Calculating Inbreeding From Common Ancestors

Why Shared Ancestors Matter

An offspring is inbred when its two alleles at a locus are identical by descent: they are copies of the same ancestral allele. The chance of this occurring depends on whether the offspring’s parents can trace alleles back through shared ancestors. A pedigree makes those routes visible, and a path calculation turns them into a probability.

The key is to count generations along each route to a common ancestor and to account for the possibility that the ancestor’s own two alleles are identical by descent. The resulting calculation is an inbreeding coefficient for the offspring—not a measure of how similar the parents look or how many ancestors they share.

Definition: The inbreeding coefficient \(F_X\) of an individual \(X\) is the probability that the two alleles at a randomly chosen autosomal locus in \(X\) are identical by descent. Equivalently, \(F_X\) is the kinship coefficient of \(X\)'s two parents: the probability that one allele chosen at random from each parent is identical by descent.

The Common-Ancestor Path Formula

Consider the two parents of \(X\). For a common ancestor \(A\), trace one path from the first parent back to \(A\), and another from the second parent back to \(A\). Let \(n_1\) and \(n_2\) be the respective numbers of parent–child links along those paths. These are generation counts: a parent who is a child of \(A\) is one link away, while a grandchild of \(A\) is two links away.

For a simple valid pair of paths, the contribution through \(A\) is \(\left(\frac12\right)^{n_1+n_2+1}(1+F_A)\), where \(F_A\) is the inbreeding coefficient of \(A\). If the parents share multiple ancestors through separate valid paths, the contributions are added. Under the usual simple-pedigree conditions—founders are unrelated and non-inbred unless otherwise specified, and the counted paths do not overlap except at their common ancestor—the calculation is:

F_X=\sum_A \left(\frac12\right)^{n_1+n_2+1}(1+F_A)

The sum is over the common ancestors being counted, with \(n_1\) and \(n_2\) determined separately for each ancestor. In the examples below, the paths meet only at their stated common ancestor or ancestors. More complicated pedigrees require care in selecting paths; counting the same route more than once changes the result.

Interpretation: Each path contribution is a probability contribution to the chance that the offspring’s two alleles are identical by descent through that ancestor. The coefficient \(F_X\) ranges from 0 to 1. For example, \(F_X=0.03125\) means a 3.125% probability of identity by descent at a randomly chosen autosomal locus, under the pedigree model.

Why the Formula Has These Factors

Along a specified path, each parent-to-child transmission has probability \(1/2\) of carrying the allele traced backward to the preceding generation. Thus, the chance that the randomly chosen allele from the first parent traces to a particular allele in \(A\) along its path is \(2^{-n_1}\); the corresponding probability for the second parent is \(2^{-n_2}\).

Once both paths reach \(A\), the two traced copies match by descent with probability \((1+F_A)/2\). To see this, with probability \(F_A\), the two alleles in \(A\) are already identical by descent, so the copies match. With probability \(1-F_A\), they are not; in that case, two transmissions chosen independently from \(A\)'s two alleles match with probability \(1/2\). Therefore:

F_A+(1-F_A)\frac12 =\frac{1+F_A}{2}

Multiplying the two path probabilities by this matching probability gives the contribution through \(A\):

2^{-n_1}2^{-n_2}\frac{1+F_A}{2} =2^{-(n_1+n_2+1)}(1+F_A)

This also explains why the exponent contains \(n_1+n_2+1\): the path transmissions contribute \(n_1+n_2\) factors of \(1/2\), and the final match between the two alleles in the common ancestor contributes one more factor, adjusted for \(F_A\).

Worked Example: One Shared Founder in Half-First Cousins

Suppose two founders, \(A\) and \(B\), have a child \(U\), and \(A\) and an unrelated founder \(C\) have a child \(V\). The individuals \(U\) and \(V\) are half-siblings. Each has a child with a different, unrelated, non-inbred partner; call those children \(P\) and \(Q\). \(P\) and \(Q\) are the parents of \(X\). Assume all founders are non-inbred and unrelated to one another. What is \(F_X\)?

The parents \(P\) and \(Q\) share the single founder \(A\). From \(P\), the path is \(P\rightarrow U\rightarrow A\), so \(n_1=2\). From \(Q\), it is \(Q\rightarrow V\rightarrow A\), so \(n_2=2\). Since \(A\) is a non-inbred founder, \(F_A=0\).

Substitute into the formula:

F_X=2^{-(2+2+1)}(1+0) =\frac{1}{32} =0.03125

Thus, under the stated pedigree assumptions, the probability that \(X\)'s two alleles at a randomly chosen autosomal locus are identical by descent is \(1/32\), or 3.125%. The fact that the parents are related does not make \(F_X\) equal to their degree of relationship; the path calculation gives the offspring’s identity-by-descent probability.

