Kinship Convention and Coefficient Errors

Why the Convention Matters

Pedigree calculations use two closely related quantities that are easy to confuse: the kinship coefficient and the additive relationship coefficient. They describe the same underlying relatedness on different scales. A factor-of-two error can therefore leave a calculation looking plausible while changing every reported coefficient or genetic covariance.

The distinction matters especially when interpreting a pedigree matrix. Its off-diagonal entries are often additive relationship coefficients, while kinship is the probability that two sampled alleles are identical by descent. The diagonal entries also have a specific interpretation: they account for an individual's inbreeding and are not simply equal to one.

IBD means that the alleles descend from the same ancestral allele copy; it does not merely mean that they have the same observed state. Kinship is a probability for a pair of sampled alleles, not the probability that two people share at least one allele. The additive relationship coefficient is a rescaled measure of relatedness, so its values need not be probabilities.

For a pair of distinct, non-inbred individuals, familiar values are often stated in either convention. For example, parent–offspring kinship is \(1/4\), whereas their additive relationship coefficient is \(1/2\). If a source says “relationship coefficient” without specifying its convention, check its definition rather than assuming which scale it uses.

The Self-Kinship Check

The diagonal is an especially useful place to detect a convention error. If the same individual is used for both allele draws, the two draws can select the same physical allele copy, which occurs with probability \(1/2\). If they select different homologous copies, those copies are IBD with probability \(F_i\). Therefore self-kinship depends on inbreeding.

For a non-inbred individual, \(F_i=0\), so \(\phi_{ii}=1/2\) and \(A_{ii}=1\). For an inbred individual, the kinship diagonal is greater than \(1/2\), and the additive relationship diagonal is greater than 1. This is not a probability exceeding one: \(A_{ii}\) is a scaled coefficient, not an IBD probability.

Familiar Relative Pairs on Both Scales

For distinct relatives, convert a kinship value to an additive relationship value by multiplying by two; convert in the other direction by dividing by two. The familiar values below assume the usual simple pedigree conditions: founders are unrelated and non-inbred, and no additional loops or relationships alter the stated pair's ancestry. They are expected pedigree coefficients, not statements about the realized alleles at every locus.

These values provide quick consistency checks. Under the stated conditions, a parent–offspring or full-sibling entry of \(1/4\) is on the kinship scale; the corresponding additive relationship entry is \(1/2\). A first-cousin entry of \(1/8\) is on the additive relationship scale, while kinship is \(1/16\). If a calculation gives one of these standard values, label its scale explicitly.

Why the Factor of Two Appears

The conversion is a definition of scale, not an extra pedigree-generation rule. Kinship records the IBD probability for one sampled allele from each individual. The additive relationship coefficient is defined to be twice that probability. This scaling gives the familiar additive coefficient \(1\) for an outbred individual's diagonal and \(1/2\) for an ordinary parent–offspring pair.

The diagonal identity gives a direct justification for the same scaling. A self-kinship value includes a \(1/2\) chance that the two draws select the same allele copy, as well as the contribution from inbreeding when different copies are drawn. Doubling that probability yields \(A_{ii}=1+F_i\), the standard additive relationship diagonal. In particular, an outbred diagonal is \(1\), while kinship on that diagonal is \(1/2\).

The scaling also explains the standard covariance convention. If \(\sigma_A^2\) denotes the additive genetic variance on the reference scale, the pedigree-based additive genetic covariance between individuals \(i\) and \(j\) is \(\operatorname{Cov}(G_i,G_j)=A_{ij}\sigma_A^2=2\phi_{ij}\sigma_A^2\). For an individual's variance, this gives \(\operatorname{Var}(G_i)=(1+F_i)\sigma_A^2\). These expressions presume the standard additive genetic model and that the same variance and relationship conventions are being used together.

A Practical Convention Check

When reading or constructing a pedigree matrix, verify the convention before interpreting any entry. A matrix called a “relationship matrix” commonly uses \(A\), whereas a kinship matrix uses \(\phi\); labels alone are not enough if the source is unclear. Compare a known pair or the diagonal with its expected value, and state the scale beside reported numbers.

Common Mistakes

• Calling a kinship value the additive relationship coefficient. A parent–offspring value of \(1/4\) is kinship under the standard assumptions; the additive relationship coefficient is \(1/2\). State the symbol and convention.
• Using \(1\) for every diagonal. The additive relationship diagonal is \(1+F_i\), so it equals \(1\) only when the individual is not inbred. The kinship diagonal is \((1+F_i)/2\), not \(1\) for an outbred individual.
• Treating an additive relationship coefficient as an IBD probability. \(A_{ii}\) may exceed \(1\) when \(F_i>0\), because it is a rescaled coefficient rather than a probability.
• Interpreting kinship as the chance of sharing any allele. Kinship is the IBD probability for a pair of sampled alleles. It is not the probability that two individuals share at least one allele.
• Doubling twice or not at all. Keep track of which scale a formula expects. If a covariance formula uses \(A_{ij}\), convert a given \(\phi_{ij}\) first; if the formula is written with \(2\phi_{ij}\), do not multiply by two again.