Deriving QTL Covariance From Identity By Descent

Why IBD Sharing Determines Locus-Specific Covariance

Relatives can resemble one another because they share genetic material, but the amount shared at a particular locus varies with their inheritance. For a quantitative-trait locus (QTL), the key link is direct: if two relatives inherit the same ancestral allele copy, that copy contributes the same allele effect to both of their trait values. The expected number of such shared copies therefore determines the covariance contribution from the locus.

This is a locus-specific statement. It does not by itself describe the entire covariance of the relatives' phenotypes, which may also reflect other genetic or environmental sources. Here the aim is to derive the covariance attributable to one additive QTL and to calculate it from IBD sharing under clearly specified pedigree assumptions.

An Additive Allele-Effect Model

Consider two relatives, \(i\) and \(j\), at a candidate QTL. Assume each is non-inbred at this locus, so each has two distinct ancestral allele copies. Under the additive model, the QTL value for person \(i\) is the sum of the effects of the two allele copies they carry. Let each distinct ancestral allele copy have a random effect with mean zero and variance \(\tau^2\). Effects of distinct ancestral copies are independent; descendants inherit an allele's effect unchanged. There is no mutation, dominance, or interaction between the two copies in this model.

Define \(K_{ij}\) as the number of allele copies that the two individuals share IBD, so \(K_{ij}\) is 0, 1, or 2 for a non-inbred pair. The quantity \(K_{ij}\) depends on inheritance at the locus; it is not the number of allele copies that happen to have the same measured value. For a given pedigree, it may be fixed or random across possible inheritance patterns.

The expectation in this expression is over the inheritance pattern under the pedigree. If the pair's IBD sharing is known, \(E(K_{ij})\) is simply its observed or specified value. If sharing varies across possible transmissions, the expected count averages over those transmissions. A useful equivalent form is \(2\tau^2\pi_{ij}\): each expected shared allele copy contributes \(\tau^2\) to covariance.

Deriving the Covariance Formula

The derivation follows by conditioning on which ancestral allele copies each relative inherited. Label each distinct ancestral allele copy by \(u\), and write its effect as \(A_u\), with \(E(A_u)=0\), \(\operatorname{Var}(A_u)=\tau^2\), and independent effects for distinct copies. Let \(S_i\) and \(S_j\) be the two-copy sets carried by individuals \(i\) and \(j\). Then \(G_i=\sum_{u\in S_i}A_u\) and \(G_j=\sum_{w\in S_j}A_w\). In the product \(G_iG_j\), terms for the same ancestral copy have expectation \(\tau^2\); terms for distinct copies have expectation zero by independence and zero means. There are \(K_{ij}\) shared copies. Thus, conditional on the inheritance pattern, the covariance is \(K_{ij}\tau^2\).

Both conditional means are zero, so the conditional expectation of each QTL value does not vary with the inheritance pattern. Averaging the conditional covariance over inheritance therefore gives the unconditional covariance. Equivalently, the law of total covariance has a zero contribution from the covariance of the conditional means:

For a non-inbred individual, the two allele effects are independent, so \(E(G_i)=0\) and \(\operatorname{Var}(G_i)=2\tau^2\). That establishes the \(v_Q\) used above. The interpretation is specific: a greater expected proportion of alleles shared IBD means a greater covariance contribution from this locus, with all other model assumptions held fixed.

Reading the Sharing Coefficient Carefully

The proportion \(\pi_{ij}\) is the expected fraction of the pair's two allele copies that are shared IBD. It is not the probability that the pair shares at least one copy. For example, in the full-sibling example, \(P(K_{ij}\geq 1)=3/4\), while \(\pi_{ij}=1/2\). These are different summaries: one is a probability of any sharing, and the other is half the expected count.

A related convention uses the kinship coefficient, the probability that one randomly selected allele copy from each relative is IBD. For non-inbred individuals, there are four equally likely cross-person allele-copy pairs, so this coefficient is \(E(K_{ij})/4\). It follows under the present model that the QTL covariance is \(4\tau^2\) times the kinship coefficient, or \(2v_Q\) times that coefficient. This is the same result in a different convention; checking which sharing quantity is being used prevents a factor-of-two error.

In a pedigree analysis, the calculation proceeds from the specified or expected IBD sharing at the candidate locus. The inheritance model and pedigree assumptions determine \(E(K_{ij})\); the additive allele-effect model then converts it to covariance by multiplying by \(\tau^2\). This relationship concerns the QTL component only, not the total phenotypic covariance.

Common Mistakes and Assumptions

• Using a relationship label as if it fixed sharing. “Full sibling,” “half-sibling,” or “parent–child” does not by itself rule out inbreeding or additional shared ancestry. State the pedigree conditions that justify the IBD distribution being used.
• Confusing IBD with matching allele values. Two allele copies can have the same measured value without descending from the same ancestral copy. The covariance derivation counts shared ancestral copies, not accidental value matches.
• Mixing up \(K_{ij}\) and \(\pi_{ij}\). \(K_{ij}\) is a count from 0 to 2 for a non-inbred pair; \(\pi_{ij}=E(K_{ij})/2\) is an expected proportion. The covariance is \(\tau^2E(K_{ij})\), equivalently \(v_Q\pi_{ij}\).
• Calling the QTL covariance the total trait covariance. The formula isolates the contribution from the specified additive locus. Other sources of covariance are not included in this derivation.