Heritability

Heritability is a term used in genetics to describe how much phenotypic variation can be explained by genetic variation.

For any phenotype, its variation $V a r (P)$ can be modeled as the combination of genetic effects $V a r (G)$ and environmental effects $V a r (E)$ .

V a r (P) = V a r (G) + V a r (E)

Broad-sense Heritability

The broad-sense heritability $H_{b r o a d - s e n s e}^{2}$ is mathmatically defined as :

H_{b r o a d - s e n s e}^{2} = \frac{V a r (G)}{V a r (P)}

Narrow-sense Heritability

Genetic effects $V a r (G)$ is composed of multiple effects including additive effects $V a r (A)$ , dominant effects, recessive effects, epistatic effects and so forth.

Narrrow-sense heritability is defined as:

h_{n a r r o w - s e n s e}^{2} = \frac{V a r (A)}{V a r (P)}

SNP Heritability

SNP heritability $h_{S N P}^{2}$ : the proportion of phenotypic variance explained by tested SNPs in a GWAS.

Common methods to estimate SNP heritability includes:

GCTA-GREML (based on Genome-based Restricted Maximum Likelihood)
LDSC (based on LD score regression)

Liability and Threshold model

Observed-scale heritability and liability-scaled heritability

Issue for binary traits :

The scale issue for binary traits

For quantitative traits the scale of measurement is the same as the scale on which heritability is expressed.
For disease traits, the phenotypes (case-control status) are measured on the 0–1 scale, but heritability is most interpretable on a scale of liability.
Reference: Lee, S. H., Wray, N. R., Goddard, M. E., & Visscher, P. M. (2011). Estimating missing heritability for disease from genome-wide association studies. The American Journal of Human Genetics, 88(3), 294-305.

Conversion formula (Equation 23 from Lee. 2011):

h_{l i a b i l i t y - s c a l e}^{2} = h_{o b s e r v e d - s c a l e}^{2} * \frac{K (1 - K)}{Z^{2}} * \frac{K (1 - K)}{P (1 - P)}

$K$ : Population disease prevalence.
$P$ : Sample disease prevalence.
$Z$ : The height of the standard normal probability density function at threshold T. scipy.stats.norm.pdf(T, loc=0, scale=1).
$T$ : The threshold. scipy.stats.norm.ppf(1 - K, loc=0, scale=1) or scipy.stats.norm.isf(K).