Why Similarity Does Not Identify Its Cause
Relatives often resemble one another, but resemblance is not a mechanism. A larger covariance between siblings than between parent and offspring could reflect dominance, epistasis, shared environments, or genetic effects that one sibling has on another. Assortative mating can also alter genetic covariance from the expectations of a random-mating model. The central task is therefore to ask not simply whether relatives resemble one another, but whether competing explanations predict different patterns across the observed relationships.
A useful comparison begins with the covariance each model predicts for each kind of pair. Family labels alone do not reveal which mechanism generated the covariance. Genomic relationships, more varied family structures, measured environments, and specific designs may provide additional contrasts, but even a rich covariance pattern may leave multiple explanations compatible with the data.
A Covariance Model for Resemblance
Let \(Y_i\) be the measured phenotype for person \(i\). A simplified variance-component model represents the phenotype as a mean plus several centered contributions:
Here \(A_i\) is an additive genetic contribution, \(D_i\) a dominance contribution, \(E_{AA,i}\) an additive-by-additive epistatic contribution, \(U_i\) an indirect genetic contribution (for example, an effect of relatives’ genotypes on person \(i\)'s phenotype), and \(\varepsilon_i\) the remaining contribution. These terms are conceptual components; a particular study may include only some of them. With random mating and mutually uncorrelated components, the covariance for a pair \(i,j\) can be written:
The variance components \(\sigma_A^2\), \(\sigma_D^2\), and \(\sigma_{AA}^2\) are on the phenotype scale (squared phenotype units). The entries \(K_{A,ij}\), \(K_{D,ij}\), and \(K_{AA,ij}\) are relationship coefficients or genomic relationship-matrix entries, scaled relative to the corresponding component. In a simple pedigree model under random mating, for example, \(K_{A,ij}\) is the expected additive covariance for that pair as a fraction of the additive variance. Genomic matrices use the observed genotypes to quantify relatedness, with the exact scaling determined by how the matrix is constructed.
The component terms need not be uncorrelated in every setting. Assortative mating, population structure, and genotype–environment correlation can create dependencies that require additional covariance terms or a model that represents those processes. The additive-plus-dominance-plus-epistasis expression is a comparison framework under stated assumptions, not a guarantee that its terms are independent in every population.
What Different Genetic Components Predict
Under the conventional random-mating pedigree model, parent–offspring pairs and full siblings have additive relationship coefficient \(1/2\). Their dominance coefficients differ: parent and offspring do not share both alleles at a locus identical by descent in the way required for dominance resemblance, whereas full siblings have expected dominance relationship coefficient \(1/4\). Thus, under this model, dominance adds expected covariance for full siblings but not for parent–offspring pairs.
For additive-by-additive epistasis at two unlinked loci, the expected relationship coefficient is based on the product of the additive coefficients at the two loci. Under the simple pedigree expectations, both parent–offspring pairs and full siblings have coefficient \(1/4\) for that component. These coefficients depend on the specified genetic coding, scaling, and assumptions; they should not be treated as universal constants for every genomic matrix or population.
| Pair type | Additive coefficient \(K_A\) | Dominance coefficient \(K_D\) | Two-locus additive-by-additive coefficient \(K_{AA}\) |
|---|---|---|---|
| Parent–offspring | 1/2 | 0 | 1/4 |
| Full siblings | 1/2 | 1/4 | 1/4 |
| Unrelated individuals, under the simple model | 0 | 0 | 0 |
This table shows why a single contrast may be informative but not decisive. A full-sibling covariance larger than a parent–offspring covariance is compatible with a dominance contribution under these assumptions. It is also compatible with sibling-shared environment or sibling effects. Moreover, these two pair types have the same listed additive-by-additive coefficient, so that component alone does not predict a difference between them.
Worked Example: A Sibling Excess Has More Than One Explanation
Suppose a hypothetical study estimates a phenotype covariance of \(0.30\) squared trait units for parent–offspring pairs and \(0.42\) squared trait units for full siblings. Assume, for illustration, that the additive variance is \(0.60\) squared trait units and that the simple pedigree coefficients above apply.
The additive-only model predicts, for either pair type:
It matches the parent–offspring covariance and falls short of the sibling covariance by \(0.42-0.30=0.12\) squared trait units. One possible model adds dominance. A dominance variance of \(0.48\) would contribute \(0.48 \times 1/4=0.12\) squared trait units for siblings and zero for parent–offspring pairs. The predicted covariances would then be \(0.30\) for parent–offspring pairs and \(0.30+0.12=0.42\) for siblings.
But a sibling-shared environmental covariance of \(0.12\) squared trait units, with no corresponding parent–offspring environmental covariance, would produce the same two predictions. The arithmetic shows that dominance can fit the contrast; the two covariances do not identify it as the cause.
Comparing Matrices Requires Variation in Relationships
With genomic data, a model may assign a relationship-matrix entry to each pair rather than using only broad family labels. For instance, a pair’s additive genomic relationship need not equal another pair’s, even if both pairs are full siblings. A dominance matrix and an epistatic matrix can therefore predict additional pair-to-pair variation. This can help distinguish components, provided the entries differ in ways that are informative and are not merely tracking the same pattern.
The key mathematical requirement is that the columns of the design formed by the candidate relationship matrices are sufficiently distinct over the observed pairs. If two matrices are proportional, or nearly so, their associated variance components cannot be separated reliably by those data. More pairs do not automatically solve the problem: many pairs with nearly identical relationship patterns may contribute little new information.
