19.05.02 · eco-evo-bio / quantitative-genetics

Twin studies and the estimation of heritability: additive versus dominance variance

stub3 tiersLean: nonepending prereqs

Anchor (Master): Falconer, D. S. & Mackay, T. F. C. — Introduction to Quantitative Genetics, 4th ed. (1996)

Intuition Beginner

Why do siblings raised in the same home differ in height, personality, and intelligence? Part of the answer lies in genetics — even full siblings share only about half their DNA. But how much of the variation we observe comes from genes, and how much from the environment?

Heritability is the fraction of the variation in a trait across individuals that is due to genetic differences. It is not about any single person. It is a statement about a population: of all the variation we see in height (or IQ, or body mass index) among people in a given environment, how much is attributable to genetic differences between them?

Twin studies provide one of the most powerful natural experiments to answer this question. Identical twins (monozygotic, MZ) originate from a single fertilised egg and share essentially 100% of their DNA. Fraternal twins (dizygotic, DZ) originate from two separate eggs and share about 50% of their DNA — the same as any pair of full siblings. Both types of twins share a womb, a household, and much of their upbringing. If identical twins are consistently more similar than fraternal twins for a given trait, the extra similarity must come from the extra genetic sharing.

Traits like human height have high heritability (roughly 80% in Western populations): identical twins are almost identical in height, while fraternal twins show moderate resemblance. Traits heavily shaped by environment — like which language a person speaks — have near-zero heritability: both types of twins are equally concordant because language is learned, not inherited.

Two critical caveats. First, heritability is specific to the population and environment in which it is measured. If everyone has identical nutrition, height heritability goes up (less environmental variance); if nutrition varies wildly, heritability goes down. Second, high heritability does not mean a trait is fixed or unchangeable. A trait with heritability 0.9 can still shift dramatically when the environment changes.

Visual Beginner

Twin studies compare the similarity (correlation) of a trait between identical (MZ) and fraternal (DZ) twin pairs. The wider the gap between the two correlations, the larger the role of genetics.

The vertical gap between the MZ bar and the DZ bar reflects the extra genetic sharing of identical twins. Falconer's formula estimates narrow-sense heritability as $h^{2} = 2 (r_{M Z} - r_{D Z})$ . If both bars are equally tall, genetics contributes nothing beyond the shared environment.

Worked example Beginner

A twin study of body mass index (BMI) measures the correlation between twin pairs:

Identical (MZ) twins: $r_{M Z} = 0.80$
Fraternal (DZ) twins: $r_{D Z} = 0.45$

Falconer's formula estimates the narrow-sense heritability:

h^{2} = 2 (r_{M Z} - r_{D Z}) = 2 (0.80 - 0.45) = 2 \times 0.35 = 0.70.

About 70% of the variation in BMI among individuals in this population is attributable to additive genetic differences. The remaining 30% comes from environmental variation and from non-additive genetic effects (dominance and epistasis).

The broad-sense heritability can also be estimated:

H^{2} = 2 r_{M Z} - r_{D Z} = 2 (0.80) - 0.45 = 1.60 - 0.45 = 1.15.

This value exceeds 1.0, which is impossible for a true heritability. The impossible estimate signals a violation of the model assumptions — most likely, identical twins share more of their environment than fraternal twins do (the equal-environments assumption fails), or gene-environment interaction inflates the MZ correlation.

Check your understanding Beginner

Exercise (easy, multiple choice).

A trait has narrow-sense heritability $h^{2} = 0.50$ in a population. Which of the following is correct?

A. Each individual's trait value is 50% determined by genes B. 50% of the variation in the trait among individuals is due to additive genetic differences C. The trait will change by 50% per generation under selection D. 50% of individuals carry the gene for this trait

Hint

Heritability is a population-level statistic about variance, not an individual-level claim.

Answer

B. Narrow-sense heritability is the ratio $V_{A} / V_{P}$ . It describes what proportion of the phenotypic variance across individuals in a specific population and environment is attributable to additive genetic variance. It says nothing about any single individual, and it is not a fixed property of the trait itself.

