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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 mathmatically 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.

Narrrow-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 includes:

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

Liability and Threshold model

image

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).

Further Reading

  • (Blog by Neale Lab) http://www.nealelab.is/blog/2017/9/13/heritability-101-what-is-heritability
  • Manolio, T. A., Collins, F. S., Cox, N. J., Goldstein, D. B., Hindorff, L. A., Hunter, D. J., ... & Visscher, P. M. (2009). Finding the missing heritability of complex diseases. Nature, 461(7265), 747-753.
  • Visscher, P. M., Hill, W. G., & Wray, N. R. (2008). Heritability in the genomics era—concepts and misconceptions. Nature reviews genetics, 9(4), 255-266.
  • Yang, J., Zeng, J., Goddard, M. E., Wray, N. R., & Visscher, P. M. (2017). Concepts, estimation and interpretation of SNP-based heritability. Nature genetics, 49(9), 1304-1310.
  • Witte, J. S., Visscher, P. M., & Wray, N. R. (2014). The contribution of genetic variants to disease depends on the ruler. Nature Reviews Genetics, 15(11), 765-776.