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 \(Var(P)\) can be modeled as the combination of genetic effects \(Var(G)\) and environmental effects \(Var(E)\).

\[ Var(P) = Var(G) + Var(E) \]

Broad-sense Heritability

The broad-sense heritability \(H^2_{broad-sense}\) is mathematically defined as :

\[ H^2_{broad-sense} = {Var(G)\over{Var(P)}} \]

Narrow-sense Heritability

Genetic effects \(Var(G)\) is composed of multiple effects including additive effects \(Var(A)\), dominant effects, recessive effects, epistatic effects and so forth.

Narrow-sense heritability is defined as:

\[ h^2_{narrow-sense} = {Var(A)\over{Var(P)}} \]

SNP Heritability

SNP heritability \(h^2_{SNP}\) : the proportion of phenotypic variance explained by tested SNPs in a GWAS.

Common methods to estimate SNP heritability include:

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

Missing heritability

Missing heritability refers to the gap between the heritability estimated from family and twin studies (narrow-sense heritability \(h^2\)) and the proportion of phenotypic variance explained by GWAS-identified variants (SNP heritability \(h^2_{SNP}\)).

For many complex traits, GWAS-identified variants explain only a fraction of the total heritability estimated from family studies. This discrepancy is known as the "missing heritability" problem. Potential explanations include:

Rare variants: Variants with low minor allele frequency that are not well captured by standard GWAS arrays
Structural variants: Large insertions, deletions, and other structural variants not well tagged by SNPs
Epistasis: Gene-gene interactions that are difficult to detect in standard GWAS
Incomplete LD: Some causal variants may not be in strong linkage disequilibrium with genotyped SNPs
Inflation of heritability estimates: Family-based estimates may be inflated by shared environmental factors

Liability and Threshold model

The liability threshold model is a conceptual framework used to model binary traits (e.g., disease status) by assuming an underlying continuous liability distribution.

In this model: - Liability is an unobserved continuous variable that represents an individual's genetic and environmental predisposition to develop a disease - The liability distribution is typically assumed to follow a normal distribution in the population - A threshold value determines disease status: individuals with liability above the threshold manifest the disease (cases), while those below remain unaffected (controls) - The threshold is determined by the population disease prevalence

This model allows us to translate heritability estimates from the observed binary scale (0/1) to a more interpretable continuous liability scale, which is important for comparing heritability estimates across traits with different prevalences.

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):

\[ h^2_{liability-scale} = h^2_{observed-scale} * {{K(1-K)}\over{Z^2}} * {{K(1-K)}\over{P(1-P)}} \]

\(K\) : 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).