Separating Dominance From Additive Relatedness

Why Relative Pairs Carry Different Information

Relatives can resemble one another for more than one genetic reason. Additive effects produce resemblance according to how much additive genetic material a pair shares. Dominance effects produce resemblance when the pair shares both alleles at a locus through descent, allowing the same genotype-specific deviations from additivity to recur in both individuals. The two sources can therefore contribute differently across kinds of relatives.

The key is not simply to collect many related individuals. It is to include pairs whose additive and dominance relationship coefficients vary in distinct ways. Parent–offspring pairs, for example, provide additive resemblance but no dominance resemblance under the standard pedigree model. Full siblings provide both. Those contrasting patterns can help separate the components, provided the covariance model and its assumptions are credible.

Definition: Let \(A_{ij}\) denote the additive relationship coefficient for individuals \(i\) and \(j\), scaled so that their additive genetic effects have covariance \(A_{ij}\sigma_A^2\). Let \(D_{ij}\) denote the dominance relationship coefficient, scaled so that their dominance genetic effects have covariance \(D_{ij}\sigma_D^2\). The variance components \(\sigma_A^2\) and \(\sigma_D^2\) are the additive and dominance variances on the modeled trait scale.

The Covariance Model and Its Assumptions

For an individual-level model, write the phenotype as a fixed-effect contribution plus additive genetic, dominance genetic, and residual effects:

y_i = \mathbf{x}_i^\mathsf{T}\boldsymbol{\beta} + g_{A,i} + g_{D,i} + e_i.

Here \(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}\) represents modeled covariates and fixed effects. The covariance model for the genetic effects is \(\operatorname{Cov}(g_{A,i},g_{A,j})=A_{ij}\sigma_A^2\) and \(\operatorname{Cov}(g_{D,i},g_{D,j})=D_{ij}\sigma_D^2\). To obtain the familiar sum of these two contributions, assume that additive and dominance effects are uncorrelated with each other, and that residuals are uncorrelated across distinct individuals and uncorrelated with both genetic effects. Fixed effects are treated as modeled mean structure, not as random covariance contributions.

Under those assumptions, for \(i\ne j\),

\operatorname{Cov}(y_i,y_j) = A_{ij}\sigma_A^2 + D_{ij}\sigma_D^2.

This is a model for covariance after accounting for the fixed effects. It is not guaranteed merely because residuals are mutually uncorrelated: residuals must also be uncorrelated with the genetic effects for the genetic covariance terms to be the whole expression. If relatives share household, neighborhood, or other environmental influences, their residuals may be correlated. Such covariance must be modeled or ruled out by design; otherwise it can be mistaken for genetic resemblance.

Pedigree Patterns: Which Pairs Inform Which Component?

Under the standard pedigree convention for noninbred individuals and unrelated founders, the additive coefficient \(A_{ij}\) is twice the kinship coefficient. For example, parent–offspring and full-sibling pairs both have expected \(A_{ij}=0.5\). The dominance coefficient \(D_{ij}\) reflects the pair's probability of sharing both alleles at a locus identical by descent, under the stated scaling. A parent and offspring share one allele by descent at a locus, not both, so their expected dominance coefficient is zero. Full siblings have probability \(0.25\) of sharing both alleles by descent, giving expected \(D_{ij}=0.25\).

Pair type	Expected \(A_{ij}\)	Expected \(D_{ij}\)	Information in the pair pattern
Parent–offspring	0.5	0	Additive covariance; no dominance covariance under this model
Full siblings	0.5	0.25	Both additive and dominance covariance
Half siblings	0.25	0	Additive covariance; no dominance covariance under this model
Unrelated founders	0	0	No expected covariance from these two components

The table gives expected pedigree coefficients, not exact realized sharing at every locus. Genomic relationship matrices can represent realized sharing and can therefore differ among pairs with the same pedigree label. Their usefulness for separating components still depends on the two matrices containing distinguishable patterns across the observed pairs.

Worked Example: Predicting Covariance for Two Relative Types

Suppose a hypothetical trait is measured in a pedigree, with additive variance \(\sigma_A^2=2.0\) trait-units squared and dominance variance \(\sigma_D^2=1.2\) trait-units squared. Assume the covariance model above, with no residual covariance between distinct individuals and no residual–genetic cross-covariance. Compare the expected covariance for a parent–offspring pair and a full-sibling pair.

For a parent–offspring pair, \(A_{ij}=0.5\) and \(D_{ij}=0\). Substitution gives

\operatorname{Cov}(y_i,y_j) = (0.5)(2.0)+(0)(1.2) = 1.0 \text{ trait-units squared}.

