Why These Kinship Coefficients Matter
A pedigree records who is related to whom, but a statistical analysis often needs a numerical summary of that relationship. The kinship coefficient provides one: it is the probability that one allele sampled at random from each of two individuals is identical by descent (IBD). IBD means the alleles are copies of the same ancestral allele, passed through the pedigree. This tutorial calculates kinship for three familiar pairs that are not parent–offspring or full siblings: half-siblings, grandparent and grandchild, and aunt or uncle with niece or nephew.
For all three calculations, begin with a simple pedigree: founders are unrelated to one another and are not inbred, and there are no additional shared ancestors or pedigree connections beyond the ones stated. These assumptions make the transmission paths easy to isolate. If a pair has additional common ancestry, the simple values below need not apply.
The Recursive Rule for a Nonfounder
A nonfounder is an individual whose parents are represented in the pedigree. At an autosomal locus, a randomly sampled allele from that individual has probability \(1/2\) of coming from either parent. Conditioning on which parent supplied the allele gives a simple recursion. If \(i\) is the child of parents \(s\) and \(d\), then, for another individual \(j\), the kinship of \(i\) and \(j\) is the average of the kinships of each of \(i\)'s parents with \(j\).
This rule follows because an allele sampled from \(i\) came from \(s\) with probability \(1/2\) and from \(d\) with probability \(1/2\). In each case, the other sampled allele is IBD to it with the corresponding parental kinship probability. The rule is most useful when one parent is related to \(j\) and the other is an unrelated founder: the unrelated parent's contribution is then zero.
We will also use the parent–offspring value \(\phi=1/4\). A sampled allele from the child comes from that parent with probability \(1/2\); conditional on that, it is a copy of either of the parent's two alleles with equal probability, so it matches the allele sampled from the parent with probability \(1/2\). Thus \( (1/2)(1/2)=1/4\). Under unrelated, non-inbred founders, two individuals with no shared ancestor in the pedigree have kinship zero.
Half-Siblings
Half-siblings share one parent and have different other parents. Under the assumptions here, the shared parent and both other parents are unrelated, non-inbred founders. The two half-siblings can be connected through the shared parent's alleles, but not through their other parents.
Worked Example: Half-Siblings With One Shared Parent
Suppose hypothetical individuals H1 and H2 share founder P. H1's other parent is founder U, and H2's other parent is founder V. Assume P, U, and V are mutually unrelated and non-inbred. Calculate \(\phi_{\mathrm{H1,H2}}\).
Apply the recursive rule to H1, whose parents are P and U:
P is a parent of H2, so \(\phi_{\mathrm{P,H2}}=1/4\). U is an unrelated founder, so \(\phi_{\mathrm{U,H2}}=0\). Substituting and calculating gives
Thus, a randomly sampled allele from each half-sibling has probability \(0.125\) of being IBD. This is a probability of shared ancestral origin, not a claim that the sampled alleles must have the same observed version.
The transmission interpretation gives the same result. Each sampled allele has probability \(1/2\) of coming from the shared parent, so the chance both come from P is \(1/2 \times 1/2=1/4\). Given that both come from P, they have probability \(1/2\) of tracing to the same one of P's two alleles. The product is \(1/4 \times 1/2=1/8\). If either sampled allele came from an unrelated other parent, it would not be IBD to the other individual's allele under these assumptions.
Grandparent and Grandchild
A grandchild's allele must pass through the intervening parent to descend from a particular grandparent. The recursive rule accounts for both transmissions without treating the grandparent and grandchild as a parent–offspring pair.
Worked Example: A Grandparent and Grandchild
Consider hypothetical grandparent G, G's child M, and M's child C. Let the other parent of M and the other parent of C be unrelated founders, neither related to G. Calculate \(\phi_{\mathrm{G,C}}\).
Apply the recursion to C, whose parents are M and the unrelated founder T:
G and M are parent and child, so \(\phi_{\mathrm{G,M}}=1/4\). G and T have no shared ancestor in this pedigree, so \(\phi_{\mathrm{G,T}}=0\). Therefore,
Under these assumptions, the kinship coefficient for a grandparent–grandchild pair is \(1/8\). The result reflects two generational transmissions: the grandchild's sampled allele must come through the intervening parent, and its IBD relationship to the allele sampled from G is then averaged over the intervening parent's inheritance.
