Heritability and Variance Components

Heritability answers a deceptively simple question: how much of the variation in a trait across a population is due to genetic differences rather than environment? It is a variance ratio, not a statement about any individual, and it anchors the study of disease genetics, breeding, and polygenic scores.

Decomposing phenotypic variance

For a trait measured across a population, the total or phenotypic variance splits into genetic and environmental parts:

$V_P = V_G + V_E,$

where $V_G$ is the variance attributable to genotype differences and $V_E$ to environment (this simple form assumes genotype and environment are uncorrelated and non-interacting). The genetic variance itself decomposes further:

$V_G = V_A + V_D + V_I,$

into additive effects $V_A$ (the average per-allele contributions that parents pass on), dominance $V_D$ (interactions between alleles at one locus), and epistatic $V_I$ (interactions across loci).

Two heritabilities

Broad-sense heritability is the fraction of phenotypic variance explained by all genetic differences:

$H^2 = \frac{V_G}{V_P}.$

Narrow-sense heritability uses only the additive part:

$h^2 = \frac{V_A}{V_P}.$

The narrow-sense $h^2$ is the more useful quantity in most of quantitative genetics because additive effects are what respond to selection and what a linear predictor built from allele dosages can capture. Both are ratios between $0$ and $1$ , and both are properties of a population in an environment, not fixed constants of a trait.

Estimating heritability

Twin studies and Falconer’s formula

Identical (monozygotic, MZ) twins share essentially all their DNA; fraternal (dizygotic, DZ) twins share on average half their segregating variants. If MZ pairs are more similar than DZ pairs, genetics is contributing. Comparing the trait correlation within MZ pairs, $r_{MZ}$ , to that within DZ pairs, $r_{DZ}$ , Falconer’s formula estimates

$h^2 \approx 2\,(r_{MZ} - r_{DZ}).$

The factor of two reflects that DZ twins share half the additive variance, so doubling the gap in correlations recovers the additive fraction. Broad-sense $H^2$ can be approximated as $2(r_{MZ}-r_{DZ})$ as well under simple assumptions, and $c^2 = 2 r_{DZ} - r_{MZ}$ estimates the shared-environment share.

SNP-heritability from genome-wide data

Modern estimates use unrelated individuals and measured genotypes directly. Methods such as GREML fit a linear mixed model in which the phenotype has a random genetic effect whose covariance between individuals is their genetic relatedness matrix (GRM), computed from genome-wide SNPs:

$\operatorname{Var}(y) = V_g\, A + V_e\, I,$

with $A$ the GRM. The estimated $V_g / (V_g + V_e)$ is SNP-heritability, the additive variance tagged by common SNPs. It is typically lower than twin-based $h^2$ , and the gap (“missing heritability”) reflects rare variants and imperfect tagging.

Worked example

A twin study of a quantitative trait reports $r_{MZ} = 0.80$ and $r_{DZ} = 0.50$ . Falconer’s formula gives

$h^2 \approx 2\,(0.80 - 0.50) = 0.60,$

so about 60% of the trait variance is additively genetic. The shared-environment estimate is $c^2 = 2(0.50) - 0.80 = 0.20$ , and the remaining $1 - 0.80 = 0.20$ is unique environment plus measurement error.

In code

R

r_mz <- 0.80; r_dz <- 0.50
h2 <- 2 * (r_mz - r_dz)      # 0.60  narrow-sense heritability
c2 <- 2 * r_dz - r_mz        # 0.20  shared environment
e2 <- 1 - r_mz               # 0.20  unique environment

# Variance-component view: V_P = V_G + V_E
V_G <- 6; V_E <- 4; V_P <- V_G + V_E
H2  <- V_G / V_P             # 0.60  broad-sense
# SNP-heritability in practice: lme4/GCTA-style mixed model with a GRM

Python

r_mz, r_dz = 0.80, 0.50
h2 = 2 * (r_mz - r_dz)       # 0.60
c2 = 2 * r_dz - r_mz         # 0.20
e2 = 1 - r_mz                # 0.20

V_G, V_E = 6.0, 4.0
V_P = V_G + V_E
H2 = V_G / V_P               # 0.60  broad-sense
# SNP-heritability: statsmodels MixedLM / GREML-style GRM model

Julia

r_mz, r_dz = 0.80, 0.50
h2 = 2 * (r_mz - r_dz)       # 0.60
c2 = 2 * r_dz - r_mz         # 0.20
e2 = 1 - r_mz                # 0.20

V_G, V_E = 6.0, 4.0
V_P = V_G + V_E
H2 = V_G / V_P               # 0.60  broad-sense
# SNP-heritability: MixedModels.jl with a genetic relatedness matrix

Why it matters

Heritability sets a ceiling on how well genetics can ever predict a trait, framing expectations for GWAS discovery and polygenic-score accuracy. It is also one of the most misunderstood numbers in science: a high $h^2$ does not mean a trait is unchangeable, nor does it explain differences between groups or within an individual — it describes the sources of variance in one population under its particular range of environments.