Formal definition Intermediate+

Phenotypic variance decomposition

The total phenotypic variance $V_{P}$ partitions into genetic, environmental, and gene-environment components:

V_{P} = V_{G} + V_{E} + V_{GE},

where $V_{G}$ is the total genetic variance, $V_{E}$ is the environmental variance, and $V_{GE}$ captures gene-environment interaction (the effect of a genotype depends on the environment). Gene-environment covariance, $Cov (G, E)$ , is often grouped into the residual or modelled separately.

The genetic variance further decomposes:

V_{G} = V_{A} + V_{D} + V_{I},

where $V_{A}$ is additive variance (the variance due to the average effects of individual alleles, transmitted predictably from parent to offspring), $V_{D}$ is dominance variance (variance from interactions between alleles at the same locus), and $V_{I}$ is epistatic variance (variance from interactions between alleles at different loci).

Heritability

Broad-sense heritability is the proportion of phenotypic variance attributable to all genetic effects:

H^{2} = \frac{V _{G}}{V _{P}} = \frac{V _{A} + V _{D} + V _{I}}{V _{P}} .

Narrow-sense heritability is the proportion attributable to additive effects alone:

h^{2} = \frac{V _{A}}{V _{P}} .

Since $V_{A} \leq V_{G}$ , we have $h^{2} \leq H^{2}$ . The gap $H^{2} - h^{2} = (V_{D} + V_{I}) / V_{P}$ reflects the contribution of non-additive genetic effects. For predicting response to selection, $h^{2}$ is the relevant quantity because only additive variance transmits faithfully across generations (dominance and epistatic contributions are reshuffled by segregation and recombination each generation).

Twin study logic

Monozygotic (MZ) twins share 100% of their genotypic value — all additive, dominance, and epistatic components are identical. Dizygotic (DZ) twins share on average 50% of their additive genetic value, 25% of their dominance deviations, and a variable fraction of epistatic effects. Both types share the same womb and, typically, the same household.

Under the standard twin model, the phenotypic correlations are:

r_{M Z} = \frac{V _{A} + V _{D} + V _{C}}{V _{P}}, r_{D Z} = \frac{\frac{1}{2} V _{A} + \frac{1}{4} V _{D} + V _{C}}{V _{P}},

where $V_{C}$ is the variance due to the shared (common) environment.

Falconer's formulas

Subtracting the DZ correlation from the MZ correlation:

r_{M Z} - r_{D Z} = \frac{\frac{1}{2} V _{A} + \frac{3}{4} V _{D}}{V _{P}} .

When $V_{D}$ is small relative to $V_{A}$ (as is typical for many complex traits):

r_{M Z} - r_{D Z} \approx \frac{V _{A}}{2 V _{P}} = \frac{h ^{2}}{2},

giving Falconer's formula for narrow-sense heritability:

h^{2} \approx 2 (r_{M Z} - r_{D Z}) .

A parallel formula estimates broad-sense heritability:

H^{2} \approx 2 r_{M Z} - r_{D Z} .

When dominance variance is non-negligible, $2 (r_{M Z} - r_{D Z})$ overestimates $h^{2}$ because the $3 V_{D} /4$ term inflates the MZ-DZ gap beyond the additive component.

The ACE model

The ACE model partitions phenotypic variance into three latent sources:

A — additive genetic variance ( $V_{A}$ )
C — shared (common) environmental variance ( $V_{C}$ )
E — unique (non-shared) environmental variance ( $V_{E}$ ), plus measurement error

The model assumes $V_{P} = V_{A} + V_{C} + V_{E}$ with $V_{D} = V_{I} = 0$ (no dominance or epistasis). The twin correlations under ACE are:

r_{M Z} = \frac{V _{A} + V _{C}}{V _{P}}, r_{D Z} = \frac{\frac{1}{2} V _{A} + V _{C}}{V _{P}} .

Structural equation modelling (SEM) fits these parameters by maximum likelihood to the observed MZ and DZ covariance matrices. The ACE model can be extended to an ADE model (replacing C with D for dominance) when the MZ correlation exceeds twice the DZ correlation — a pattern indicating dominance variance.

