Deriving Cross-Trait Covariance From Genetic Relatedness

Why Relatedness Scales Cross-Trait Covariance

Suppose two traits are measured on two different people. Their measurements can covary because the people share genetic effects: relatives tend to share more of their additive genetic variation than unrelated people. In a bivariate genomic mixed model, this idea has a precise implication. The genetic covariance between the traits is multiplied by the people’s genomic relatedness to give the model-implied covariance between their measurements.

The result is a covariance, not a claim that one trait causes the other. It also does not say that every pair of relatives has exactly the same genetic sharing. The genomic relationship matrix records pair-specific relatedness; familiar values such as one-half for siblings or parent–offspring pairs are expectations under a pedigree model.

Model and Assumptions

Let \(y_{it}\) be the measurement of trait \(t\) on person \(i\), where \(t\) is 1 or 2. Write the measurement as a fixed mean or covariate contribution, an additive genetic effect, and a residual:

Here, \(g_{it}\) is the additive genetic effect for person \(i\) and trait \(t\), and \(e_{it}\) is the residual. The fixed part is not random in the covariance calculation, conditional on the covariates. Let \(K\) be the genomic relationship matrix, with \(K_{ij}\) denoting the modeled relatedness between people \(i\) and \(j\). We use the convention that its diagonal is approximately 1 and that its entries are on the additive-genetic relationship scale.

The genetic covariance matrix is

Its diagonal entries are the genetic variances of the two traits, and its off-diagonal entry \(\sigma_{g,12}\) is their genetic covariance. The residual covariance matrix has corresponding entries \(\sigma_{e,ts}\). These parameters are on the measurement scales of the traits: for example, if trait 1 is in centimetres and trait 2 is in kilograms, \(\sigma_{g,12}\) has units of centimetre-kilograms.

Deriving the Cross-Person Covariance

Consider trait 1 on person \(i\) and trait 2 on a different person \(j\). Once fixed contributions are held constant, the covariance between their measurements is the covariance between their genetic effects plus the covariance between their residuals, along with any genetic–residual cross terms. Under the model assumptions, the residual covariance and the cross terms are zero for distinct people. Therefore:

The same reasoning applies to any pair of traits \(t\) and \(s\): the model-implied genetic contribution to the covariance across people is \(K_{ij}\sigma_{g,ts}\). In particular, the covariance between trait 2 on person \(i\) and trait 1 on person \(j\) is also \(K_{ij}\sigma_{g,12}\), since \(K\) is symmetric and the genetic covariance matrix is symmetric.

For measurements on the same person, the covariance has a different form. The genetic effects contribute \(\sigma_{g,12}\), and the within-person residuals may contribute \(\sigma_{e,12}\):

If \(K_{ii}=1\), this is \(\sigma_{g,12}+\sigma_{e,12}\). Thus the covariance across two different relatives is not generally the same estimand as the phenotypic covariance between two traits recorded on one person. The distinction follows from which genetic effects are shared and from the residual-covariance assumption.

A Marker-Level Justification

The scaling by \(K_{ij}\) can also be seen from a simple marker-effect representation. Suppose \(m\) standardized marker values are collected in the rows of a matrix \(Z\), and let \(z_{i\ell}\) be person \(i\)'s value at marker \(\ell\). Write the genetic effect on each trait as a sum of marker contributions, \(g_{it}=\sum_{\ell=1}^{m}z_{i\ell}\beta_{\ell t}\). Assume marker effects at different loci are independent, and that the two trait effects at each locus have covariance \(\Omega_{12}/m\). Define \(K_{ij}=m^{-1}\sum_{\ell=1}^{m}z_{i\ell}z_{j\ell}\). Then:

In the corresponding mixed-model notation, \(\Omega_{12}\) is the genetic covariance parameter \(\sigma_{g,12}\). The argument relies on the specified marker scaling and effect assumptions; it is a justification for the covariance structure, not a claim that every analysis uses this exact marker-level parameterization.

Expected Pedigree Relatedness and Realized Relatedness

Under the usual additive relationship convention, unrelated people have expected relatedness 0, while full siblings and parent–offspring pairs have expected relatedness 0.5. The model’s expected cross-trait covariance for a pair follows by multiplying that value by \(\sigma_{g,12}\). For a negative genetic covariance, the expected cross-person covariance is negative at positive relatedness; for a zero genetic covariance, it is zero at every relatedness value under this model.

A genomic relationship matrix is estimated from observed genotypes. Its off-diagonal entries can differ from pedigree expectations: two full siblings may have different realized values, for instance, because they inherit different combinations of parental genetic material. When an observed \(K_{ij}\) is available, use that value in the model-implied covariance calculation rather than substituting the pedigree average. The expected relationship remains useful when the question specifically asks for a pedigree-level expectation.

Worked Marker Calculation

Scope of the Result

The cross-person formula is conditional on the covariance structure. If residuals for different people are correlated—for example, because of shared household or other modeled environmental effects—then an additional cross-person term may be needed. In that setting, the covariance is not just \(K_{ij}\sigma_{g,12}\). Likewise, the result describes covariance under the stated additive genetic model; it does not establish a causal pathway between the traits or identify which variants contribute to their covariance.

The parameter \(\sigma_{g,12}\) is on the product scale of the two traits, while \(K_{ij}\) is a dimensionless relationship coefficient. Consequently, multiplying by relatedness changes the magnitude of the covariance but not its units. A positive covariance indicates that the model predicts same-direction deviations across the two measurements on average; a negative covariance indicates opposite-direction deviations on average. Neither sign alone specifies the covariance for every individual pair.

Common Pitfall

• Wrong reasoning: “Siblings share half their genes, so the covariance between their two trait measurements is one-half of the phenotypic covariance between the traits.” This uses the wrong quantity: the relationship coefficient multiplies the genetic covariance, not the total within-person phenotypic covariance.
• Correction: For different people, apply \(\operatorname{Cov}(y_{i1},y_{j2})=K_{ij}\sigma_{g,12}\) under the zero cross-person residual assumption. The within-person residual covariance \(\sigma_{e,12}\) belongs in the covariance of two traits measured on the same person, not in this cross-person formula.
• Another caution: Do not treat 0.5 as the exact genomic relatedness of every sibling or parent–offspring pair. It is the usual expected value; use pair-specific \(K_{ij}\) for a calculation based on a genomic relationship matrix.