For a full-sibling pair, \(A_{ij}=0.5\) and \(D_{ij}=0.25\), so

\operatorname{Cov}(y_i,y_j) = (0.5)(2.0)+(0.25)(1.2) = 1.0+0.3 = 1.3 \text{ trait-units squared}.

The additive contribution is the same for these two pair types, while the full-sibling covariance has an additional \(0.3\) trait-units squared from dominance. This contrast is informative only if other sources of covariance, such as shared rearing environment, are controlled or adequately modeled.

Identifiability: A Requirement on the Relationship Patterns

A covariance component is identifiable from pair covariances only when the observed relationship patterns supply enough distinct information. For each observed pair, the model is a linear equation in the two unknown variance components, with coefficients \(A_{ij}\) and \(D_{ij}\). In matrix terms, separate estimation requires the rows \((A_{ij},D_{ij})\) across the analyzed pairs to span two dimensions: the two columns must have rank two.

If every observed pair has dominance coefficients proportional to additive coefficients, \(D_{ij}=cA_{ij}\) for the same constant \(c\), then each covariance reduces to

\operatorname{Cov}(y_i,y_j) = A_{ij}\bigl(\sigma_A^2+c\sigma_D^2\bigr).

If at least one observed \(A_{ij}\) is nonzero, these off-diagonal covariances can inform the combined quantity \(\sigma_A^2+c\sigma_D^2\), but they cannot separate its two terms. If all observed \(A_{ij}\) are zero, then proportionality also makes every observed \(D_{ij}\) zero. The off-diagonal covariances are then zero under this model regardless of either variance component, so they identify neither component nor a combination of them. This degenerate case is important: a formula for a combined quantity does not imply that the data contain information about it when all its coefficients are zero.

In practice, rank is a minimum requirement, not a guarantee of precise estimation. If the additive and dominance relationship patterns are nearly proportional, the components may be weakly distinguished: small changes in estimated covariances can produce large changes in the separate component estimates. A useful design includes enough pair types and enough independent variation in the relationship matrices to make the distinction stable.

Worked Example: Solving for Both Variance Components

Consider a hypothetical analysis in which covariances have been estimated for parent–offspring and full-sibling pairs. Suppose their estimated values are \(1.2\) and \(1.5\) trait-units squared, respectively. Assume the standard coefficients in the table, and assume the residual and cross-covariance conditions stated above. Use the pair patterns to solve for the two components.

For parent–offspring pairs, the model gives

1.2=(0.5)\sigma_A^2+(0)\sigma_D^2, \qquad \sigma_A^2=\frac{1.2}{0.5}=2.4.

For full siblings, the equation is

1.5=(0.5)(2.4)+(0.25)\sigma_D^2.

The additive contribution is \((0.5)(2.4)=1.2\) trait-units squared, leaving \(1.5-1.2=0.3\) for the dominance contribution. Thus,

\sigma_D^2=\frac{0.3}{0.25}=1.2 \text{ trait-units squared}.

The estimates are \(\sigma_A^2=2.4\) and \(\sigma_D^2=1.2\) trait-units squared. The two pair types provide different equations because parent–offspring pairs have \(D_{ij}=0\), whereas full siblings have \(D_{ij}=0.25\). These calculations illustrate identification from the assumed covariance pattern; an actual variance-component analysis would also account for sampling uncertainty and the full set of individuals and relationships.

A Short Identifiability Derivation

The rank condition follows directly by considering two pair types. Let their expected covariances be \(C_1\) and \(C_2\), with relationship coefficients \((A_1,D_1)\) and \((A_2,D_2)\). The model yields

\begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = \begin{pmatrix} A_1 & D_1 \\ A_2 & D_2 \end{pmatrix} \begin{pmatrix} \sigma_A^2 \\ \sigma_D^2 \end{pmatrix}.

The two components can be uniquely recovered from these two equations when the coefficient matrix has nonzero determinant:

A_1D_2-A_2D_1 \ne 0.

If the determinant is zero, the rows are proportional (or one row is zero), so the equations do not supply two independent directions of information. With many pairs, the corresponding condition is that the matrix of observed additive and dominance coefficients has rank two. This establishes why differing pedigree patterns matter; simply increasing the number of pairs that all have the same coefficient ratio does not create the missing direction.

