Cousin Kinship Through Shared-Ancestor Paths

Why Cousin Kinship Is a Path Calculation

Cousins may inherit alleles from the same earlier ancestor, but their relationship is not summarized by counting how many ancestors they have in common. The relevant question is whether an allele sampled from one cousin and an allele sampled from the other can trace back to the same ancestral allele. A pedigree makes those routes visible: each parent-to-child transmission is one step, and each valid shared-ancestor route contributes a probability of identity by descent.

For ordinary first cousins, the two parents are siblings. The cousins therefore share two grandparents. Each shared grandparent can provide a route by which the cousins’ sampled alleles are identical by descent. The calculation adds the contributions of those routes, subject to assumptions that make the routes distinct and the shared ancestors noninbred.

Definition: The kinship coefficient between two individuals is the probability that one allele sampled at random from one individual and one allele sampled at random from the other, at the same locus, are identical by descent (IBD). IBD means that the two alleles descend from the same ancestral allele. It does not merely mean that their observed states are the same.

Kinship is often written $\phi$. It is a probability for a pair of sampled alleles, not the probability that the two individuals share at least one IBD allele across both copies of their genotype. Under the usual convention, the relatedness coefficient is $2\phi$ for two distinct, non-inbred individuals. This tutorial calculates kinship coefficients.

The Contribution From One Shared Ancestor

Consider two descendants, $X$ and $Y$, who have a particular common ancestor $A$. Suppose the path from $X$ to $A$ has $m$ parent-to-child transmissions, and the path from $Y$ to $A$ has $n$ transmissions. For example, a grandchild is two transmissions from a grandparent, so a path from a grandchild to that grandparent has length 2.

For the simple path calculation used here, assume Mendelian transmission at a single autosomal locus, no mutation, a non-inbred common ancestor, and distinct routes through the ancestor. Assume also that unrelated founders do not introduce other shared ancestral alleles. A transmission follows one of the transmitting parent's two alleles, each with probability $1/2$. Once both sampled alleles have traced back to $A$, they must also trace to the same one of $A$'s two alleles; for a non-inbred $A$, that match has probability $1/2$.

Path contribution: Under these assumptions, the contribution to kinship through one common ancestor $A$ is $$ \left(\frac{1}{2}\right)^{m+n+1}. $$ The $m+n$ transmissions give the first factors of $1/2$; the final factor of $1/2$ is the chance that the two lineages select the same ancestral allele in $A$.

Deriving the Path Contribution

The path expression follows by multiplying the probabilities of the required transmission events. This is a probability for the specified route: both sampled alleles must trace along their respective paths to $A$, and the alleles reached in $A$ must match.

P(\text{both lineages reach }A\text{ and match}) = \underbrace{\left(\frac{1}{2}\right)^m}_{X\text{ path}} \underbrace{\left(\frac{1}{2}\right)^n}_{Y\text{ path}} \underbrace{\left(\frac{1}{2}\right)}_{\text{same allele in }A} = \left(\frac{1}{2}\right)^{m+n+1}.

This calculation relies on the stated pedigree assumptions: at each transmission, either parental allele is passed with probability $1/2$, and a non-inbred common ancestor has two distinct alleles at the locus for the final matching step. When several distinct, valid routes give IBD, their contributions can be added. That addition is the key to ordinary first-cousin kinship.

Worked Example: Ordinary First Cousins

Suppose hypothetical individuals Cora and Davi have parents who are full siblings. Those parents share two grandparents, Ada and Ben. Assume Ada and Ben are non-inbred, their founders are unrelated, and there are no other shared ancestors. Find the kinship coefficient between Cora and Davi.

For the route through Ada, Cora is two transmissions from Ada, and Davi is also two transmissions from Ada. Thus $m=2$ and $n=2$, giving

\left(\frac{1}{2}\right)^{2+2+1} = \left(\frac{1}{2}\right)^5 = \frac{1}{32} = 0.03125.

The route through Ben has the same lengths and contributes another $1/32$. Since these are two distinct shared grandparents and the assumptions exclude other shared routes, add the contributions:

\phi_{\text{Cora,Davi}} = \frac{1}{32}+\frac{1}{32} = \frac{2}{32} = \frac{1}{16} = 0.0625.

The kinship coefficient is $0.0625$: at the specified locus, a randomly sampled allele from Cora and one from Davi have a 6.25% probability of being IBD. This is not a statement that the cousins have a 6.25% probability of sharing any allele by state.

How Additional Shared Ancestors Change the Calculation

A single valid route through a shared ancestor contributes according to the lengths of its two paths. If an otherwise separate route through another shared ancestor is present, it supplies another contribution. In an ordinary first-cousin pedigree, the shared grandparents give two such routes. Other pedigree structures may create more routes, but the sum is straightforward only when the routes being added are distinct and the assumptions behind the path formula hold.

Path length matters. A more distant common ancestor contributes less through any one route because more transmissions must occur. For instance, if each relative is three transmissions from a shared ancestor, the route contributes $(1/2)^{3+3+1}=1/128$, rather than $1/32$ for two grandchild-to-grandparent paths. Having more shared ancestors can offset some of that reduction, but the number and lengths of the valid routes both matter.

