Heritability

Standard Definition

Heritability is typically defined in the context of an additive genetic model. Let $Y$ denote the phenotype of interest. Let $G$ denote the genetic component of the phenotype and $E$ the environmental component. Assume the components are additively separable, so that:

\[ \begin{align} Y=E+G. \label{model} \end{align} \]

Assume also that the random variables $E$ and $G$ are uncorrelated.

Under these assumptions, define heritability:

\[ h^2:= \frac{\mathrm{Var}(G)}{\mathrm{Var}(Y)}. \]

Note that in the uncorrelated-additive model above, $h^2$ is equal to the coefficient of determination in the linear regression of $Y$ on $G$. To see this, observe that if we run a simple linear regression of $Y$ on $G$, then the coefficient of determination is given by the formula:

\[ \begin{align} \mathrm{cor}(G,Y)^2&=\frac{\mathrm{Cov}(Y,G)^2}{\mathrm{Var}(G) \mathrm{Var(Y)}}\\ &=\frac{\mathrm{Cov}(E+G,G)^2}{\mathrm{Var}(G) \mathrm{Var(Y)}}& \text{ by }(\ref{model})\\ &=\frac{\mathrm{Var}(G)^2}{\mathrm{Var}(G) \mathrm{Var(Y)}} & \text{$G$ and $E$ uncorrelated}\\ &=\frac{\mathrm{Var}(G)}{\mathrm{Var}(Y)}\\ &=:h^2 \end{align} \]

$h^2$ thus expresses the proportion of phenotypic variation attributable to genetics.

Note on assumption

The assumption that $G$ and $E$ are uncorrelated is unrealistic in some contexts. This is problematic for two reasons:

If $\mathrm{Cov}(G,E)\ne 0$, then $h^2$ loses much of its interpretive value. It is no longer the coefficient of determination of a regression.
According to Visscher et al.¹, if $\mathrm{cov}(G,E)\ne 0$, then statistical estimates of $h^2$ will tend to be inflated.

As I understand it, practitioners typically argue that any correlation between $G$ and $E$ is not strong, so that the above effects are not severe, and the heritability concept remains a useful.

Heritability Depends on the Population Context

The heritability of a trait depends on the studied population.

For instance, suppose we study a trait in two genetically identical populations $A$ and $B$, but population $B$ lives in a much more variable environment than population $A$, and that this variability affects the trait under study. Then $\mathrm{Var}(Y_B)>\mathrm{Var}(Y_A)$ . Consequently, $h^2_A> h^2_B$, despite the genetics of the two populations being the same.

As a more specific example, note that (a) many recent GWAS have shown that obesity is highly heritable, and that (b) there has been a large increase in obesity rates in recent history, indicating a strong environmental influence. These two facts do not contradict one another. The obesity GWAS have studied cross-sectional populations drawn from a single snapshot in time, and have shown that a large portion of the variability in obesity in such populations is attributable to genetics. On the other hand, if the GWAS samples had included individuals from a range of historical periods, environmental influence would have been proportionately larger, and calculated heritability would have been smaller.

For a popular science book that discusses subtle issues in the definition of heredity, see Harden 2021².

Alternative definitions of Heritability

The standard definition of heritability stated above has certain unsatisfactory aspects:

The random variables $G$ and $E$ are not precisely defined.
Even if we $G$ and $E$ were precisely defined, heritability also assumes the additive model $Y=G+E$. Given the incredible complexity of biology, this is implausible. It might, however, be reasonable to argue that the additive model reflects a first order Taylor expansion of the true phenotype-determining function.
Even if we accept an additive model, the lack of correlation between $G$ and $E$ is also problematic. Different individuals with different genes inhabit different environments.

The above unsatisfactory aspects make heritability a somewhat fuzzy concept, and motivate the development of alternative definitions.

Definition via conditional expectation

In the definition above, we first specified the relationship between $G$, $E$, and $Y$ , and then defined heritability in the context of this relationship. It is possible to reverse this.

As before, let the random variable $Y$ denote the phenotypic trait of interest. Let $g$ denote genotype. Define:

\[ \begin{align} G&:=\mathbb{E} (Y|g),\\ E&:=Y-G,\\ h^2&:= \frac{\mathbb{Var}(G) }{\mathbb{Var}(Y)}. \end{align} \]

We have

\[ \begin{align} \mathbb{Cov}(G,E)&=\mathbb{E}( \mathbb{E}(Y|g) -\mathbb{E}Y )( Y- \mathbb{E}(Y|g) )\\ &=0, \end{align} \]

where the last line follows from the Projection Theorem (pg. 345 in Grimmet and Stirzaker³). Where before we needed to assume $\mathbb{Cov}(E,G)=0$, here this property is automatic.

This approach has the advantage of its mathematical clarity. Whereas the standard definition of heritability requires some fairly restrictive assumptions, this alternative definition is applicable to any phenotype representable by a random variable in $L_2$. Mathematically, it is now crystal clear what we mean when we speak of $G$ and $E$.
On the other hand, the conditional expectation definition of heritability has the disadvantage of reduced interpretability. In particular, $G$ now incorporates all aspects of the phenotype that can be predicted from the genotype, including artifacts of population stratification.

Counterfactual heritability

Todo⁴

Links

Broad Institute Primer Talk on Heritability

References

Balding, David J., Ida Moltke, and John Marioni, eds. Handbook of statistical genomics. John Wiley & Sons, 2019. (Chapter 15)
Visscher, Peter M., William G. Hill, and Naomi R. Wray. "Heritability in the genomics era—concepts and misconceptions." Nature reviews genetics 9.4 (2008): 255-266.

Peter M Visscher, William G Hill, and Naomi R Wray. Heritability in the genomics era—concepts and misconceptions. Nature Reviews Genetics, 9(4):255–266, 2008. URL: https://www.researchgate.net/profile/Naomi-Wray/publication/5532009_Visscher_PM_Hill_WG_Wray_NR_Heritability_in_the_genomics_era-concepts_and_misconceptions_Nat_Rev_Genet_9_255-266/links/54176be90cf203f155ad57c4/Visscher-PM-Hill-WG-Wray-NR-Heritability-in-the-genomics-era-concepts-and-misconceptions-Nat-Rev-Genet-9-255-266.pdf. ↩
Kathryn Paige Harden. The genetic lottery: Why DNA matters for social equality. Princeton University Press, 2021. ↩
Geoffrey Grimmett and David Stirzaker. Probability and Random Processes: Third Edition. Oxford university press, 2001. URL: https://www.amazon.ca/Probability-Random-Processes-3th-third/dp/B006QYQVTS. ↩
Haochen Leia, Jieru Shib, Hongyuan Caoa, and Qingyuan Zhaoc. Heritability: a counterfactual perspective. preprint, 2025. URL: https://www.statslab.cam.ac.uk/~qz280/publication/counterfactual-heritability/paper.pdf. ↩