Heritability of specific traits

Representative narrow-sense heritability estimates from twin studies:

Trait	$h^{2}$ (approximate)
Height	0.80
BMI	0.50
IQ	0.50 - 0.80 (increases with age)
Type 2 diabetes (liability)	0.30 - 0.40
Schizophrenia (liability)	0.80

These estimates vary across populations and environmental conditions. IQ heritability increases from about 0.20 in early childhood to 0.60-0.80 in adulthood in Western samples, likely because adults select environments that amplify genetic predispositions.

Breeding values and response to selection

The breeding value of an individual is twice the expected deviation of its offspring from the population mean when mated to a random member of the population. The response to selection is predicted by the breeder's equation ^{[Falconer & Mackay 1996]}:

R = h^{2} S,

where $S$ is the selection differential. Only $V_{A}$ contributes to $R$ , which is why distinguishing $h^{2}$ from $H^{2}$ matters: a trait with high $H^{2}$ but low $h^{2}$ (most genetic variance is dominance or epistatic) responds poorly to selection. In animal breeding, this distinction directly affects genetic gain per generation.

Counterexamples to common slips

Falconer's formula does not give the "true" $h^{2}$ . It is an approximation that assumes no dominance, no gene-environment interaction, equal environments for MZ and DZ twins, and random mating. Violations of any assumption bias the estimate.
Heritability is not an individual-level quantity. Saying "your height is 80% genetic" is meaningless. Heritability describes variance across a population, not the genetic contribution to any single person's phenotype.
High heritability does not prevent environmental change. Height ( $h^{2} \approx 0.8$ ) increased 10-15 cm across a few generations due to improved nutrition. The breeder's equation $R = h^{2} S$ predicts genetic response to selection, not phenotypic change from environmental shifts.

Key theorem with proof Intermediate+

Theorem (Falconer's formula from twin correlations). Under the equal-environments assumption (MZ and DZ twins share environments to the same degree), no gene-environment interaction, and random mating, the narrow-sense heritability is $h^{2} = 2 (r_{M Z} - r_{D Z})$ when dominance variance is negligible.

Proof. Let $z_{1}, z_{2}$ be the phenotypes of a twin pair. Each phenotype decomposes as $z_{i} = μ + A_{i} + C_{i} + E_{i}$ , where $A_{i}$ is the additive genetic value, $C_{i}$ is the shared environmental deviation, and $E_{i}$ is the unique environmental deviation. Assume $A$ , $C$ , and $E$ are mutually uncorrelated with mean zero.

For MZ twins, who share 100% of their additive genetic value and 100% of their shared environment: $Cov (A_{1}^{M Z}, A_{2}^{M Z}) = V_{A}$ and $Cov (C_{1}, C_{2}) = V_{C}$ . The MZ phenotypic correlation is:

r_{M Z} = \frac{Cov ( z _{1}^{M Z} , z _{2}^{M Z} )}{V _{P}} = \frac{V _{A} + V _{C}}{V _{P}} .

For DZ twins, who share on average 50% of their additive genetic value: $Cov (A_{1}^{D Z}, A_{2}^{D Z}) = V_{A} /2$ . They share the same common environment: $Cov (C_{1}, C_{2}) = V_{C}$ . The DZ phenotypic correlation is:

r_{D Z} = \frac{V _{A} /2 + V _{C}}{V _{P}} .

Subtracting:

r_{M Z} - r_{D Z} = \frac{V _{A} + V _{C} - V _{A} /2 - V _{C}}{V _{P}} = \frac{V _{A} /2}{V _{P}} = \frac{h ^{2}}{2} .

Therefore:

h^{2} = 2 (r_{M Z} - r_{D Z}) . □

The key insight is that the shared environment $V_{C}$ cancels in the subtraction. This is why twin studies are powerful: the common-environment component is differenced out, isolating the additive genetic contribution. The proof fails when the equal-environments assumption is violated — if MZ twins experience more similar environments than DZ twins, the MZ correlation is inflated beyond what genetics alone would produce, and $h^{2}$ is overestimated.

Bridge. Falconer's formula is the simplest estimator of $h^{2}$ from twin data, but it is a point estimate from a pair of correlations. The ACE model ^{[Falconer & Mackay 1996]} generalises this to full structural equation modelling, fitting $V_{A}$ , $V_{C}$ , and $V_{E}$ simultaneously by maximum likelihood. This in turn connects to the animal model (the mixed-model framework covered in 19.05.01), which estimates $h^{2}$ from arbitrary pedigrees rather than just twin pairs. The progression is from a simple formula (Falconer) to a model (ACE) to a general framework (animal model), each relaxing assumptions and expanding the data requirements.

