Kinship Coefficients as Pairwise IBD Probabilities

From IBD to a Pairwise Probability

Identity by descent (IBD) concerns the history of allele copies: two copies are IBD when they descend from the same ancestral copy. A pedigree describes possible paths of descent, but for a pair of individuals there can be uncertainty about which copies share that ancestry. The kinship coefficient summarizes that uncertainty as a probability for a carefully specified pair of allele draws.

At an autosomal locus in a diploid individual, there are two homologous gene copies, one on each homologous chromosome. To define kinship for two individuals, select one of the two copies from each individual at random, giving each copy probability \(1/2\). The probability that the selected copies are IBD is the kinship coefficient. For a pair of distinct individuals, the selections are made independently. The same sampling convention also defines self-kinship, but in that case the two selections are from one individual and may select the same physical copy.

This definition makes the sampling operation part of the meaning of the coefficient. For distinct individuals, there are four equally likely pairings of their homologous copies. Kinship averages the IBD probabilities over those four pairings. For self-kinship, the two independent draws can pick the same copy or different copies; those possibilities must both be included.

A kinship coefficient is a probability, so it lies between 0 and 1. It is symmetric: \(\phi_{ij}=\phi_{ji}\), because swapping the order of the two independently sampled copies does not change whether they are IBD. Kinship is usually determined from the pedigree and its specified assumptions about descent, rather than read directly from a pair of observed genotypes.

Writing the Probability for Distinct Individuals

For distinct individuals \(i\) and \(j\), label their two homologous copies \(i_1,i_2\) and \(j_1,j_2\). Let \(I_{ab}\) be the event that copies \(i_a\) and \(j_b\) are IBD. Each of the four pairings has probability \(1/4\) under the random-copy sampling rule. Therefore, the kinship coefficient is the average of the four IBD probabilities:

The probabilities in this expression are probabilities over the pedigree’s possible descent histories, including Mendelian transmission where relevant. A particular pairing may have a higher IBD probability than another, but the random-copy definition gives all four pairings equal weight. This is why kinship is not simply the probability that two people share at least one IBD copy somewhere in their genomes: it concerns one copy sampled from each person.

Self-Kinship

When both labels refer to the same individual, the sampling rule still makes two independent selections, each uniform over that person’s two gene copies. There is a probability \(1/2\) that both selections pick the same copy. In that event, the selected copy is IBD with itself. There is also a probability \(1/2\) that the selections pick the two different homologous copies. The probability that those two homologous copies are IBD is conventionally denoted by the individual’s inbreeding coefficient \(F_i\).

The formula follows by conditioning on whether the two draws select the same copy or different copies. The same-copy event has probability \(1/2\) and is certainly IBD; the different-copy event has probability \(1/2\) and is IBD with probability \(F_i\). Adding the contributions gives:

This derivation depends on independent sampling with replacement: after the first draw, the second draw can select that same physical copy. If the two homologs were instead required to be different, the probability of IBD would be \(F_i\), not \(\phi_{ii}\). That is a different sampling question.

If \(F_i=0\), the individual’s two homologous copies are not IBD under the specified model, but self-kinship is still \(0.50\): the two independent draws can select the same copy. If \(F_i=1\), the homologs are IBD, and either draw pattern produces IBD, so self-kinship is \(1\). These boundary cases help check the sampling interpretation.

IBD Is Not the Same as Matching Allele States

IBD is a statement about descent, not merely about whether two copies carry the same allele state. Two copies can have the same state without descending from the same ancestral copy; such a match is not, by itself, evidence that the copies are IBD. Conversely, the kinship coefficient is defined through IBD events, even when genotype data are unavailable or allele states do not uniquely reveal ancestry.

In practice, translating observed genetic data into information about IBD requires an appropriate model for allele states and descent. The definition of kinship remains the probability of IBD for random copy draws; it is not redefined as the observed fraction of matching alleles.

How to Read and Use a Kinship Coefficient

A value such as \(\phi_{ij}=0.125\) says that, under the specified pedigree and founder assumptions, the probability of IBD is 12.5% for one copy sampled at random from each of \(i\) and \(j\). The coefficient has no physical units. It is not a count of shared copies, a probability that the people themselves are related, or a statement that their complete genotypes match with that probability.

A kinship value is conditional on the pedigree model used to define descent. Assumptions about founder relationships matter because they determine whether copies entering the pedigree can already be IBD. Mendelian transmission determines how ancestral copies can be passed through the pedigree. The random-copy definition then specifies exactly which probability is summarized. A numerical kinship value is therefore meaningful only alongside a clear locus, sampling convention, and pedigree model.

Common Mistakes

• Counting shared copies instead of defining a random draw. Kinship concerns one randomly selected copy from each individual. The four equally likely pairings for distinct diploid individuals are averaged; the coefficient is not simply a count of shared-looking alleles.
• Equating allele-state agreement with IBD. A match in state does not establish shared descent. IBD is about a common ancestral copy, so the two events must be kept distinct.
• Using \(F_i\) as self-kinship. \(F_i\) is the IBD probability for the two different homologous copies within individual \(i\). Under independent sampling, self-kinship also includes the chance of selecting the same copy, giving \(\phi_{ii}=(1+F_i)/2\).
• Sampling the two copies without replacement in the self case. That procedure forces different homologs and gives probability \(F_i\). The conventional self-kinship definition uses independent draws, so the same copy can be selected twice.