Worked Example: First Cousins With One Shared Grandparent

Imagine hypothetical individuals Elise and Farid whose parents are half-siblings: they share one parent, but not both. Assume the shared parent, Morgan, is non-inbred, and assume there are no other shared ancestors. Elise and Farid are each two transmissions from Morgan. Calculate their kinship.

There is one shared-grandparent route, with $m=2$ and $n=2$. Its contribution is

\phi_{\text{Elise,Farid}} = \left(\frac{1}{2}\right)^{2+2+1} = \frac{1}{32} = 0.03125.

The half-sibling parents share one parent rather than two, so there is one shared-grandparent contribution rather than the two contributions for ordinary first cousins. Under the stated assumptions, the kinship coefficient is $0.03125$, half the ordinary first-cousin value.

Worked Example: Double First Cousins

Suppose hypothetical individuals Hana and Ivo have parents who are full siblings in two separate pairs: Hana’s mother and Ivo’s mother are siblings, and Hana’s father and Ivo’s father are also siblings. Assume these two sibling pairs descend from four distinct, non-inbred grandparents, with no additional shared ancestry. This is a double-first-cousin pedigree. Find the kinship coefficient.

For each of the four shared grandparents, Hana and Ivo are each two transmissions away. Therefore, each route contributes

\left(\frac{1}{2}\right)^{2+2+1} = \frac{1}{32} = 0.03125.

There are four distinct routes under the assumptions, so

\phi_{\text{Hana,Ivo}} = 4\left(\frac{1}{32}\right) = \frac{4}{32} = \frac{1}{8} = 0.125.

The additional shared grandparent pair raises kinship from the ordinary first-cousin value $1/16$ to $1/8$. The coefficient is larger because the pedigree supplies more distinct routes for the sampled alleles to trace to the same ancestral allele.

Adding Contributions Without Confusing the Quantities

A path contribution is the probability associated with one specified route. When routes are distinct in the simple pedigree setting, kinship is the sum of their contributions. A useful way to organize the calculation is to list each common ancestor, record the number of transmissions from each relative, compute the route contribution, and then sum. This makes it easier to see whether a grandparent has been omitted or counted twice.

Identify the common ancestors.
Use the pedigree to name the ancestors through whom the pair may share ancestral alleles.

Count transmissions on each side.
For each ancestor, count parent-to-child steps from the first relative and from the second.

Calculate each route's contribution.
For path lengths $m$ and $n$, calculate $(1/2)^{m+n+1}$, using the stated non-inbreeding and pedigree assumptions.

Sum valid distinct contributions.
Add the route contributions to obtain kinship, and state the result as an IBD probability for a pair of sampled alleles.

Kinship is not the same as the probability of sharing at least one IBD allele in a two-allele genotype comparison. Nor is it the probability of matching allele states: two alleles can have the same state without being IBD. Keeping the definition tied to one sampled allele from each person prevents these quantities from being confused.

Common Mistake: Counting Generations or Ancestors Incorrectly

Wrong reasoning: “First cousins share two grandparents, so their kinship is $2(1/2)^4=1/8$.” This omits the final $1/2$ probability that the two lineages reach the same allele within a non-inbred common ancestor.
Correction: Each grandparent route has four transmissions in total and one additional matching step, so its contribution is $(1/2)^5=1/32$. Two routes give $1/16$.
Wrong reasoning: “The cousins have a 6.25% chance of sharing an allele.” That wording does not identify whether the claim concerns IBD, matching allele states, or at least one match across both genotype copies.
Correction: State precisely that their kinship coefficient is $1/16$: the probability that one allele sampled from each cousin at the locus is IBD, under the stated assumptions.

The simple addition rule should not be applied mechanically when the same ancestral allele can be reached through overlapping routes or when the shared ancestors are themselves related or inbred. In those situations, a pedigree requires more careful treatment than the distinct-route examples here; merely listing the names of common ancestors does not guarantee that each term can be added independently.

Key takeaway: Under the simple non-inbred, unrelated-founder assumptions, a shared ancestor contributes $(1/2)^{m+n+1}$ to kinship, where $m$ and $n$ count transmissions from the two relatives. Ordinary first cousins have two grandparent routes, each contributing $1/32$, for a total kinship of $1/16$.

Check Your Understanding

For each question, keep the path lengths and the definition of kinship explicit.

Two relatives each have a path of two transmissions to a non-inbred shared ancestor. What is the contribution from that route?
Why does an ordinary first-cousin calculation include two grandparent contributions rather than one?
Suppose a pair has three distinct shared ancestors, and each is three transmissions away from each relative. Under the simple path assumptions, what is the total kinship contribution from these routes?
State what the kinship coefficient $1/16$ means for two ordinary first cousins at a locus.
Why is it unsafe to add every named common ancestor's route contribution without checking whether the routes are distinct and whether the assumptions hold?