Exercises Intermediate+

Exercise 3 (medium, numeric).

For a trait with $r_{M Z} = 0.90$ and $r_{D Z} = 0.35$ , compute $h^{2}$ using Falconer's formula. Observe that $r_{M Z} > 2 \times r_{D Z}$ . What does this pattern suggest about dominance variance?

Hint

Under the ACE model (no dominance), $r_{M Z} = (V_{A} + V_{C}) / V_{P}$ and $r_{D Z} = (V_{A} /2 + V_{C}) / V_{P}$ . If $r_{M Z} > 2 r_{D Z}$ , then $V_{A} + V_{C} > V_{A} + 2 V_{C}$ , which implies $V_{C} < 0$ — impossible. What additional variance component could explain this?

Answer

$h^{2} = 2 (0.90 - 0.35) = 1.10$ . This exceeds 1.0, which is impossible. The pattern $r_{M Z} > 2 r_{D Z}$ signals that the ACE model is inadequate. The correct model is the ADE model, which includes dominance variance. Under the ADE model, the MZ correlation includes $V_{D}$ while the DZ correlation includes $V_{D} /4$ . The large MZ-DZ gap relative to the DZ correlation is the signature of dominance: MZ twins share all dominance interactions, while DZ twins share only 25%. An ADE fit would partition the variance into $V_{A}$ , $V_{D}$ , and $V_{E}$ instead of $V_{A}$ , $V_{C}$ , and $V_{E}$ .

Exercise 5 (medium, numeric).

A population has $V_{P} = 200$ , $V_{A} = 100$ , $V_{D} = 20$ , and $V_{I} = 10$ . Calculate $h^{2}$ , $H^{2}$ , and the expected MZ and DZ correlations under the standard twin model with $V_{C} = 30$ .

Hint

$h^{2} = V_{A} / V_{P}$ , $H^{2} = V_{G} / V_{P}$ . For correlations: $r_{M Z} = (V_{A} + V_{D} + V_{I} + V_{C}) / V_{P}$ (MZ twins share all genetic effects), $r_{D Z} = (V_{A} /2 + V_{D} /4 + V_{I}^{*} /4 + V_{C}) / V_{P}$ , where the epistatic sharing is approximately $V_{I} /4$ for two-locus interactions.

Answer

$V_{G} = 100 + 20 + 10 = 130$ . $h^{2} = 100/200 = 0.50$ . $H^{2} = 130/200 = 0.65$ .

$r_{M Z} = (100 + 20 + 10 + 30) /200 = 160/200 = 0.80$ .

$r_{D Z} = (50 + 5 + 2.5 + 30) /200 = 87.5/200 = 0.4375$ .

Falconer's formula gives $h_{es t}^{2} = 2 (0.80 - 0.4375) = 0.725$ , which overestimates the true $h^{2} = 0.50$ because the dominance and epistatic components inflate the MZ-DZ gap. This demonstrates the bias in Falconer's formula when non-additive genetic variance is present.

Exercise 6 (hard, symbolic).

Show that under assortative mating with spousal correlation $ρ$ for the trait, the DZ twin correlation for additive genetic effects increases from $V_{A} /2$ to $V_{A} (1 + ρ) /2$ . What effect does this have on Falconer's estimate of $h^{2}$ ?

Hint

Under assortative mating, the midparent breeding value has variance larger than under random mating. The genetic covariance between DZ twins increases because the parents are correlated on the trait.

Answer

Under random mating, the additive genetic covariance between DZ twins is $V_{A} /2$ because each twin inherits half of each parent's breeding value and the parental breeding values are uncorrelated. Under assortative mating with spousal correlation $ρ$ , the covariance between parental breeding values is $ρ V_{A}$ . The genetic covariance between DZ twins becomes:

Cov (A_{1}^{D Z}, A_{2}^{D Z}) = \frac{V _{A}}{2} + \frac{ρ V _{A}}{4} = \frac{V _{A} ( 2 + ρ )}{4} \approx \frac{V _{A} ( 1 + ρ )}{2} .

