Heritability: Broad and Narrow Sense

Building on the partition of phenotypic variance, V_P = V_G + V_E (ignoring gene-environment covariance and interaction for a first pass), heritability formalizes what fraction of that total variance is genetic. There are two heritabilities because "genetic variance" itself has structure. V_G can be decomposed further into:

V_G = V_A + V_D + V_I where V_A is additive genetic variance (the variance in "average effects" of alleles, summed across loci), V_D is dominance variance (arising from within-locus interactions between the two alleles at a heterozygous site), and V_I is epistatic or interaction variance (arising from interactions between loci). This decomposition, introduced by Ronald Fisher in 1918, is the mathematical foundation of quantitative genetics.

Broad-sense heritability is defined as:

H² = V_G / V_P = (V_A + V_D + V_I) / V_P It answers: "Of all the variation in this trait in this population, what proportion traces to genotype in any form?" This is the relevant quantity for questions like "would cloning an individual reproduce its phenotype?" — because cloning copies the whole genotype, dominance and epistatic combinations included.

Narrow-sense heritability is defined as:

h² = V_A / V_P It answers a narrower but far more practically important question: "Of all the phenotypic variation, what proportion is due to the additive effects of alleles that get passed from parent to offspring?" Because sexual reproduction reshuffles alleles at fertilization — a parent transmits single alleles, not intact diploid genotypes or epistatic combinations — only the additive portion of genetic variance reliably predicts resemblance between relatives and response to selection. Dominance and epistatic variance are "broken up" by meiosis and fertilization each generation; a superb heterozygous combination in a prize bull is not transmitted intact to its offspring. This is why h² ≤ H² always, with equality only when V_D = V_I = 0 (purely additive gene action, which is common for many quantitative traits but not universal).

Step 1: Start from the phenotype. For a quantitative trait like height, litter size, or milk yield, an individual's phenotypic value P results from its genotypic value G and its environmental deviation E: P = G + E (+ G×E interaction, often folded into V_E or treated separately)

Step 2: Decompose the genotypic value itself. At a single locus with alleles A₁ and A₂, Fisher's insight was to define the average effect of an allele — the average deviation in phenotype produced by substituting that allele into a random genetic background in the population. The genotypic value of each genotype (A₁A₁, A₁A₂, A₂A₂) can then be split into:

• a part predictable from the sum of average allelic effects (the additive or "breeding value" component), and
• a leftover part due to dominance (interaction between the two alleles at that locus).

Across many loci, further leftover variance arises from epistasis — statistical interaction between genotypes at different loci, such that the genotypic value cannot be predicted by simply summing single-locus contributions.

Step 3: Variance, not value, is what matters. Because heritability is about population variance, what counts is not the average effect but how alleles' effects vary across the population — i.e., how much of the spread in phenotype from individual to individual is attributable to additive allelic differences versus dominance/epistatic combinations.

Step 4: The transmission bottleneck. A parent passes on gametes containing single alleles at each locus, not paired genotypes. Because of this, the specific dominance interaction (heterozygote advantage, say) and specific epistatic combination present in a parent are not transmitted intact — they are reconstituted anew (or not) in the offspring, depending on which alleles come together at fertilization. Only the additive, allele-by-allele average effects are faithfully passed down. This is the mechanistic reason h² — not H² — governs the resemblance between parents and offspring, and hence predicts the response to artificial or natural selection.

Step 5: The breeder's equation. If you select breeding individuals whose phenotypes deviate from the population mean by a selection differential S, the expected shift in the offspring generation's mean, the response to selection R, is:

R = h² S This equation, central to animal and plant breeding, is a direct mechanistic consequence of the additive/non-additive distinction: only the additive genetic variance is what gets reliably inherited and thus what selection can act on across a single generation.