Worked Example: Two Shared Grandparents

Suppose \(P\) and \(Q\), the parents of \(X\), are first cousins. Their respective parents \(U\) and \(V\) are full siblings whose parents are the two non-inbred, unrelated founders \(G_1\) and \(G_2\). Assume no other shared ancestry. Calculate \(F_X\).

For each shared grandparent, the path from \(P\) goes through one parent to that grandparent, giving \(n_1=2\). The path from \(Q\) also has two links, so \(n_2=2\). Each grandparent is a non-inbred founder, and therefore contributes:

2^{-(2+2+1)}(1+0) =\frac{1}{32} =0.03125

There are two distinct shared grandparents, so add their contributions:

F_X=\frac{1}{32}+\frac{1}{32} =\frac{2}{32} =\frac{1}{16} =0.0625

The result is a 6.25% probability of identity by descent for \(X\) at a randomly chosen autosomal locus, given this pedigree. Each shared grandparent contributes separately because the stated paths are distinct and have the same lengths.

Worked Example: A Shared Ancestor Who Is Inbred

Suppose the parents \(P\) and \(Q\) of \(X\) are half-siblings whose shared parent is \(A\). Assume \(A\) has inbreeding coefficient \(F_A=0.25\), and that \(P\) and \(Q\) have no other shared ancestry except through \(A\). Calculate \(F_X\).

Each parent is one generation below \(A\), so \(n_1=n_2=1\). The inbreeding coefficient of \(A\) is not zero, so it must remain in the formula:

F_X=2^{-(1+1+1)}(1+0.25) =\frac18(1.25) =0.15625

Thus, the offspring’s inbreeding coefficient is 0.15625, or 15.625%. This is larger than the contribution \(1/8=0.125\) that would result from a non-inbred shared parent, because \(A\)'s two alleles have an additional chance of already being identical by descent.

A Reliable Calculation Sequence

For a small pedigree, it helps to separate drawing the routes from doing the arithmetic. Keep the two parental paths for each common ancestor distinct, and write down their lengths before substituting into the formula.

Identify the offspring’s parents.
The coefficient being calculated belongs to their child, so the relevant paths begin at the two parents.

Find each stated common ancestor.
For each ancestor, trace one route from each parent and count parent–child links to that ancestor.

Record the ancestor’s inbreeding coefficient.
Use \(F_A=0\) for a non-inbred founder under the stated assumptions; otherwise use the known value.

Calculate each path contribution and add.
Substitute \(n_1\), \(n_2\), and \(F_A\) into the formula for each valid path, then sum the contributions.

Common Mistakes and Why the Calculation Matters

Counting generations from the offspring. The path lengths in the formula start at the offspring’s parents, not at the offspring. First identify the two parents, then count links from each parent to the common ancestor.
Leaving out the final exponent term. The contribution is \(2^{-(n_1+n_2+1)}(1+F_A)\), not \(2^{-(n_1+n_2)}(1+F_A)\). The additional factor accounts for whether the two ancestral copies match.
Assuming every common ancestor is non-inbred. Use \(F_A=0\) only when that is supported by the pedigree assumptions. If \(F_A\) is positive, the term \(1+F_A\) increases the contribution.
Confusing kinship and inbreeding coefficients. \(F_X\) is the probability that the two alleles within \(X\) are identical by descent. The kinship coefficient of two individuals is the probability that one allele from each is identical by descent. \(F_X\) equals the kinship coefficient of \(X\)'s parents, not the kinship coefficient of \(X\) with either parent.
Adding paths without checking whether they are distinct. A common ancestor should contribute through a valid pair of paths, and the same route must not be counted twice. In more complex pedigrees, path overlap needs explicit attention; blindly listing every shared name can inflate the result.

Inbreeding coefficients are useful because they summarize a pedigree’s implications for identity by descent at a locus. They can inform genetic models that distinguish individuals with different probabilities of homozygosity by descent. The coefficient describes a probability under the pedigree assumptions; it does not establish that a particular individual carries a harmful variant or has a particular phenotype.

Key takeaway: For each valid common-ancestor path, count generations from each of the offspring’s parents, calculate \(2^{-(n_1+n_2+1)}(1+F_A)\), and add the distinct contributions. State the pedigree assumptions, especially whether founders are unrelated and non-inbred.

Check Your Understanding

For each question, identify the paths and explain the result using the common-ancestor formula.

Two parents of \(X\) are half-first cousins who share one non-inbred founder. Each parent is two generations from that founder. What is \(F_X\)?
In a pedigree with two non-inbred shared grandparents, both paths have lengths \(n_1=2\) and \(n_2=2\). What is the contribution from each grandparent, and what is the total?
What change does a positive \(F_A\) make to the path contribution through ancestor \(A\)?
Why does the formula include a \(+1\) in the exponent \(n_1+n_2+1\)?
In one sentence, distinguish the inbreeding coefficient of an offspring from the kinship coefficient between its parents.