This inflates $r_{D Z}$ relative to the random-mating expectation, reducing the MZ-DZ gap and causing Falconer's formula to underestimate $h^{2}$ . For IQ, where assortative mating is substantial ( $ρ \approx 0.3$ - $0.5$ ), this bias is nontrivial. The correction requires estimating $ρ$ and adjusting the formula accordingly.

Exercise 7 (hard, conceptual).

A researcher finds that MZ twins reared apart have a correlation of 0.75 for a trait, while MZ twins reared together have a correlation of 0.78. DZ twins reared together have a correlation of 0.40. Compare the heritability estimate from the standard Falconer formula with the estimate from MZ twins reared apart. Which is more trustworthy and why?

Hint

MZ twins reared apart share 100% of their genes but different environments, so their correlation directly estimates $(V_{A} + V_{D}) / V_{P}$ without the shared-environment component. The standard Falconer formula uses MZ and DZ twins reared together.

Answer

Falconer estimate (twins reared together): $h^{2} = 2 (0.78 - 0.40) = 0.76$ .

MZ-apart estimate: since MZ twins reared apart share all their genetics but no shared environment, $r_{M Z apart} \approx (V_{A} + V_{D}) / V_{P} = H^{2} \approx 0.75$ . Under the assumption that $V_{D}$ is small, this gives $h^{2} \approx 0.75$ .

The MZ-apart estimate is more trustworthy because it bypasses the equal-environments assumption entirely. The near-agreement of the two estimates (0.76 vs 0.75) suggests that the equal-environments assumption is approximately valid for this trait — the shared environment contributes little ( $V_{C}$ is small). If the MZ-apart correlation were substantially lower than the Falconer estimate, that would indicate shared-environment inflation of the MZ-together correlation.

Studies of MZ twins reared apart (e.g., the Minnesota Study of Twins Reared Apart, Bouchard et al. 1990) have been influential precisely because they provide this internal check on the twin model.

Lean formalization Intermediate+

Mathlib has no formalisation of variance partitioning, heritability estimation, or twin study statistics. The specific gaps include: phenotypic variance decomposition into additive, dominance, and epistatic components; heritability as a ratio of variance components; Falconer's formula as a derived result from twin correlation expectations; the ACE model as a structural equation model; and the distinction between broad-sense and narrow-sense heritability. The foundational prerequisite is the variance-component algebra covered in 19.05.01; once that infrastructure exists in Mathlib, the twin-study derivations in this unit follow as straightforward corollaries. This unit ships without a Lean module and is reviewer-attested, following the prose-first contract for biology units.

Advanced heritability estimation: beyond the classical twin design Master

The classical twin design — comparing MZ and DZ correlations via Falconer's formula or the ACE model — is the entry point for heritability estimation, but modern quantitative genetics has extended the approach in several directions, each addressing limitations of the basic design.

SNP-based heritability and GCTA

Yang et al. (2010) ^{[Futuyma & Kirkpatrick 2017]} introduced GCTA (Genome-wide Complex Trait Analysis), which estimates heritability from genome-wide SNP data in unrelated individuals. The method constructs a genetic relatedness matrix (GRM) from all typed SNPs, then uses restricted maximum likelihood (REML) to partition phenotypic variance into a component tagged by the SNPs ( $h_{S N P}^{2}$ ) and a residual. The resulting $h_{S N P}^{2}$ is a lower bound on $h^{2}$ because SNPs capture only common variants and imperfectly tag causal variants through linkage disequilibrium.

For human height, GCTA estimates $h_{S N P}^{2} \approx 0.45$ - $0.55$ , compared to twin-study estimates of $h^{2} \approx 0.80$ . The missing heritability gap — the difference between family-based $h^{2}$ and SNP-based $h^{2}$ — has multiple contributors: rare variants poorly tagged by SNP arrays, structural variants, gene-gene and gene-environment interactions, and potential inflation of twin estimates by shared environmental effects.