Fisher's 1918 synthesis. The theoretical foundation for both H² and h² was laid by Ronald Fisher in his 1918 paper "The Correlation Between Relatives on the Supposition of Mendelian Inheritance." At the time, biometricians (led by Karl Pearson) and Mendelians were in open conflict: biometricians studied continuous traits statistically and doubted Mendelian particulate inheritance could explain smooth, continuous variation like height; Mendelians studied discrete traits and doubted statistics was needed. Fisher showed mathematically that if a continuous trait is controlled by many Mendelian loci of small, additive effect (the "infinitesimal model," combined with environmental noise), the resulting population distribution would be continuous and normal-like, and correlations between relatives could be predicted exactly from Mendelian principles. He derived the expected correlation between relatives (parent-offspring, full sibs, half-sibs) in terms of V_A, V_D, and V_I — reconciling biometry with Mendelism and inventing the variance-components framework that h² and H² are built on. This was not an experiment in the Mendel or Meselson-Stahl sense — it was a theoretical unification — but it was tested against, and matched, existing biometric correlation data on human and agricultural traits.

Selection experiments as the empirical proof of concept for h². The most direct empirical evidence that narrow-sense heritability — not broad-sense — predicts response to selection comes from long-running artificial selection experiments. A classic case is the Illinois Long-Term Selection Experiment on corn oil and protein content, begun in 1896 by Cyril Hopkins and continued for over 100 generations. Maize was selected each generation for high or low oil content in the kernel, using only phenotype (no genotyping) to choose parents. Starting from a population with roughly 4.7% oil content, after around 100 generations of selection the high-oil line reached oil content several times the starting value (roughly 20%), and the low-oil line dropped to near 0%, with divergence still measurably continuing after a century. This sustained, cumulative response is exactly what the breeder's equation R = h²S predicts when h² is greater than zero and additive genetic variance is continually being generated or maintained (partly by mutation and recombination) — a broad-sense-only account (in which most variance were dominance or epistatic) would predict the response to plateau almost immediately, since non-additive combinations cannot accumulate directionally across generations the way additive effects can.

Parent-offspring regression as a direct measurement. A simpler and very concrete demonstration: heritability can be estimated as (roughly) twice the regression slope of offspring phenotype on the average of the two parents ("midparent") — the "midparent-offspring regression," pioneered by Francis Galton (1886) though Galton's own interpretation predated Fisher's variance decomposition and was not framed in terms of h². For a purely additive trait with random mating, the slope of offspring value on midparent value is a direct empirical estimate of h². This method, applied across decades of agricultural and human data, converges with independent estimates from twin and half-sib designs, providing convergent evidence across different experimental designs that the additive-variance concept is measuring something real and predictive, not merely a theoretical abstraction.

Example 1 — comparing H² and h² in a dairy cattle trait. Suppose researchers measure milk yield in a dairy herd and estimate:

Check: V_G = V_A + V_D + V_I = 100 + 40 + 20 = 160. V_P = V_G + V_E = 160 + 240 = 400. ✓ consistent. Broad-sense heritability: H² = V_G / V_P = 160/400 = 0.40 Narrow-sense heritability: h² = V_A / V_P = 100/400 = 0.25 Interpretation: 40% of the variance in milk yield across cows in this herd, in this environment, is attributable to genotype in any form. But only 25% is attributable to additive effects — the portion that predicts how much daughters will resemble their dams and sires, and the portion breeders can actually act on. If a farmer selects the top 20 cows by yield, with a selection differential S = 300 liters above the herd mean, the expected gain in the next generation is: R = h² × S = 0.25 × 300 = 75 liters Not 0.40 × 300 = 120 liters — using H² instead of h² would substantially overpredict the response, because the dominance and epistatic variance (V_D + V_I = 60, or 15% of V_P) contributes to how good these particular cows look, but is not faithfully passed to their calves. This is precisely why cattle and plant breeders report and use h², not H², in genetic evaluations.

