Inbreeding Coefficients as Parental Kinship

Why Parental Relatedness Determines Offspring Inbreeding

An offspring receives one allele at a locus from each parent. The two inherited alleles may be copies of the same ancestral allele, passed down through separate branches of the pedigree. When that happens, the offspring’s two alleles are identical by descent (IBD). The probability of this event is determined by how likely it is that a randomly transmitted allele from one parent and a randomly transmitted allele from the other parent share an ancestral origin.

The key result is concise: an individual's inbreeding coefficient equals the kinship coefficient between that individual's parents. The equality is useful because kinship describes a pair of individuals, while inbreeding describes the two alleles within one individual. The result connects these two quantities directly.

IBD is about shared ancestral origin, not merely matching allele states. For example, two alleles may both be labelled A but have come from different ancestral copies; they match in state without being IBD. Conversely, if two alleles are IBD, they descend from the same ancestral allele copy, assuming no mutation in the pedigree model.

The standard pedigree calculation assumes diploid inheritance at the locus, one allele transmitted from each parent, and Mendelian transmission: each of a parent's two alleles has probability \(1/2\) of being transmitted. The transmissions from the two parents are independent conditional on the pedigree. We also use the specified founder assumptions, such as treating unrelated founders as non-IBD. The result below does not require the parents to be noninbred; their kinship coefficient already accounts for the pedigree-defined IBD probabilities between their alleles.

Deriving the Equality

Let \(s\) and \(d\) denote the two parents of individual \(i\). Label the two alleles of \(s\) at the locus \(A_{s1}\) and \(A_{s2}\), and those of \(d\) \(A_{d1}\) and \(A_{d2}\). The offspring receives one of the two alleles from \(s\) and one of the two from \(d\). There are four equally likely pairs of parental allele choices.

For each possible pair of choices, ask whether the selected allele from \(s\) and the selected allele from \(d\) are IBD. Averaging that probability over the four equally likely choices gives the chance that the offspring's two inherited alleles are IBD. This average is also exactly the definition of the parents' kinship coefficient: it is the IBD probability for one randomly chosen allele from each parent.

The second line partitions the event according to which allele each parent transmits. Under Mendelian transmission, each specific pair of choices has probability \((1/2)(1/2)=1/4\). The final line follows because a random allele from \(s\) and a random allele from \(d\) also select each of these four pairs with probability \(1/4\). Thus the two probabilities average over exactly the same possibilities.

This is an equality of probabilities, not a claim that the offspring's alleles are already known to be IBD. At a particular locus, the two alleles either are or are not IBD in the realized pedigree transmission. The coefficient summarizes the probability of the IBD event under the pedigree model.

Worked Applications

The calculations below use kinship coefficients established by the relevant pedigree relationships. In each case, the coefficient for the offspring is obtained by identifying the parents and applying \(F_i=\phi_{sd}\). The arithmetic does not require a separate calculation of the offspring's two inherited allele origins.

Interpreting the Coefficients Carefully

The equality helps keep the two coefficients conceptually separate. Kinship is defined for a pair of individuals by sampling one allele from each. Inbreeding is defined within one individual by considering that individual's two alleles. For a child, the pair of alleles is formed by one transmission from each parent, so the two definitions refer to the same pair of parental allele copies.

It is also important to distinguish a kinship coefficient from a relatedness coefficient sometimes reported on a different scale. Here, \(\phi_{sd}\) is the probability that one randomly sampled allele from each parent is IBD. The theorem uses this kinship convention directly:

Do not multiply the parental kinship coefficient by two when finding \(F_i\). For example, if half-sibling parents have \(\phi_{sd}=1/8\), their child's inbreeding coefficient is \(1/8\), not \(1/4\). The scale matters: the answer is a probability of IBD for the offspring's two alleles.

Common Mistakes and Why the Result Matters

A frequent error is to calculate the offspring's inbreeding coefficient using the offspring's own kinship with a parent. That is the wrong pair: the offspring's two alleles came from the two parents, so the relevant pair is the parents themselves. Another error is to interpret \(F_i\) as the probability of homozygosity. IBD alleles necessarily have the same ancestral copy, but matching allele states can also arise by chance from distinct ancestral copies. Therefore, IBD and homozygosity are not interchangeable concepts.

The relationship is useful because it turns an within-individual probability into a pairwise pedigree calculation. Once the kinship of a parental pair is known, the offspring's inbreeding coefficient follows immediately. In a pedigree, this provides a clear bridge between questions about relatedness among parents and questions about shared ancestral origin within their offspring.

• Wrong: “The offspring's inbreeding coefficient is twice the parents' kinship because the offspring has two alleles.”
• Correction: Each of the two parental allele choices is already included in the kinship definition, which samples one allele from each parent. Therefore \(F_i=\phi_{sd}\), with no extra factor of two.
• Wrong: “An inbreeding coefficient of zero proves the parents share no ancestry anywhere.”
• Correction: It means the IBD probability is zero under the specified pedigree and founder assumptions. Unrecorded ancestry is outside that model.