Recursive Kinship Calculation From Parents

Why Calculate Kinship Recursively?

In a pedigree, two individuals may be related through several generations of parents and offspring. A direct calculation can quickly become difficult if it requires listing every possible path by which alleles were transmitted. A simpler strategy is to calculate kinship in pedigree order: first establish kinship among founders, then use the known kinship values of a person's parents to calculate that person's kinship with other individuals.

The key is to condition on the parental origin of a randomly sampled allele. For a diploid individual, the sampled allele came from one parent or the other, each with probability one half under Mendelian transmission. This gives a recursion that works for any pair of distinct individuals when the parents of one member of the pair are known.

Kinship is a probability, so its value lies between 0 and 1. It concerns IBD: the alleles are copies of the same ancestral allele. Merely having the same allele state does not establish that two alleles are IBD. The recursion below calculates the probability of IBD using the pedigree's parentage and transmission structure.

Deriving the Parental Recursion

Let \(j\) be an individual with known parents \(s\) and \(d\), and let \(i\) be a distinct individual. Sample one allele from \(i\) and one from \(j\). The allele sampled from \(j\) was transmitted by \(s\) with probability \(1/2\), or by \(d\) with probability \(1/2\). Conditional on its parental origin, the transmitted allele is a randomly selected allele from that parent. Therefore, the probability that it is IBD with the sampled allele from \(i\) is \(\phi(i,s)\) or \(\phi(i,d)\), respectively.

The law of total probability combines those two mutually exclusive transmission events:

The argument uses only the parental origin of the sampled allele and equal Mendelian transmission from the two parents. It does not require the parents to be unrelated. If a pedigree includes related founders, their kinship values must be used rather than set to zero. If parentage is unknown, the recursion cannot be applied to that individual until the missing parental information is supplied or modeled.

There is an important boundary case. The displayed recursion is for distinct individuals; it must not be applied by setting \(i=j\). Self-kinship is a diagonal value defined by sampling two alleles within the same individual. For a non-inbred founder, the two homologous alleles are not IBD, so its self-kinship is \(1/2\). More generally, a diagonal value must be supplied using the appropriate self-kinship rule. In the examples below, all founders are assumed non-inbred.

Setting the Starting Values and Choosing an Order

A recursion needs starting values. A common pedigree convention is to treat the listed founders as unrelated and non-inbred. For distinct founders \(u\) and \(v\), this gives \(\phi(u,v)=0\); for each such founder \(u\), \(\phi(u,u)=1/2\). These are assumptions about the pedigree boundary, not consequences of the recursion. If the founders are known to be related, their nonzero kinship values replace the unrelated-founder values.

Calculate values in an order that ensures the parents' kinship with the individual of interest are already known. For each non-founder, apply the recursion to the pair of interest and that non-founder, using its two parents. If one parent is the queried individual, the calculation may require that individual's diagonal self-kinship. Symmetry, \(\phi(i,j)=\phi(j,i)\), lets you choose which member of a pair to expand when both have known parents.

Worked Calculations

Assumptions, Interpretation, and a Common Pitfall

The recursion is a standard consequence of Mendelian transmission, but its use still depends on the pedigree model. The individuals must be diploid at the autosomal locus under consideration; the listed parents must be the genetic parents; and, conditional on a parent transmitting an allele, each of that parent's two alleles must have probability \(1/2\) of transmission. The calculation also treats transmission as having no mutation that changes the allele's genealogical identity. These assumptions define the transmission model behind the IBD probability.

The kinship coefficient is an expected pedigree-based probability, not a statement that the two observed genotypes must match. Two alleles can have the same state without being IBD, and IBD alleles need not be distinguishable from genotype data alone. The recursion calculates kinship from parentage and starting kinship values; it does not itself estimate kinship from observed genotypes.

Common mistake: A learner sees that \(j\) has parents \(s,d\) and writes \(\phi(i,j)=\phi(i,s)+\phi(i,d)\), reasoning that either parent might transmit an IBD allele. This adds the two conditional probabilities without weighting them by the probabilities of the two transmission events, and can even produce a value greater than 1. The correction is to use the law of total probability: each parent transmits the sampled allele with probability \(1/2\), so the two parental kinships are averaged. Another related error is to use this recursion for \(\phi(i,i)\); diagonal self-kinship is a separate boundary value, not a distinct-individual recursion.

Recursive calculation is useful because each result can be checked locally: identify the two parents, retrieve the two relevant kinship values, and average. This structure also makes assumptions visible. A changed founder relationship or a corrected parentage can affect values down the pedigree, while the recursion itself remains the same.