Variance Decomposition for Quantitative Family Traits

Why Decompose Variation in Family Traits?

When relatives resemble one another in a quantitative trait, their resemblance may have several sources. They can share alleles across the genome, share alleles at a particular quantitative locus, or experience a common household or other family-level environment. In addition, each individual has influences that are not shared with relatives under the model. A variance decomposition separates these sources so that similarity between relatives is not automatically attributed to a single cause.

The central distinction is between variance within an individual and covariance between individuals. Each variance component contributes to an individual's total trait variance. Its contribution to covariance between a pair depends on whether, and how, that source is shared by the pair. The model below is a framework for describing these contributions; it does not by itself establish that a particular component has been estimated accurately or that an association is causal.

A Four-Component Model

Let \(Y_i\) be the quantitative trait for individual \(i\), and let \(\mu_i\) denote the individual's mean given specified fixed effects, such as age or sex. Write the trait after accounting for that mean as a sum of four random contributions:

Here \(A_i\) is the additive polygenic contribution from many genetic effects across the genome, excluding the focal quantitative locus represented separately by \(Q_i\). The term \(C_i\) is a shared-environmental contribution, such as the effect of a common household in a model with one family-level environmental effect. The residual \(E_i\) collects individual-specific variation not represented by the other terms, including measurement error under this model.

For the basic decomposition, assume the random contributions have mean zero, the stated variances are the same across individuals, and distinct components are uncorrelated with one another. Assume also that the residuals of different individuals are uncorrelated. These assumptions make the decomposition additive. If, for example, genetic effects and environments are correlated, extra covariance terms are needed and the simple sums below do not describe the full variance or covariance.

The left side is the variance around the specified mean, not the variance of the fixed mean itself. If all components are measured in units of the trait, their variances are in squared units. For example, if the trait is height in centimetres, the variance components are in square centimetres.

How Each Component Creates Covariance

For two distinct individuals \(i\) and \(j\), the covariance of their traits is the sum of the covariances contributed by the four components, provided that different components are uncorrelated. The components do not all contribute in the same way.

The polygenic term uses genome-wide pedigree relatedness. The QTL term instead depends on IBD sharing at the particular locus. For a pair with uncertain locus-specific sharing, the formula describes covariance conditional on the specified \(\pi_{ij}\); averaging across possible inheritance states replaces \(\pi_{ij}\) by its expectation. At an individual's own locus, \(\pi_{ii}=1\), so the QTL contribution to that individual's variance is \(\sigma_Q^2\).

For a simple shared-environment model, let \(H_{ij}=1\) if individuals \(i\) and \(j\) share the modeled environment, and \(H_{ij}=0\) otherwise. The same environmental effect applies to both members of a sharing pair, so their covariance contribution is \(\sigma_C^2 H_{ij}\). This term is not determined by genetic relatedness: siblings raised apart need not share this effect, while unrelated people in the same household may share it. The residual contributes \(\sigma_E^2\) to an individual's variance but, under the independent-residual assumption, contributes no covariance between distinct individuals.

This formula is for a model that separates the focal locus from the remaining additive polygenic effects. It also assumes there are no additional covariance sources, such as a modeled shared effect beyond \(C_i\), dominance, or correlated residuals. The covariance is not the same as the correlation: for individuals with equal trait variance \(V\), the correlation is the covariance divided by \(V\); with unequal variances, it is divided by the square root of their variance product.

Worked Variance and Covariance Calculations

Why the Decomposition Matters

The decomposition clarifies what family resemblance can and cannot tell us. Pedigree relatedness predicts a polygenic covariance contribution, but common environments can also make relatives similar. A focal QTL can contribute covariance in proportion to locus-specific IBD sharing, which may vary among pairs with the same pedigree relationship. Separating these sources gives a model a way to describe those distinct patterns rather than treating all family resemblance as interchangeable.

The variance components also depend on the model's definitions. In this tutorial, \(\sigma_A^2\) excludes the focal QTL so that its contribution can be represented separately by \(\sigma_Q^2\). If the polygenic component were defined to include that locus, adding \(\sigma_Q^2\) again would count some genetic variance twice. Similarly, the shared-environment component represents only the environmental sharing encoded by \(H_{ij}\); it is not a general label for every resemblance that remains unexplained.

A Short Justification of the Addition Rules

The variance and covariance formulas follow from the algebra of sums and the assumption that distinct components are uncorrelated. For an individual, subtract its fixed mean to obtain \(A_i+Q_i+C_i+E_i\). Expanding the variance gives the four component variances plus twice each pairwise cross-covariance. Each cross-covariance is zero by assumption, leaving the sum of four variances.

For two individuals, expand the covariance of their component sums. The model assumes cross-covariances between different component types vanish. The remaining same-component terms are \(\operatorname{Cov}(A_i,A_j)=r_{ij}\sigma_A^2\), \(\operatorname{Cov}(Q_i,Q_j)=\pi_{ij}\sigma_Q^2\), and \(\operatorname{Cov}(C_i,C_j)=H_{ij}\sigma_C^2\). Independent residuals give \(\operatorname{Cov}(E_i,E_j)=0\) for \(i\ne j\). Adding these surviving terms yields the stated pairwise covariance.

Common Pitfall

A common error is to see that two siblings resemble one another and attribute their entire covariance to genetic sharing. That reasoning omits the possibility of a common environment. The correction is to identify each modeled source and calculate its contribution separately: genetic terms depend on pedigree or locus-specific sharing, whereas the shared-environment term depends on whether the pair shares the specified environment. Another error is to treat the residual variance as covariance between relatives; under the independent-residual assumption it contributes to each individual's variance, not to pairwise covariance.