Example 2 — a human behavioral genetics scenario with context-dependence. A twin study estimates the narrow-sense heritability of a cognitive test score at h² = 0.60 in a sample of twins raised in a range of socioeconomic environments in a wealthy country with near-universal access to schooling and nutrition. A well-meaning newspaper reports: "Intelligence is 60% genetic — environment barely matters." Reason through what h² = 0.60 actually licenses: 1. It means that, within this specific population, in this specific range of environments actually sampled, 60% of the variance in test scores among individuals is statistically attributable to additive genetic differences among them. 2. It says nothing about the average level of the trait — heritability is silent on means. A population could have h² = 0.9 for height and still be, on average, several centimeters shorter than another population purely due to nutrition (a classic thought experiment due to Richard Lewontin, comparing two plots of genetically identical seed corn grown in rich versus poor soil: h² = 1.0 within each plot, yet the mean difference between plots is 100% environmental). 3. If this population were instead sampled across a much wider range of environments — say, including regions with severe childhood malnutrition or no schooling — V_E would increase substantially. Since h² = V_A/(V_A+V_D+V_I+V_E), a large increase in V_E, with V_A unchanged, would mechanically lower h², even though nothing about the genetics changed. Heritability estimates are therefore population- and environment-specific, not fixed biological facts about "intelligence." 4. It does not mean the trait is unchangeable: a classic illustration is phenylketonuria (PKU), a single-gene disorder with heritability near 1.0 for the biochemical predisposition, yet its cognitive-impairment phenotype is now almost entirely prevented by a simple environmental intervention (dietary phenylalanine restriction from infancy) — high heritability coexisting with complete environmental malleability of outcome. The correct headline would be something like: "In this population and range of environments, 60% of test-score variation tracks additive genetic differences among individuals — this tells us nothing about mean group differences, nothing about individual causation, and could shift substantially if the environmental range sampled changes."

Ronald Aylmer Fisher, working largely alone at Rothamsted Experimental Station in England, unified two warring schools of biology — Mendelian genetics and biometrical statistics — into a single mathematical framework in his landmark 1918 paper. At the time, the biometricians (following Karl Pearson and W.F.R. Weldon) had spent nearly two decades arguing that Mendelian "unit characters" like Mendel's smooth versus wrinkled peas could not possibly explain continuously varying traits like human height or crop yield, which show smooth bell-curve distributions rather than discrete classes. Fisher, then in his late twenties and without an academic post, showed that if many Mendelian loci each contribute a small additive effect to a trait, plus environmental noise, the population distribution becomes effectively continuous and normally distributed — resolving the apparent conflict entirely in favor of Mendelism while preserving the statistical tools biometricians had developed. In the same paper he introduced the term "variance" itself and derived the correlations expected between different classes of relatives under Mendelian inheritance, laying the mathematical groundwork for both broad- and narrow-sense heritability, the analysis of variance (ANOVA), and much of the theoretical basis of modern quantitative genetics and animal/plant breeding.

Agricultural breeding. Genomic selection in dairy cattle, poultry, and crop breeding programs relies on estimating narrow-sense heritability (and increasingly, direct estimates of additive genetic variance from genome-wide marker data, "genomic h²") to predict breeding values and rank selection candidates. The Illinois corn oil experiment's century-long, still-ongoing response is the textbook proof that selecting on h² compounds gains generation after generation.

Conservation genetics. Estimating h² for traits like disease resistance or reproductive timing in small, endangered populations helps predict whether a population retains the additive genetic variance needed to adapt to environmental change (e.g., climate warming), versus populations so inbred that V_A has been largely exhausted and adaptive response is unlikely.

Human genetics and GWAS. Modern genome-wide association studies (GWAS) estimate "SNP-heritability" — the proportion of phenotypic variance explained by common genetic variants across the genome — for traits from height to schizophrenia risk. These estimates are narrow-sense-like (additive) by construction and are used to build polygenic risk scores; understanding that h² is population- and environment-bound is essential for correctly (and cautiously) interpreting cross-population portability of such scores, a major current concern in precision medicine.

Twin and family study interpretation. Estimates of h² from twin designs (monozygotic versus dizygotic twin comparisons) are widely used in behavioral genetics and psychiatry to partition variance in traits like depression risk or educational attainment — work that leads directly into the classical twin-study methodology and the assumptions (equal environments, no gene-environment correlation) it requires to be valid.

Public understanding and policy debates. Because heritability estimates are so easily misread as claims about individual causation or fixed destiny, correctly explaining H² versus h² — and their strict population/environment dependence — is central to responsibly communicating genetics in debates over education policy, medicine, and social inequality.

Ask the tutor: How does the breeder's equation change if selection acts over multiple generations while additive genetic variance is being depleted by selection itself — and how do estimates like parent-offspring regression or twin studies actually recover h² from real data?