How Inbreeding Changes Diagonal Genetic Covariance

Why Inbreeding Changes a Diagonal Entry

An additive genetic covariance matrix records how additive genetic values vary together across individuals. Its diagonal entries have a direct interpretation: each is the variance of one individual's additive genetic value. Inbreeding affects that variance because it increases the chance that the two alleles an individual carries at a locus are identical by descent (IBD). Under a standard additive allele-effect model, that shared ancestry contributes covariance between the two allele effects.

The result is a simple scaling rule: if \(\sigma_A^2\) is the additive genetic variance for a noninbred reference individual, then an individual with inbreeding coefficient \(F_i\) has diagonal additive genetic covariance \((1+F_i)\sigma_A^2\). This is a model-based statement about additive genetic variance, not a claim that every component of phenotypic variance, or every real population's variance, changes by exactly that factor.

The Allele-Effect Model

Consider one autosomal locus. Write the additive genetic value at that locus for individual \(i\) as \(g_i=a_{i1}+a_{i2}\), where \(a_{i1}\) and \(a_{i2}\) are the effects of the individual's two allele copies. Use a centered random-effect model: each allele effect has mean zero and variance \(v\); effects of alleles that are not IBD are independent; and copies that are IBD have the same effect. This model treats effects as additive, with no dominance or epistatic terms.

If the two copies are not IBD, their covariance is zero under the model. If they are IBD, their effects are the same random effect, so their covariance is \(v\). Since the probability of IBD is \(F_i\), the unconditional covariance between the two effects is \(F_i v\). The variance of their sum is therefore the sum of the two individual variances plus twice their covariance.

For a noninbred reference individual, \(F_i=0\), so the variance at this locus is \(2v\). For multiple loci, assume the effects at different loci have no covariance and add their variances. If \(v_\ell\) is the allele-effect variance at locus \(\ell\), the noninbred reference variance is \(\sigma_A^2=\sum_\ell 2v_\ell\). Applying the same inbreeding coefficient across the autosomal loci gives the diagonal relationship below.

Connection to Kinship and the Relationship Matrix

The same result can be expressed using the kinship coefficient \(\phi_{ij}\), the probability that a randomly selected allele from individual \(i\) and one from individual \(j\) are IBD. For two distinct individuals, the standard additive covariance model gives \(\operatorname{Cov}(g_i,g_j)=2\phi_{ij}\sigma_A^2\). On the diagonal, selecting one allele from individual \(i\) twice requires accounting for the chance that the selections choose the same physical copy.

Two independent selections from the two allele copies of individual \(i\) choose the same physical copy with probability \(1/2\); those selections are necessarily IBD. They choose different copies with probability \(1/2\), and those copies are IBD with probability \(F_i\). Thus \(\phi_{ii}=(1+F_i)/2\). Substituting this diagonal kinship value into the additive covariance formula yields the diagonal scaling rule.

This establishes why the diagonal entry is not simply \(\sigma_A^2\) when \(F_i>0\). The IBD copies are not two independent draws of allele effect: the same inherited effect appears twice in the additive sum. The additional covariance is \(F_i\sigma_A^2\). When \(F_i=0\), the diagonal returns to the noninbred reference value.

Worked Calculations

Interpretation and Limits

A diagonal additive genetic covariance is an individual's modeled additive genetic variance. It is not a covariance between distinct relatives: those entries describe how two individuals' additive genetic values vary together. It is also not the total phenotypic variance. Environmental variation and other genetic components are outside the quantity calculated here.

The multiplier \(1+F_i\) has a clear reference point. If \(F_i=0\), the individual's two allele copies are not IBD under the specified pedigree model, and the diagonal equals \(\sigma_A^2\). If \(F_i=0.10\), the model predicts a variance \(1.10\sigma_A^2\), or 10% above that reference. Since \(F_i\) is a probability, it lies between 0 and 1; within this model, the corresponding multiplier lies between 1 and 2.

The result relies on the chosen reference population and on the allele-effect assumptions: additive effects, identical effects for IBD copies, independent effects for non-IBD copies, and no covariance across loci. The formula is not a universal adjustment that can be applied without checking the model. For example, if the analysis includes nonadditive genetic effects or uses a different variance parameterization, those components require their own covariance model.

Common Mistake

• Wrong reasoning: “The inbreeding coefficient is \(F_i\), so the diagonal covariance is \(F_i\sigma_A^2\).” This keeps only the added IBD contribution and drops the noninbred baseline.
• Correction: The baseline variance is \(\sigma_A^2\), and IBD contributes an additional \(F_i\sigma_A^2\). Therefore the full diagonal is \(\sigma_A^2+F_i\sigma_A^2=(1+F_i)\sigma_A^2\).
• Another error: Treating the multiplier as a change in the individual's trait value. The multiplier changes modeled variance; it does not specify a directional change in that individual's additive genetic value.