Aunt or Uncle and Niece or Nephew
An avuncular pair consists of an aunt or uncle and a niece or nephew. In the standard pedigree considered here, the aunt or uncle A and the niece or nephew's parent P are full siblings, and the niece or nephew N is the child of P and an unrelated founder. The aunt or uncle and the child's other parent are unrelated.
The relevant full-sibling kinship is \(1/4\). To see why, the two siblings each contribute a randomly sampled allele. Each allele is equally likely to have come from either of their two shared parents. The alleles come from the same parent with probability \(1/2\); conditional on that, they trace to the same allele in that parent with probability \(1/2\). Since the parents are unrelated, alleles from different parents do not add another IBD route. The resulting kinship is \((1/2)(1/2)=1/4\).
Worked Example: An Aunt and Her Niece
Suppose hypothetical A and P are full siblings, and P is a parent of N. Let N's other parent be unrelated to A and P. Assume the two parents of A and P are unrelated, non-inbred founders, with no additional shared ancestry. Calculate \(\phi_{\mathrm{A,N}}\).
Apply the recursive rule to N, whose parents are P and the unrelated founder T:
A and P are full siblings, so \(\phi_{\mathrm{A,P}}=1/4\), as shown above. T is unrelated to A, so \(\phi_{\mathrm{A,T}}=0\). Substitution gives
The aunt–niece kinship is therefore \(1/8\) in this pedigree. The same calculation applies to an uncle and nephew, or to other aunt-or-uncle–niece-or-nephew combinations, provided the stated pedigree and founder assumptions hold.
Comparing the Three Results
The three relationships have the same kinship coefficient under the simple pedigrees considered here, even though the connections differ. In each case, the recursive rule averages a nonzero kinship contribution with a zero contribution from an unrelated parent, or traces shared inheritance through the appropriate common parent.
| Pair | Pedigree condition | Kinship coefficient |
|---|---|---|
| Half-siblings | One shared parent; other parents unrelated | \(1/8=0.125\) |
| Grandparent–grandchild | One intervening parent; no additional shared ancestry | \(1/8=0.125\) |
| Aunt or uncle–niece or nephew | Aunt or uncle is a full sibling of the child's parent | \(1/8=0.125\) |
Equal coefficients do not mean that the pedigrees are interchangeable. They summarize the IBD probability for a pair under specified assumptions; the underlying routes of transmission differ. The coefficient is useful precisely because it compresses that pedigree information into a pairwise probability, while the pedigree remains essential for checking whether the assumptions are appropriate.
Assumptions and a Common Pitfall
The values \(1/8\) are not unconditional labels that apply whenever two people are described as half-siblings, grandparent and grandchild, or avuncular relatives. They rely on the simple pedigree specified above. If founders are related, an individual is inbred, or the pair shares another ancestor, additional IBD routes can change the calculation. Before using the values, identify the parents in the pedigree and check which founders are assumed unrelated.
A common mistake is to report \(1/4\) for half-siblings, grandparent–grandchild pairs, or avuncular pairs because \(1/4\) is familiar from parent–offspring and full-sibling kinship. That substitutes a different relationship's coefficient. For each pair, follow the inheritance through the actual parents: for example, half-siblings receive only one shared parent contribution, and the recursion gives \((1/2)(1/4)+(1/2)(0)=1/8\). The result is a kinship coefficient of \(1/8\), not \(1/4\).
Check Your Understanding
For each question, state the relevant pedigree assumption as well as the calculation or interpretation.
- Two hypothetical half-siblings share founder R; their other parents are unrelated founders. Use the recursive rule to calculate their kinship coefficient.
- A grandchild's parents are the child of grandparent G and an unrelated founder. What kinship value between G and the grandchild follows under the assumptions in this tutorial?
- An aunt is a full sibling of a child's parent, and the child's other parent is unrelated to her. Show the recursion that gives the aunt–child kinship coefficient.
- Why does the probability that two sampled alleles have the same observed variant not, by itself, establish that they are IBD?
- Name one pedigree feature that could make the value \(1/8\) inappropriate for one of these pairs.