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Pedigrees and Genetic Relatedness · Tutorial 74 of 989

Deriving Genetic Covariance From Kinship

See why the covariance between two relatives’ additive genetic values is twice their kinship coefficient times the reference additive genetic variance.

Beginner 10 min read

What You'll Learn

  • Define kinship as the IBD probability for a randomly selected allele-copy pair.
  • State the allele-effect assumptions that connect IBD to genetic covariance.
  • Derive the covariance formula by summing over four allele-copy pairings.
  • Calculate covariance for parent–offspring, sibling, and cousin pairs.
  • Distinguish an average IBD probability from the probability for a fixed pairing.

Why Kinship Determines Additive Genetic Covariance

Kinship summarizes how likely two relatives are to carry allele copies that descend from the same ancestral copy. Additive genetic covariance asks a related but different question: how much do their additive genetic values tend to vary together? Under a clear set of allele-effect assumptions, kinship connects these quantities through a simple formula. The key is to use the kinship coefficient as an average across allele-copy pairings, not as the IBD probability for every particular pairing.

We consider two distinct individuals, \(i\) and \(j\), and their additive genetic values across a set of loci. The derivation uses the pedigree meaning of kinship established in earlier material: \(\phi_{ij}\) is the probability that a randomly selected allele copy from \(i\) and a randomly selected allele copy from \(j\) are identical by descent (IBD). “Randomly selected” means each of the two copies in each individual has equal selection probability.

Definition: At locus \(\ell\), let \(Z_{i\ell k}\) be the additive effect carried by copy \(k\) (\(k=1,2\)) in individual \(i\). Write the additive genetic value as \(G_i=\sum_{\ell}\sum_{k=1}^{2} Z_{i\ell k}\). The covariance \(\operatorname{Cov}(G_i,G_j)\) measures how the two individuals’ additive genetic values vary together across the assumed genetic model; its units are squared units of genetic effect.

Assumptions About Allele Effects

IBD is a statement about descent, whereas covariance is a statement about values. To turn one into the other, we need assumptions about how allelic effects behave. The following reference model is sufficient for the derivation.

  • At each locus \(\ell\), an allele copy has an additive effect with mean zero and variance \(v_\ell\) in the reference population.
  • Copies that are IBD carry the same inherited allelic effect. Copies that are not IBD have uncorrelated effects under the reference model. This represents unrelated reference founders and excludes extra correlation from population structure or other shared ancestry outside the pedigree.
  • Effects at different loci are uncorrelated, so covariance across loci contributes no additional terms. There are no dominance or epistatic terms in \(G_i\); it is the sum of the allele effects alone.
  • The reference additive genetic variance is defined for a non-inbred individual with two independently sampled reference copies at each locus. Environmental effects are not part of \(G_i\).

These are modeling assumptions, not consequences of kinship alone. In particular, shared environment or population structure can create covariance in observed traits without creating the pedigree-based additive genetic covariance derived here. The result concerns additive genetic values under the stated reference model.

Definition: The reference additive genetic variance is \(V_A^{\mathrm{ref}}=2\sum_{\ell}v_\ell\). The factor of two arises because a non-inbred reference individual has two allele copies at each locus, each with variance \(v_\ell\), and the two reference copies are uncorrelated.

From IBD Probabilities to Covariance

At a particular locus, consider a fixed pairing: copy \(k\) in individual \(i\) and copy \(m\) in individual \(j\). Let \(q_{i\ell k,j\ell m}\) denote the probability that these two specified copies are IBD, with the probability taken over the pedigree’s transmission process. If they are IBD, they carry the same mean-zero effect, so their covariance contribution is \(v_\ell\). If they are not IBD, the model assumes their effects are uncorrelated. Therefore,

$$ \operatorname{Cov}(Z_{i\ell k},Z_{j\ell m}) = q_{i\ell k,j\ell m}v_\ell. $$

There are four possible pairings of the two copies in \(i\) with the two copies in \(j\). Kinship is the average of the IBD probabilities for those four pairings. Thus the sum of their probabilities is \(4\phi_{ij}\). Summing the four copy-pair covariances gives the covariance contribution at locus \(\ell\).

$$ \begin{aligned} \operatorname{Cov}\left(\sum_{k=1}^{2}Z_{i\ell k}, \sum_{m=1}^{2}Z_{j\ell m}\right) &= \sum_{k=1}^{2}\sum_{m=1}^{2} q_{i\ell k,j\ell m}v_\ell \\ &= v_\ell(4\phi_{ij}) \\ &= 4\phi_{ij}v_\ell. \end{aligned} $$

Because effects at different loci are uncorrelated, the total covariance is the sum of these locus-specific contributions. Substituting the definition of \(V_A^{\mathrm{ref}}\) yields the result.

Theorem: Under the stated additive allele-effect assumptions, for two distinct individuals \(i\) and \(j\), $$ \operatorname{Cov}(G_i,G_j) = \sum_{\ell}4\phi_{ij}v_\ell = 2\phi_{ij}V_A^{\mathrm{ref}}. $$ The kinship coefficient is an average over the four cross-individual copy pairings. It is generally not the IBD probability for each fixed pairing.

Worked Applications

Worked Example: Parent and Offspring

Suppose a hypothetical reference model has two loci with \(v_1=3\) and \(v_2=7\) effect-units squared. A parent and child have kinship \(\phi_{ij}=1/4\). Assume the other parent is unrelated to the first parent, and apply the allele-effect assumptions above. Find their additive genetic covariance.

