Variance Components

Total phenotypic variance $V_P$ is decomposed as:

\[V_P = V_A + V_D + V_I + V_E\]

where $V_A$ is additive genetic variance, $V_D$ dominance variance, $V_I$ epistatic (interaction) variance, and $V_E$ environmental variance. The two main heritability measures are:

\[h^2 = \frac{V_A}{V_P} \quad \text{(narrow-sense)}, \qquad H^2 = \frac{V_G}{V_P} = \frac{V_A + V_D + V_I}{V_P} \quad \text{(broad-sense)}\]

Only $h^2$ predicts response to selection; $H^2$ is relevant for clonal propagation.

Parent–Offspring Regression

The simplest estimator regresses offspring phenotype on mid-parent value:

\[y_{\text{offspring}} = \alpha + h^2 \bar{y}_{\text{parents}} + e\]

The slope equals $h^2$ because the parent-offspring phenotypic covariance equals $V_A/2$ per parent, and $V_A$ for the mid-parent.

GREML / REML Estimation

Genomic REML (GREML) fits the mixed model:

\[\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{g} + \boldsymbol{\varepsilon}, \quad \mathbf{g} \sim \mathcal{N}(\mathbf{0}, h^2 \mathbf{G}), \quad \boldsymbol{\varepsilon} \sim \mathcal{N}(\mathbf{0}, (1-h^2)\mathbf{I})\]

where $\mathbf{G}$ is the GRM constructed from SNP data. REML maximises the restricted log-likelihood, removing fixed-effect bias. SNP-based $h^2$ estimates the proportion of variance tagged by common SNPs, typically lower than twin-based estimates (“missing heritability”).

Common Estimates

Trait	$h^2_{\text{SNP}}$	$h^2_{\text{twin}}$
Height	0.50	0.80
BMI	0.27	0.75
Educational attainment	0.11	0.40
Schizophrenia	0.23	0.80