Gene-environment correlation (rGE)

Gene-environment correlation occurs when individuals' genotypes are non-randomly associated with the environments they experience. Three types are distinguished:

Passive rGE: Parents transmit both genes and environments. A child of tall parents inherits height-promoting alleles and grows up in a nutrition-rich household. The genetic and environmental effects are confounded because the parents provide both.
Evocative rGE: An individual's genetically influenced traits evoke specific environmental responses. A child with a genetic predisposition to outgoing behaviour elicits more social interaction from peers and teachers, amplifying the genetic effect through the environment.
Active rGE (niche picking): Individuals actively select environments that match their genetic predispositions. A person with genetic aptitude for analytical thinking gravitates toward intellectually stimulating activities and careers, creating a gene-environment correlation that strengthens with age.

rGE inflates heritability estimates because the environmental component is partially genetic in origin. The increasing heritability of IQ with age (from ~0.20 in early childhood to ~0.60-0.80 in adulthood in Western samples) likely reflects active rGE: as individuals gain autonomy, they increasingly select environments that amplify their genetic predispositions.

Gene-environment interaction (GxE)

Gene-environment interaction occurs when the effect of a genotype depends on the environment, or equivalently, when environmental effects differ across genotypes. In the variance decomposition, GxE contributes to $V_{GE}$ , which the standard twin model subsumes into the residual.

GxE has critical implications for heritability estimation. A trait may have high $h^{2}$ in one environment and low $h^{2}$ in another — not because the genes changed, but because the environmental variance changed. The heritability of BMI is higher in populations with abundant food and low physical-activity demands than in populations with food scarcity and manual labour, because in the latter, environmental constraints dominate genetic variation.

Formally, if the phenotypic variance in environment $e$ is $V_{P} (e) = V_{G} + V_{E} (e)$ (ignoring GxE for simplicity), then $h^{2} (e) = V_{G} / (V_{G} + V_{E} (e))$ . As $V_{E} (e)$ increases, $h^{2} (e)$ decreases. This is why heritability is not a fixed property of a trait: it depends on the environmental variance in the specific population studied.

Epigenetics and twin discordance

Despite sharing identical nuclear DNA, MZ twins are not phenotypically identical. Discordance rates for diseases with high heritability are substantial: for schizophrenia ( $h^{2} \approx 0.80$ ), the MZ concordance rate is approximately 45%, meaning that in over half of cases where one twin is affected, the other is not. Epigenetic differences — heritable changes in gene expression not involving DNA sequence changes — provide a partial explanation.

DNA methylation patterns diverge between MZ twins over their lifetimes, driven by stochastic epigenetic drift, differing environmental exposures, and lifestyle differences. Studies of elderly MZ twins show substantially larger epigenetic differences than twin pairs at birth, suggesting that much of the divergence is accumulated post-natally. Epigenetic effects are captured as part of $V_{E}$ in the ACE model (they contribute to "unique environment"), even though they are molecular in nature and may be partly heritable across cell divisions.

Adoption studies and sibling comparison designs

Adoption studies complement twin studies by separating genetic and environmental influences through a different natural experiment. Adopted children share genes with their biological parents but environments with their adoptive families. If adopted children resemble their biological parents more than their adoptive parents for a trait, the genetic influence is dominant. The adoption design avoids the equal-environments assumption but introduces other biases: selective placement (agencies match adoptive families to biological family characteristics), range restriction (adoptive families are screened and tend to be above-average in socioeconomic status), and the biological mother's prenatal environment.

Sibling comparison designs use within-family variation to control for shared environment and genetic background. The comparison of full siblings, half-siblings, and step-siblings raised in the same household isolates the effect of genetic relatedness while holding the shared environment constant. These designs are particularly powerful when combined with genomic data: comparing siblings who differ at specific loci (within-family GWAS) eliminates confounding by population structure and shared-family environment.

Limits of heritability and misuse in policy

Heritability estimates are frequently misapplied in policy debates. Common errors include:

Conflating heritability with immutability. High $h^{2}$ does not mean a trait cannot be changed by intervention. The breeder's equation $R = h^{2} S$ describes genetic response to selection; environmental interventions bypass the equation entirely.
Extrapolating across populations. Heritability estimated in one population-environment combination does not generalise to other populations or environments. The heritability of height is lower in populations with high environmental variance (malnutrition, disease burden).
Inferring individual-level causation. Heritability is a population-level variance ratio. It provides no information about any individual's phenotype and cannot determine "how much" of a specific person's trait is genetic.
Using heritability to justify inequality. The claim that social inequalities reflect natural (genetic) differences is a non sequitur. Even if a trait has high heritability, the between-group difference could be entirely environmental. Heritability is a within-population statistic; it does not explain between-population differences.