First calculate the reference additive genetic variance:

$$ V_A^{\mathrm{ref}}=2(v_1+v_2)=2(3+7)=20 $$

The covariance formula gives:

$$ \operatorname{Cov}(G_i,G_j) =2\left(\frac{1}{4}\right)(20)=10 $$

The result is 10 effect-units squared. We can check it by examining fixed pairings at either locus. Label the child’s copy inherited from this parent as the inherited copy. Each of the parent’s two copies has probability \(1/2\) of being the one transmitted, so its IBD probability with that child copy is \(1/2\). The parent’s copies have IBD probability zero with the child’s copy from the unrelated other parent. The four probabilities are therefore \(1/2, 1/2, 0, 0\): their average is \(1/4\), and their sum is 1. The locus-\(\ell\) covariance is \(v_\ell\), so the two loci contribute \(3+7=10\), as before. This explicitly shows why a fixed pairing need not have IBD probability \(\phi_{ij}\).

Worked Example: Full Siblings

Consider two hypothetical full siblings with two loci, each having allele-effect variance \(v_\ell=4\) effect-units squared. Their kinship coefficient is \(1/4\). Assume the reference founders are unrelated and the additive model applies. Calculate their covariance and check the four copy pairings at one locus.

The reference variance is \(2(4+4)=16\) effect-units squared. Hence:

$$ \operatorname{Cov}(G_i,G_j) =2\left(\frac{1}{4}\right)(16)=8 $$

At a locus, label each sibling’s copies by parental origin. The probability that the two paternal-origin copies are IBD is \(1/2\), and the probability that the two maternal-origin copies are IBD is also \(1/2\). A paternal-origin copy paired with a maternal-origin copy has IBD probability zero under the unrelated-founder assumption. The four probabilities sum to 1, with average \(1/4\). Thus the locus contributes \(4(1/4)(4)=4\) effect-units squared; the two loci contribute \(4+4=8\). The calculation depends on the total across pairings, not on assigning \(1/4\) to each fixed pairing.

Worked Example: First Cousins

Suppose two hypothetical first cousins descend from a pair of unrelated, non-inbred grandparents through two siblings. The standard pedigree kinship calculation gives \(\phi_{ij}=1/16\). Let the reference additive genetic variance across the loci under study be \(V_A^{\mathrm{ref}}=24\) effect-units squared. Find the cousins’ additive genetic covariance.

The model assumptions are the same: effects are additive, IBD copies share effects, non-IBD reference copies are uncorrelated, and effects at different loci are uncorrelated. Apply the formula:

$$ \operatorname{Cov}(G_i,G_j) =2\left(\frac{1}{16}\right)(24) =3 $$

The covariance is 3 effect-units squared. It is smaller than the parent–offspring covariance in the first example because the cousins’ kinship is smaller, while the reference additive genetic variance is held fixed. This comparison concerns covariance under the model; it does not say that either individual’s genetic value determines the other’s.

Why the Formula Is Useful

The formula translates a pedigree summary into a covariance on the scale of additive genetic values. Kinship is dimensionless, while \(V_A^{\mathrm{ref}}\) supplies the variance scale, so the product has the correct squared-effect units. For a fixed reference variance, pairs with larger kinship have larger expected additive genetic covariance. This relationship is one reason kinship calculations are useful in statistical genetics: they express how pedigree relatedness enters a covariance model without enumerating every possible allele transmission anew for each pair.

The formula also clarifies the role of the reference population. \(V_A^{\mathrm{ref}}\) is not an arbitrary variance for whichever individual happens to be under study; it is defined from the reference allele-effect variances. The present derivation uses that baseline to scale pairwise covariance. Inbreeding can alter an individual’s diagonal genetic covariance, so diagonal terms require careful treatment rather than an unexamined use of the distinct-individual argument; that issue is considered separately.

Common Mistake: Giving Every Pairing Probability \(\phi_{ij}\)

A tempting but incorrect shortcut is: “There are four copy pairings, and each has IBD probability \(\phi_{ij}\).” Kinship does not make that claim. It says that the average of the four IBD probabilities is \(\phi_{ij}\). For a parent and child, for example, two specified pairings have probability \(1/2\) and two have probability zero. Their average is \(1/4\), the parent–offspring kinship coefficient.

The correction is to use a separate IBD probability for each fixed pairing, multiply that probability by \(v_\ell\), and then sum over all four pairings. The sum of probabilities is \(4\phi_{ij}\), which gives the locus covariance \(4\phi_{ij}v_\ell\). Replacing all four probabilities by \(\phi_{ij}\) happens to preserve the sum, but falsely describes the individual pairings and can lead to errors when a problem asks about a specified pair of copies.

Key takeaway: With additive allele effects of reference variance \(v_\ell\), a fixed copy pairing contributes its own IBD probability times \(v_\ell\). Averaging the four probabilities gives kinship, and summing them gives the covariance formula \(\operatorname{Cov}(G_i,G_j)=2\phi_{ij}V_A^{\mathrm{ref}}\).

Check Your Understanding

Use the stated reference model when answering. Keep fixed-pair probabilities distinct from their average.

  1. At one locus, the four fixed-pair IBD probabilities are \(1/2, 1/2, 0, 0\). What is the kinship coefficient, and what is the locus covariance if \(v_\ell=6\)?
  2. If \(\phi_{ij}=1/8\) and \(V_A^{\mathrm{ref}}=32\) effect-units squared, calculate the additive genetic covariance and state its units.
  3. Explain why the IBD probability for each fixed pairing cannot generally be set equal to the kinship coefficient.
  4. Name two assumptions about allele effects needed to derive covariance from IBD probabilities.
  5. Would shared environment alone change the pedigree-based additive genetic covariance in this model? Explain the distinction.

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