Defining SNP Heritability as a Variant-Set Estimand

Why the Variant Set Is Part of the Definition

A heritability estimate is not a universal property of a trait. It refers to a particular population, phenotype scale, and definition of genetic variation. For SNP heritability, one more choice is essential: which variants are included? A set of common measured variants, a set enriched by imputation, and a set restricted to one genomic region can tag different amounts of phenotypic variation.

This matters when interpreting an estimate. A value of 0.30 does not, by itself, mean that 30% of the trait is caused by the particular variants in a data file. The variants may tag genetic effects through correlation with other variants, and the estimate depends on how the phenotype and genetic component are defined. The estimand must be stated before discussing how it is estimated.

Definition: For a specified target population, phenotype scale, covariate adjustment, and variant set $S$, the SNP heritability for $S$, written $h^2_S$, is the proportion of phenotypic variance represented by the additive genetic signal tagged by the variants in $S$. In a variance-component formulation, $$ h^2_S = \frac{\operatorname{Var}(G_S)}{\operatorname{Var}(Y)}. $$ Here $Y$ is the phenotype on the specified scale after the stated covariate adjustment, and $G_S$ is the additive genetic component represented or tagged by set $S$. Both variances refer to the target population.

The denominator is the variance of the phenotype as defined for the analysis. If the phenotype is transformed, adjusted for covariates, or measured on a different scale, its variance can change, and so can the ratio. The numerator is also specific: changing $S$ changes which genetic signal is counted as tagged. In applications, the target population and variant set should therefore be reported along with the estimate.

One population-level way to make “tagged” precise is to define $G_S$ as the best additive linear predictor of the adjusted phenotype from the variants in $S$. This is a projection onto the linear span of those variants in the target population. The definition does not require every variant in $S$ to be causal. A variant can help predict the phenotype because it is correlated with a causal variant that is not in $S$. Conversely, a causal variant absent from $S$ may contribute little or no signal tagged by that set.

Relationship to Narrow-Sense and Broad-Sense Heritability

Narrow-sense heritability refers to all additive genetic variance in the population, not just the additive signal represented by a particular measured variant set. Broad-sense heritability refers to total genetic variance, including additive and nonadditive genetic contributions, under the variance decomposition being used.

Definition: With phenotypic variance $V_P$, additive genetic variance $V_A$, and total genetic variance $V_G$, the conventional ratios are $$ h^2 = \frac{V_A}{V_P} \qquad\text{and}\qquad H^2 = \frac{V_G}{V_P}. $$ The first is narrow-sense heritability; the second is broad-sense heritability. By contrast, $h^2_S$ refers to the additive signal tagged by the specified variant set $S$, divided by the phenotypic variance.

Under a clean additive variance partition in which the set captures only part of the total additive signal, SNP heritability for $S$ is no greater than narrow-sense heritability, and narrow-sense heritability is no greater than broad-sense heritability. This ordering is not a license to compare numbers from different populations, phenotype scales, or adjustment schemes as if they shared a denominator. Nor should it be assumed when the component definitions or modeling assumptions differ.

The word “SNP” identifies the marker set used to represent genetic signal; it does not imply that the estimate counts only effects of variants that are themselves causal. The distinction is between causal genetic effects and genetic variation tagged by the chosen markers. These can coincide under suitable conditions, but they are not identical by definition.

Worked Example: Comparing the Three Heritabilities

Suppose a hypothetical population has an adjusted phenotype with variance $V_P=1.00$ squared trait units. Its additive genetic variance is $V_A=0.45$, including $0.30$ squared units tagged by a specified measured-variant set $S$. Nonadditive genetic variance is $0.10$, and the remaining variance is $0.45$. Assume these components form an orthogonal variance partition, so they sum to the total.

For the specified set, the SNP heritability is $$ h^2_S = \frac{\operatorname{Var}(G_S)}{\operatorname{Var}(Y)} = \frac{0.30}{1.00} = 0.30. $$

Narrow-sense heritability counts all additive variance, including the $0.15$ squared units not tagged by $S$: $$ h^2 = \frac{V_A}{V_P} = \frac{0.45}{1.00} = 0.45. $$

Broad-sense heritability counts additive and nonadditive genetic variance: $$ H^2 = \frac{V_G}{V_P} = \frac{0.45+0.10}{1.00} = 0.55. $$

Thus the set-specific value is 0.30, narrow-sense heritability is 0.45, and broad-sense heritability is 0.55. In this hypothetical variance partition, the differences have a direct interpretation: the specified set tags only part of the additive variance, and the broad-sense ratio also includes nonadditive genetic variance.

The Set-Specific Estimand in Practice

A useful way to think about $h^2_S$ is as a target that is fixed by the scientific question, rather than as a number supplied by a particular estimation method. Before estimation, specify the phenotype, its scale, the target population, the covariate adjustment, and the membership of $S$. Estimation methods may differ in how they use genotype data and model covariance, but the interpretation still depends on what genetic component and denominator they target.

For example, if set $S$ is contained in a larger set $T$, then a population linear predictor using $T$ can use all predictors available to $S$, as well as additional ones. Under the same target population and projection definition, adding predictors cannot reduce the variance explained. The set-specific estimand can therefore change even when the phenotype and population do not.