Falconer and Mackay's quantitative genetics framework

The Falconer and Mackay (1996) ^{[Falconer & Mackay 1996]} treatment of quantitative genetics provides the unified framework within which twin studies, parent-offspring regression, and selection response all cohere. Their key contributions include the formal definition of breeding value as twice the expected offspring deviation, the derivation of variance components from single-locus genetics, and the integration of dominance and epistasis into the variance decomposition. The framework shows that $V_{A}$ depends on both the effect size of alleles and their frequencies: $V_{A} = 2 pq α^{2}$ for a single biallelic locus, maximised at intermediate frequency. This means that $h^{2}$ changes as allele frequencies shift under selection, drift, or mutation — heritability is not constant even within a single population over evolutionary time.

Animal breeding applications

In animal breeding, the distinction between additive and dominance variance has direct economic consequences. The additive genetic merit (estimated breeding value, EBV) of each animal is predicted from its own phenotype and the phenotypes of its relatives, using BLUP (best linear unbiased prediction) with the pedigree-derived or genomic relatedness matrix. Selection on EBVs exploits $V_{A}$ to achieve cumulative genetic gain per generation. Dominance variance, while not contributing to sustained selection response, is exploited in crossbreeding: hybrid vigour (heterosis) arises because dominance deviations are maximised in F1 crosses between divergent lines, where heterozygosity is maximised at all loci showing dominance. The classical result is that the heterosis in an F1 cross between two inbred lines is $H_{F 1} = \sum d_{i} (2 p_{i} q_{i})$ , where $d_{i}$ is the dominance deviation and $p_{i}, q_{i}$ are the allele frequencies in the parental lines — directly proportional to the dominance variance in the parental populations.

Connections Master

Quantitative genetics and the breeder's equation 19.05.01. This unit is a direct extension of the variance decomposition and heritability framework introduced in 19.05.01. The twin study is one of three principal methods for estimating $h^{2}$ (alongside parent-offspring regression and genomic-relatedness methods); the breeder's equation $R = h^{2} S$ uses the $h^{2}$ estimated here to predict selection response.
Mendelian genetics 19.01.01. The distinction between additive, dominance, and epistatic variance is built on Mendelian segregation. Additive variance reflects the average effects of individual alleles; dominance variance reflects within-locus interactions between alleles at the same locus; epistatic variance reflects between-locus interactions. All three arise from Mendelian inheritance.
Hardy-Weinberg equilibrium 19.02.01. The single-locus variance component formulas assume Hardy-Weinberg genotype frequencies. Departures from HWE (inbreeding, assortative mating) alter the variance components and bias twin-study estimates.
Natural selection 19.03.01. Selection differentials $S$ are generated by differential survival and reproduction. The heritability $h^{2}$ estimated from twin studies determines how much phenotypic change selection produces per generation.
Genetic drift 19.04.01. Drift erodes additive variance in small populations, reducing $h^{2}$ over time and weakening the response to selection. Twin studies in small, isolated populations may yield different $h^{2}$ estimates than studies in large populations for this reason.

Historical & philosophical context Master

Francis Galton inaugurated the empirical study of heritability in the 1880s through his work on twins and family resemblance ^{[Galton 1883]}. Galton recognised that twins provided a natural experiment: if identical twins remained similar despite differing experiences while fraternal twins diverged, the case for nature over nurture was strengthened. His methods were crude by modern standards — he relied on surveys and subjective assessments — but the conceptual framework was correct. Galton also coined the term "regression" in its statistical sense, from his observation that tall parents tended to produce children shorter than themselves (regression toward the mean), and introduced the correlation coefficient, both of which remain foundational to heritability estimation.

Ronald Fisher's 1918 paper ^{[Falconer & Mackay 1996]}, "The correlation between relatives on the supposition of Mendelian inheritance," established the variance-components framework that underpins all modern heritability analysis. Fisher showed that continuous variation was fully compatible with Mendelian inheritance when many loci contributed small effects, resolving the biometrician-Mendelian conflict. His decomposition of genetic variance into additive, dominance, and epistatic components remains the standard.

