance from dominance deviations, which shuffle each generation.

The key insight is that meiosis transmits single alleles, not genotypes. A heterozygote's Aa combination is broken up when gametes form, so its dominance advantage is not passed on; only the average effect α of each allele carries forward. This is why the breeding value — twice the mean deviation of an individual's offspring from the population mean — depends solely on V_A.

Epistatic variance V_I (additive-by-additive, additive-by-dominance, dominance-by-dominance) is likewise mostly reshuffled by recombination. So h² is the fraction of phenotypic variance that behaves predictably across generations.

Breeding Values, the Breeder's Equation, and Real Numbers

The practical payoff of h² is the breeder's equation: R = h²·S, where S is the selection differential (how much selected parents deviate from the population mean) and R is the response (how much the offspring mean shifts). This is a direct consequence of the parent-offspring regression having slope h².

Concrete values from decades of quantitative genetics:

• Human height: h² ≈ 0.8 (twin/pedigree estimates); SNP-based estimates ≈ 0.5, and ~3,300 GWAS loci explain much of it.
• Dairy cattle milk yield: h² ≈ 0.25–0.30 — modest, yet decades of selection roughly doubled yield.
• Human IQ: H² ≈ 0.5–0.8 in adults, rising with age.
• Body mass index: h² ≈ 0.4–0.7.
• Fitness-related traits (litter size, viability): low h², often < 0.2, because selection erodes V_A.

Example: if h² = 0.3 for milk yield and breeders select cows averaging 500 kg above the mean (S = 500), expected gain is R = 0.3 × 500 = 150 kg per generation.

How Heritability Is Estimated

No single experiment measures V_A directly; it is inferred from resemblance between relatives, whose expected covariance is a known function of the variance components.

• Parent-offspring regression: slope of offspring on mid-parent value estimates h² directly.
• Twin studies (Falconer's formula): H² ≈ 2(r_MZ − r_DZ), comparing monozygotic (100% shared genes) with dizygotic (~50%) correlations. If MZ correlate 0.9 and DZ 0.6, H² ≈ 0.6.
• Sib and half-sib designs: ANOVA partitions between- and within-family variance; the sire component isolates ¼V_A.
• GREML / GCTA: uses genome-wide SNP relatedness in unrelated individuals to estimate SNP heritability (h²_SNP), the additive variance tagged by common variants.

The classic landmark is R.A. Fisher's 1918 paper "The Correlation between Relatives on the Supposition of Mendelian Inheritance," which reconciled Mendelism with continuous variation and defined V_A. Sewall Wright and later Douglas Falconer (whose 1960 textbook standardized the field) built the estimation machinery. Jay Lush applied it to animal breeding, coining "breeding value."

Broad- vs Narrow-Sense and Their Cousins

The two heritabilities diverge exactly when non-additive genetics is large. For identical (clonal) plants or MZ twins, H² is what matters because whole genotypes are copied, but for breeding and evolutionary prediction, only h² works.

• H² > h² whenever dominance or epistasis contributes (V_D, V_I > 0). The gap is the "non-additive" genetic variance.
• Realized heritability is h² back-calculated from an actual selection experiment via R/S, a check on predicted response.
• Repeatability sets an upper bound on H² by including permanent environmental effects across repeated measures.
• Liability-scale heritability converts the observed 0/1 heritability of a disease to an underlying continuous liability, needed for comparing across prevalences.

Distinguish heritability from heredity (transmission itself) and from Hardy-Weinberg (allele-frequency equilibrium, which supplies the p, q terms in the variance formulas). A trait can be 100% genetically determined (a Mendelian disease) yet have low heritability in a population where its causal allele is rare or fixed.

Why It Matters, Pitfalls, and Open Questions

Heritability underpins modern breeding, agriculture, and human genetics, but it is chronically misread. It does not say how genetic a trait is in a fixed sense, nor how modifiable it is. A trait with h² = 0.8 can still be shifted entirely by environment: PKU (phenylketonuria) is highly heritable yet fully preventable with a low-phenylalanine diet. Heritability is bounded to the population and environment measured — change either and the number changes.

• Missing heritability: for height, GWAS long explained a fraction of pedigree h² (~0.8); the gap has narrowed with larger samples, rarer variants, and better tagging, but remains an active puzzle across traits.
• Gene-environment interaction and covariance (V_GxE, Cov(G,E)) can inflate or bias estimates.
• Assortative mating and shared family environment can masquerade as additive variance in twin designs.
• Portability: polygenic scores built from one ancestry predict poorly in others, exposing how population-specific h² is.

Open questions include how much epistasis truly contributes to complex-trait variance and how to disentangle direct genetic effects from genetic nurture.

Narrow-sense vs broad-sense heritability and their variance components

Property	Narrow-sense h²	Broad-sense H²
Formula	V_A / V_P	V_G / V_P
Variance included	Additive only (V_A)	Additive + dominance + epistasis (V_A + V_D + V_I)
Transmitted parent→offspring	Yes — predicts breeding value	No — dominance/epistasis reshuffled each generation
Predicts response to selection	Yes, via R = h²·S	No
Best estimated by	Parent-offspring regression, sib designs, GREML	MZ vs DZ twin comparison, clonal replicates
Human height (typical)	≈ 0.5–0.8	≈ 0.8