Worked Example: Expanding a Variant Set

In a hypothetical population, an adjusted phenotype has variance $1.25$ squared units. A population linear predictor based on measured variants $S$ has variance $0.25$ squared units. An expanded set $T$, which contains $S$ plus additional imputed variants, has a predictor variance of $0.40$ squared units. Assume the population, phenotype scale, covariates, and projection definition are unchanged.

For the original set, $$ h^2_S = \frac{0.25}{1.25} = 0.20. $$ For the expanded set, $$ h^2_T = \frac{0.40}{1.25} = 0.32. $$

The expanded set tags an additional $0.40-0.25=0.15$ squared units of phenotypic variance, corresponding to an increase of $0.15/1.25=0.12$ in the ratio. The interpretation is not that the added variants have necessarily caused those additional effects. They may provide information about genetic signal through their own effects, through correlation with other variants, or both.

A Short Justification for the Nested-Set Property

The non-decrease just described follows from the definition by population projection. Assume the phenotype and predictors are centered and have finite second moments. Let $P_S Y$ denote the best linear predictor of $Y$ from set $S$, and let $P_T Y$ be the predictor from a larger set $T$, with $S\subseteq T$. Because every linear combination of predictors in $S$ is also available in $T$, the larger set’s best predictor cannot have greater mean squared prediction error.

\operatorname{Var}(Y-P_TY) \leq \operatorname{Var}(Y-P_SY).

For a population least-squares projection, each prediction error is orthogonal to its predictor. Consequently, $$ \operatorname{Var}(Y) = \operatorname{Var}(P_SY)+\operatorname{Var}(Y-P_SY), $$ and the same identity holds for $T$. Subtracting the two residual-variance expressions gives $$ \operatorname{Var}(P_TY)-\operatorname{Var}(P_SY) = \operatorname{Var}(Y-P_SY)-\operatorname{Var}(Y-P_TY) \geq 0. $$ Thus the variance tagged by the larger set cannot be smaller under this projection definition. The argument does not say how much the variance will increase, nor does it say that every added variant is causal.

Tagging Is Not the Same as Causation

A variant set can tag phenotypic variance even when its members are not themselves causal. The following deliberately simple example illustrates the distinction. Let $Z$ be a standardized causal additive genotype and $X$ a standardized measured marker, with correlation $0.80$. Suppose the centered phenotype is $Y=0.5Z+e$, where $e$ is independent of both genotypes and has variance $0.75$. The marker’s value for prediction comes from its correlation with $Z$, not from a direct effect of $X$ in this model.

Worked Example: A Marker Tags a Causal Effect

The variance due to the causal additive component is $$ \operatorname{Var}(0.5Z)=0.5^2\operatorname{Var}(Z) =0.25(1)=0.25. $$ The total phenotypic variance is $$ \operatorname{Var}(Y)=0.25+0.75=1.00, $$ because the environmental error is independent of $Z$.

The best linear predictor from the single standardized marker $X$ has coefficient $$ \frac{\operatorname{Cov}(Y,X)}{\operatorname{Var}(X)} = \frac{0.5\operatorname{Cov}(Z,X)}{1} = 0.5(0.80)=0.40. $$ Its tagged variance is therefore $$ \operatorname{Var}(0.40X)=0.40^2(1)=0.16, $$ and the SNP heritability for the set containing $X$ is $0.16/1.00=0.16$. In this example, the causal additive variance ratio is $0.25$, while the single-marker set tags $0.16$ of phenotypic variance. The marker tags part of the causal signal because it is correlated with $Z$; the calculation does not establish that $X$ itself causes the phenotype.

This example relies on specified assumptions: centered standardized genotypes, the stated correlation, an additive phenotype model, and an error independent of genotype. In real analyses, population structure, relatedness, environmental correlations with genotype, and model misspecification can complicate the interpretation. Adjustments and model choices are intended to address relevant structure, but an estimate should not be interpreted as a causal fraction merely because it is called heritability.

Common Mistakes and Interpretation

Calling SNP heritability the heritability of the trait. The correction is to state the variant set, population, phenotype scale, and adjustment. The estimate is an estimand for those choices.
Equating the analyzed variants with causal variants. A set can tag effects through correlations with variants outside the set. Tagged variance and variance caused directly by set members are different concepts.
Treating SNP heritability, narrow-sense heritability, and broad-sense heritability as interchangeable. They count different genetic components: signal tagged by a specified set, all additive genetic variance, and total genetic variance, respectively.
Comparing ratios without checking the denominator. Phenotype scale and covariate adjustment can change the phenotypic variance. A numerical difference between estimates may therefore reflect different targets, not a change in genetic architecture alone.
Reading a heritability ratio as an individual-level percentage. A population variance ratio describes variation among individuals in the stated population; it does not give the fraction of one person’s phenotype caused by genes.

Key takeaway: SNP heritability is defined relative to a specified variant set: it is the proportion of phenotypic variance represented by the additive genetic signal that the set tags. It is not automatically the total additive variance, the total genetic variance, or the proportion caused by the measured variants themselves.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following questions.

What population, phenotype, and variant-set details are needed to interpret a SNP heritability estimate?
A variant set tags 0.24 squared units of variance in a phenotype whose population variance is 0.80 squared units. Calculate its SNP heritability and state what the result means.
How does narrow-sense heritability differ from SNP heritability for a specified set?
Can a noncausal measured marker contribute to SNP heritability for its set? Explain how.
Under the population projection definition, why can adding predictors to a nested variant set not reduce the variance tagged?