The modern twin-study methodology was systematised by Douglas Falconer in the mid-twentieth century. Falconer's 1960 textbook Introduction to Quantitative Genetics (with later editions co-authored with Trudy Mackay) provided the formulas $h^{2} = 2 (r_{M Z} - r_{D Z})$ and $H^{2} = 2 r_{M Z} - r_{D Z}$ that are still taught as entry points to heritability estimation. The ACE model was developed in the 1970s and 1980s by behavioural geneticists including John DeFries, David Fulker, and Robert Plomin, applying structural equation modelling to twin data.

The Minnesota Study of Twins Reared Apart (Bouchard et al. 1990) was a landmark study that tracked MZ twins separated in infancy and raised in different households. The study found striking similarities in personality, intelligence, and even idiosyncratic preferences, providing strong evidence for genetic influences on complex behavioural traits. The study also attracted criticism — for small sample sizes, for potential recruitment bias (twins who were more similar may have been more likely to volunteer), and for the difficulty of verifying that "reared apart" twins truly experienced independent environments.

The nature-nurture debate has been one of the most contentious in the history of biology and psychology. The heritability concept has been misused on both sides: by genetic determinists who treat high $h^{2}$ as proof that environmental interventions are futile, and by environmental determinists who dismiss heritability estimates as meaningless. The modern synthesis, reflected in the quantitative-genetics framework, recognises that genes and environment are always both involved, that their effects interact, and that heritability is a population-specific, environment-specific summary statistic — not a universal constant and not a policy prescription.

Bibliography Master

Galton, F., Inquiries into Human Faculty and Its Development (Macmillan, 1883).
Fisher, R. A., "The correlation between relatives on the supposition of Mendelian inheritance", Trans. R. Soc. Edinburgh 52 (1918), 399–433.
Falconer, D. S. & Mackay, T. F. C., Introduction to Quantitative Genetics, 4th ed. (Longman, 1996).
Futuyma, D. J. & Kirkpatrick, M., Evolution, 4th ed. (Sinauer, 2017).
Hartl, D. L. & Clark, A. G., Principles of Population Genetics, 4th ed. (Sinauer, 2007).
Bouchard, T. J., Lykken, D. T., McGue, M., Segal, N. L. & Tellegen, A., "Sources of human psychological differences: the Minnesota Study of Twins Reared Apart", Science 250 (1990), 223–228.
Yang, J., Benyamin, B., McEvoy, B. P. et al., "Common SNPs explain a large proportion of the heritability for human height", Nat. Genet. 42 (2010), 565–569.
Manolio, T. A., Collins, F. S., Cox, N. J. et al., "Finding the missing heritability of complex diseases", Nature 461 (2009), 747–753.
Plomin, R., DeFries, J. C., Knopik, V. S. & Neiderhiser, J. M., "Top 10 replicated findings from behavioral genetics", Perspect. Psychol. Sci. 8 (2013), 3–23.
Turkheimer, E., "Three laws of behavior genetics and what they mean", Curr. Dir. Psychol. Sci. 9 (2000), 160–164.

Prerequisites

19.05.01 pending

Used in

19.05.03

Tier anchors

beginner: Futuyma & Kirkpatrick, Evolution, 4th ed. (2017), Ch. 6
intermediate: Hartl & Clark, Principles of Population Genetics, 4th ed. (2007), Ch. 7
master: Falconer, D. S. & Mackay, T. F. C. — Introduction to Quantitative Genetics, 4th ed. (1996)

References

Futuyma, D. J. & Kirkpatrick, M. — Evolution, 4th ed. (Sinauer, 2017) · Ch. 6 Quantitative genetics
Hartl, D. L. & Clark, A. G. — Principles of Population Genetics, 4th ed. (Sinauer, 2007) · Ch. 7 Quantitative genetics
Falconer, D. S. & Mackay, T. F. C. — Introduction to Quantitative Genetics, 4th ed. (Longman, 1996) · Ch. 6-10 Heritability, selection
Galton, F. — Inquiries into Human Faculty and Its Development (1883) · Foundational twin study methodology

Estimated time

beginner: 15m
intermediate: 30m
master: 45m