Kinship Coefficients and Genetic Relationships

Why Genetic Relationships Need a Probability Scale

Labels such as “first-degree relative” or “cousin” summarize a pedigree, but they do not specify the probability that two alleles at a locus descend from the same ancestral allele. Kinship coefficients put that probability on a numerical scale. This makes relationships comparable and clarifies what a pedigree-based value does—and does not—say about genetic similarity at observed markers.

The key concept is identity by descent (IBD): two allele copies are IBD if they are copies of the same allele in a shared ancestor, passed down through generations. Two copies can have the same allelic state without being IBD. For example, two people may each carry allele A because A is common in their population, even if their copies came from different ancestors.

These are probabilities defined by a pedigree and its ancestry assumptions, not direct measures of observed genotype matching. The coefficient refers to a locus under the pedigree model; genome-wide interpretation concerns the expected sharing across loci, while realized sharing varies across the genome because inheritance is random.

Assumptions Behind Standard Pedigree Values

The familiar values below assume autosomal inheritance, correct recorded parentage, and Mendelian transmission: each parent passes one of its two allele copies at random. Unless stated otherwise, founders are assumed to be unrelated to one another and non-inbred. Thus, two founder alleles are not IBD merely because they happen to have the same allelic state. These assumptions define a reference pedigree calculation; they may not describe a population with shared ancestry or unrecorded relationships.

A useful way to calculate a coefficient is to trace the two sampled alleles back through their pedigree. Each meiosis along a lineage contributes a probability of \(1/2\) that the transmitted allele is the ancestral copy of interest. If the lineages meet at a non-inbred common ancestor, the chance they refer to the same one of that ancestor’s two copies is also \(1/2\).

The factor \(2^{-(m+n)}\) accounts for the transmissions along the two paths. The remaining factor \((1+F_A)/2\) is the probability that the two lineages correspond to the same ancestral allele copy: it is \(1/2\) when \(A\) is non-inbred, and larger when \(A\)'s own alleles may be IBD. This argument justifies the standard values without treating a relationship label as a probability by itself. More complicated pedigrees with overlapping paths require care to avoid counting the same inheritance event incorrectly.

Expected Values for Common Relationships

For a parent and child, the paths to the parent are zero and one meiosis. The kinship coefficient is \(2^{-(0+1+1)}=1/4\), so the relationship coefficient is \(1/2\). For a grandparent and grandchild, the path lengths are zero and two, giving kinship \(1/8\) and relationship \(1/4\).

A full-sibling pair has two shared parents. Each shared-parent path has one meiosis from each sibling, so each contributes \(2^{-(1+1+1)}=1/8\) to kinship. Adding the contributions gives kinship \(1/4\), and relationship \(1/2\). Half-siblings share just one parent, so their kinship is \(1/8\) and their relationship is \(1/4\). These matching values for parent–offspring and full siblings do not imply the same pedigree structure; they happen to have the same expected coefficients under the stated assumptions.

Inbreeding and Self-Kinship

The relationship between two distinct individuals is not the only useful coefficient. For an individual \(i\), the inbreeding coefficient \(F_i\) is the probability that the two allele copies carried by \(i\) at a locus are IBD. Under the standard kinship convention, \(\phi_{ii}\) is the probability of IBD for two allele copies sampled independently from \(i\). With probability \(1/2\), the draws select the same physical copy; with probability \(1/2\), they select the two different copies, which are IBD with probability \(F_i\). Therefore \(\phi_{ii}=(1+F_i)/2\), and the diagonal relationship value \(2\phi_{ii}\) is \(1+F_i\).

Interpretation, Diagnostics, and Common Pitfalls

The relationship coefficient \(r=2\phi\) is often described as the expected proportion of alleles shared IBD for a pair of non-inbred individuals. It is an expectation under the pedigree and inheritance model, not a guarantee about the realized proportion in one pair. Recombination and random transmission cause segment sharing to vary, especially for more distant relatives. The kinship coefficient is the probability for one randomly selected allele from each person; it is half the conventional relationship coefficient for distinct individuals.

A pedigree calculation should be checked against its assumptions and its paths. Confirm that the pair’s shared ancestors have all been included, that founder relationships have not been silently assumed away, and that a path has not been counted twice. If a population has substantial shared ancestry, treating all founders as unrelated can understate pedigree kinship. Likewise, observed matching alleles alone do not establish IBD: common alleles can match by state without a recent shared ancestral copy. The introductory material on population stratification explains why ancestry-related allele-frequency differences can also shape genomic similarity.

A common mistake is to say, “Full siblings have kinship \(1/2\), because they share half their genes.” That statement uses the relationship coefficient while labeling it kinship. The correction is to state the convention: for full siblings, \(\phi=1/4\) and \(r=2\phi=1/2\). Another mistake is to interpret \(r=1/2\) as an exact fraction of the genome shared IBD by every parent–child pair. It is the expected relationship coefficient under Mendelian inheritance; realized sharing across genomic segments is not identical for every pair.

Pedigree coefficients and genomic estimates answer related but distinct questions. A pedigree coefficient is calculated from recorded ancestry and assumptions about founders. A genomic estimate uses observed markers to summarize sharing, and can be affected by allele frequencies, population structure, marker quality, and sampling variation. Neither number should be called a kinship coefficient without making clear which definition and scale are being used.