Worked Example: Proportional Patterns and the Zero-Coefficient Case

Suppose a hypothetical sample contains only full-sibling pairs under the standard pedigree coefficients. Every pair has \(A_{ij}=0.5\) and \(D_{ij}=0.25=0.5A_{ij}\). If the estimated pair covariance is \(1.5\) trait-units squared, then

1.5=(0.5)\sigma_A^2+(0.25)\sigma_D^2 =(0.5)\bigl(\sigma_A^2+0.5\sigma_D^2\bigr).

Dividing by \(0.5\) identifies the combination \(\sigma_A^2+0.5\sigma_D^2=3.0\) trait-units squared. It does not identify the two variances separately. For instance, \((\sigma_A^2,\sigma_D^2)=(3.0,0)\) and \((2.0,2.0)\), both in trait-units squared, produce the same modeled covariance of \(1.5\).

Now consider the degenerate case in which every observed pair has \(A_{ij}=0\), and suppose the observed dominance coefficients satisfy \(D_{ij}=0.5A_{ij}\). Then every \(D_{ij}\) is also zero. For every observed pair,

\operatorname{Cov}(y_i,y_j) = (0)\sigma_A^2+(0)\sigma_D^2=0.

These off-diagonal covariances contain no information about either variance component or about a nonzero combination of them. The distinction between this case and the full-sibling case is whether the proportional coefficients include any nonzero values.

Using Genomic Covariance Components Carefully

With genomic data, the same logic applies to additive and dominance relationship matrices. The entries \(A_{ij}\) and \(D_{ij}\) are then computed from genotypes using the chosen matrix conventions. The off-diagonal phenotype covariance is modeled as their weighted sum, with \(\sigma_A^2\) and \(\sigma_D^2\) as the weights. The relevant question is not just whether both matrices can be computed, but whether their patterns across individuals differ enough to support separate estimates.

A genomic matrix may contain realized variation that is absent from a simple pedigree classification. That variation can help distinguish components, but it does not by itself guarantee identification. Investigators should examine whether the additive and dominance matrices are nearly collinear in the analyzed sample and should evaluate the uncertainty of the resulting estimates. Matrix scaling also matters: changing a relationship matrix's scale changes the numerical variance component paired with it. The matrix definition and scale must therefore be reported alongside the component estimates.

The interpretation is conditional on the model. A fitted dominance component describes covariance captured by the specified dominance matrix, given the additive term, residual structure, and other modeled effects. It is not automatically proof that a biological mechanism is dominance, especially if environmental covariance or other misspecification can produce a similar pattern.

Common Pitfalls and Practical Checks

Assuming uncorrelated residuals are enough. Residuals being uncorrelated with one another does not rule out covariance between a residual and a genetic effect in another individual. State residual–genetic uncorrelatedness, or explicitly posit the covariance model that excludes those cross-covariances.
Treating full-sibling resemblance as purely additive. Under the standard pedigree coefficients, full siblings have both additive and dominance covariance. Parent–offspring pairs can help anchor the additive contribution because their dominance coefficient is zero under this model.
Claiming separate estimates from many similar pairs. Many pairs with proportional additive and dominance coefficients still provide only one independent pattern. Check matrix rank and near-collinearity, not just the number of pairs.
Overlooking shared environment. If siblings share environmental influences, their covariance can be attributed incorrectly to dominance unless that environmental structure is addressed. A genetic relationship pattern is informative only relative to plausible alternatives.
Reporting a combined quantity when all coefficients vanish. Under proportionality, a nonzero additive coefficient allows the associated component combination to be informed. If every observed additive coefficient is zero, and dominance coefficients are proportional to them, the off-diagonal pairs provide no information about either component.

Key takeaway: Additive and dominance variances are separated by contrasts in relationship patterns, not by labels alone. Parent–offspring pairs have additive but no expected dominance covariance under the standard pedigree model; full siblings have both. Separate estimation requires rank-two relationship information, a correctly specified covariance model, and careful attention to environmental and residual cross-covariances.

Check Your Understanding

Use the covariance model and assumptions stated above to answer these questions.

Why can parent–offspring and full-sibling covariance estimates help distinguish additive from dominance resemblance?
If \(D_{ij}=cA_{ij}\) for every observed pair and at least one \(A_{ij}\) is nonzero, what quantity do the off-diagonal covariances identify?
What can be inferred from those covariances if all observed \(A_{ij}\) are zero and \(D_{ij}=cA_{ij}\)?
Why does uncorrelatedness among residuals alone not justify the two-component covariance equation?
How could shared household environment affect an estimate interpreted as dominance covariance?