Worked Example: Calculating a Genomic-Model Covariance
Consider two hypothetical individuals with \(K_{A,ij}=0.40\), \(K_{D,ij}=0.20\), and \(K_{AA,ij}=0.10\). Suppose a prespecified model uses \(\sigma_A^2=0.50\), \(\sigma_D^2=0.30\), and \(\sigma_{AA}^2=0.20\), all in squared trait units, and assumes the components are uncorrelated. Assume also that the pair has no modeled shared residual covariance.
The predicted additive contribution is \(0.50 \times 0.40=0.20\) squared trait units. The dominance contribution is \(0.30 \times 0.20=0.06\), and the epistatic contribution is \(0.20 \times 0.10=0.02\). Thus:
This is the covariance predicted by the stated model and entries, not evidence that each biological mechanism has been separately established. If the observed pairs had \(K_{D,ij}=0.5K_{A,ij}\) throughout the sample, for example, the dominance term would have the same pairwise pattern as an additive term with a rescaled coefficient. The two variance components would then be difficult or impossible to distinguish from covariance data alone.
Assortative Mating and Indirect Effects
Assortative mating matters because standard pedigree expectations often rely on random mating at the modeled trait or genotype. When mates are correlated, their genetic contributions can also become correlated, and gametic disequilibrium can affect the distribution of alleles passed to offspring. Consequently, the expected genetic covariance among relatives can depart from the simple coefficients in the table. A spouse covariance by itself does not prove genetic assortment: shared environment, social pairing, and other sources can also make spouses similar. Distinguishing these possibilities calls for information on mating, genotypes, and the relevant population structure, not just a spouse covariance.
Indirect genetic effects create another route to resemblance. A person’s outcome may depend on their own genotype and on genotypes of people around them. In a simple two-person illustration, let each person’s phenotype contribution be their direct genetic value plus \(\lambda\) times the other person’s genetic value. If the direct genetic values \(G_1,G_2\) have equal variance \(\sigma_G^2\) and covariance \(c_G\), then:
The extra covariance depends on the social effect \(\lambda\); it is not simply a dominance or epistatic contribution. But if the study observes only the pair’s outcome covariance, an indirect effect may resemble an added genetic relationship component. Measuring or exploiting who interacts with whom, and whose genotype is transmitted, can provide contrasts that a covariance for relatives alone does not.
Worked Example: A Social Genetic Effect Can Mimic Extra Sibling Resemblance
Suppose two hypothetical siblings have direct genetic values with variance \(\sigma_G^2=0.60\) and covariance \(c_G=0.30\), in squared trait units. Assume, just for this illustration, that each sibling’s phenotype includes their own direct genetic value plus \(\lambda=0.10\) times the other sibling’s direct genetic value. Ignore other phenotype components.
Their covariance under this pair-level model is:
Without the indirect effect, the covariance would be \(0.30\) squared trait units. The additional \(0.123\) is generated here by the modeled social effect. A covariance of roughly this size could also be produced by other components or by shared conditions, so this calculation illustrates a competing prediction rather than a way to diagnose indirect effects from one sibling covariance.
A Practical Comparison Strategy
A disciplined comparison makes the assumptions and limits visible before interpreting fitted components:
State whether the model includes additive, dominance, epistatic, assortative-mating, shared-environment, or indirect-effect contributions. Do not treat unexplained resemblance as a named mechanism.
Use the pedigree or genomic relationships and state assumptions about mating, genetic coding, and component covariance. Include environmental or social covariance when the design makes it plausible.
Ask whether the relationship matrices vary independently across the pairs observed. If patterns are proportional or nearly proportional, report that the components are confounded or weakly identified.
A fitted variance component describes the covariance assigned to that component under the model and assumptions. It does not, by itself, establish a unique biological mechanism.
A strong analysis may combine several kinds of information: relatives with different degrees of relatedness, genotyped individuals, relatives who do not share a household, nontransmitted parental alleles, or measurements of social contact. These designs can add contrasts that help distinguish explanations. They do not remove the need to state which assumptions make those contrasts informative.
Common Mistake: Naming a Mechanism From One Contrast
- Wrong reasoning: “Full siblings have more covariance than parent–offspring pairs, so the excess must be dominance.” Correction: dominance is one explanation under the simple model, but sibling-shared environment and indirect sibling effects can predict a similar excess.
- Wrong reasoning: “A fitted epistatic component proves that the measured phenotype is caused by biological interaction between loci.” Correction: the component is a statistical allocation of covariance under a specified model and scaling. Its interpretation depends on whether alternative covariance patterns have been represented and separated.
- Wrong reasoning: “Spouses resemble one another, so assortative mating has been established.” Correction: spouse resemblance is compatible with assortment but also with shared environment or other social processes; additional data are needed to distinguish them.
- Wrong reasoning: “A large sample guarantees that every variance component is identifiable.” Correction: precision requires informative variation in the design. If candidate relationship patterns are collinear, increasing sample size alone cannot make them distinct.
Check Your Understanding
For each question, focus on the predicted covariance pattern and the assumptions needed to interpret it.
- Under the simple random-mating pedigree model, why can dominance contribute to full-sibling covariance but not parent–offspring covariance?
- What does it imply for identifiability if two relationship matrices are proportional across all observed pairs?
- A spouse covariance is positive. Name two explanations other than genetic assortment and state what additional information could help distinguish them.
- In the two-person indirect-effect model, which terms contribute to the covariance beyond the direct genetic covariance?
- Why is a fitted epistatic variance component not, by itself, proof of a unique biological interaction mechanism?