Why Loops Need Careful Counting
When an individual's parents are related, an allele inherited from one parent may be identical by descent (IBD) to an allele inherited from the other. A pedigree loop is one way to trace such a possibility: start at one parent, follow parentage links back to a common ancestor, and then follow a different line of descent to the other parent. The loop calculation is useful only when each route is counted appropriately. If a purported route goes through the same person twice, it is not a separate valid loop and can inflate the result.
This tutorial focuses on identifying the loops that contribute to an individual's inbreeding coefficient and on avoiding duplicate counting. Let \(F_O\) denote the inbreeding coefficient of offspring \(O\), and let \(P\) and \(Q\) denote the offspring's two parents. By definition, \(F_O\) is the probability that the two alleles at a randomly chosen autosomal locus in \(O\) are IBD. Equivalently, it is the kinship coefficient of \(P\) and \(Q\).
Contribution From One Valid Loop
Suppose \(A\) is a common ancestor of parents \(P\) and \(Q\). Let \(n_P\) be the number of parent-child links from \(P\) up to \(A\), and \(n_Q\) the corresponding number from \(Q\). For example, a parent is one generation from their own parent, and a grandchild is two generations from a grandparent. The contribution of this loop to \(F_O\) is
where \(F_A\) is the inbreeding coefficient of common ancestor \(A\). If \(A\) is a noninbred founder, then \(F_A=0\), and the contribution reduces to \((1/2)^{n_P+n_Q+1}\). If there are several valid loops, their contributions can be added under the path method's conditions: each loop must represent a distinct valid route, with no repeated individual within the loop. Ancestral routes that only appear to be additional loops because they retrace a person already on both arms must not be added.
The factor \(1/2\) for each generation reflects Mendelian transmission: at each parent-child link, a specified allele has probability \(1/2\) of being transmitted. The additional factor and the \(F_A\) term account for whether the two lineages select the same allele copy in \(A\), including the possibility that \(A\)'s own two alleles are IBD.
Worked Examples
Worked Example: Offspring of Full-Sibling Parents
Suppose \(P\) and \(Q\) are full siblings, both children of the same two noninbred, unrelated founders \(A\) and \(B\). They have a child \(O\). Find \(F_O\) using the valid loops.
There are two common ancestors: \(A\) and \(B\). For each ancestor, the path from each parent is one generation long, so \(n_P=n_Q=1\). Both ancestors are noninbred, so each loop contributes
The two loops are \(P\)-\(A\)-\(Q\) and \(P\)-\(B\)-\(Q\). They represent the two distinct shared-parent routes in this pedigree. Thus
The calculated inbreeding coefficient is \(0.25\), a probability with no physical unit. The calculation assumes the stated founders are unrelated and noninbred and that there are no other shared ancestors. It does not count either founder's route twice: each founder is a distinct common ancestor.
Worked Example: A Shared Parent and an Invalid Extra Route
Suppose \(P\) and \(Q\) are both children of \(A\), with different, unrelated other parents. Assume \(A\) is noninbred. The pedigree also shows that \(A\)'s parents are \(G\) and \(H\). Find the contribution to the inbreeding coefficient of a child of \(P\) and \(Q\), and explain why \(G\) and \(H\) do not provide two additional valid loops.
The valid loop is \(P\)-\(A\)-\(Q\). Both arms have length one, and \(F_A=0\), so its contribution is
One might mistakenly list \(G\) and \(H\) as further common ancestors. Each is two generations from both \(P\) and \(Q\), which, if inserted mechanically, would give \(1/32\) per ancestor. That would produce the incorrect sum \(1/8+1/32+1/32=3/16\).
The supposed route through \(G\) goes from \(P\) to \(A\), then to \(G\), then back down through \(A\) to \(Q\). It repeats \(A\), so it is not a valid loop. The same problem occurs for \(H\). These are not distinct additional routes between the parents; they retrace the already-counted connection through \(A\). The valid path calculation therefore gives \(F_O=1/8\), not \(3/16\). This example assumes there are no other shared ancestors or sources of relatedness.
Worked Example: Two Distinct Loops With Different Path Lengths
Consider parents \(P\) and \(Q\) with two shared, noninbred common ancestors, \(A\) and \(B\). Assume the pedigree has no other common ancestors and that the following paths are valid: \(P\) is two generations from \(A\), while \(Q\) is three generations from \(A\); \(P\) is three generations from \(B\), while \(Q\) is two generations from \(B\). Within each loop no person is repeated.
For \(A\), the generation distances sum to five, so the exponent in the loop formula is six. Its contribution is
For \(B\), the distances also sum to five, giving
Because these are two distinct valid common-ancestor loops, the offspring's inbreeding coefficient is
The result is a probability, not a percentage of the offspring's genome that must be homozygous. It is the probability that the two alleles at a randomly chosen autosomal locus are IBD under the stated pedigree assumptions.
Why the Formula Has This Form
For a particular common ancestor \(A\), trace one allele copy in each of the offspring's parents back along the specified arms. The chance that the allele copy in \(P\) descends from a specified allele copy in \(A\) is \((1/2)^{n_P}\); for \(Q\), it is \((1/2)^{n_Q}\). In \(A\), two independently selected allele copies are IBD with probability \((1+F_A)/2\): with probability \(1/2\) the selections are the same copy, and with probability \(1/2\) they are different copies, which are IBD with probability \(F_A\). Multiplying these factors gives the single-loop contribution:
This calculation explains both the generation counts and why an ancestor's inbreeding coefficient appears in the formula. It also shows why an apparent route that goes up from one parent to an ancestor and then retraces a person on the other arm cannot simply be treated as a new, independent contribution.
A Practical Loop-Checking Procedure
Inbreeding depends on whether the alleles they transmit can be IBD, so trace the routes connecting those parents.
Record the number of parent-child links from each parent to that ancestor. Keep the two arms distinct.
If a person appears more than once in the proposed loop, such as on both arms before the designated common ancestor, do not count that route as an additional valid loop.
Use the two generation distances and the common ancestor's \(F_A\), then add the contributions of distinct valid loops.
Confirm that the founder relationships and any other shared ancestry are represented. If paths overlap in a way that makes the route accounting unclear, do not force a simple sum; use the parent-based kinship recursion to calculate the relationship without duplicating a route.
Common Mistakes and Why the Distinction Matters
A common mistake is to treat every ancestor appearing in both parental pedigrees as an automatic extra loop. But a person can appear in both pedigrees because a route passes through a closer shared ancestor and then retraces the same individual. Counting that more distant ancestor as another loop can count the same inheritance history twice, as in the parent-sharing example.
A different mistake is to count only one common ancestor when the parents genuinely share more than one distinct ancestor. In the full-sibling example, the two shared parents each provide a valid loop, and both contributions are needed. The right question is not simply, “How many shared names are in the pedigree?” It is, “Which distinct, valid routes connect the two parents through common ancestors?”
The loop method makes the structure behind an inbreeding calculation visible: the path lengths control how likely an allele is to travel from a common ancestor to both parents, while \(F_A\) adjusts for whether the ancestor's own alleles are IBD. Careful loop identification matters because even correct arithmetic applied to duplicated paths gives a wrong genetic probability.
- Do not count a repeated person as a new route. A loop must not revisit an individual before returning to its starting parent.
- Do not omit distinct common ancestors. Separate valid loops can each contribute, even when they are close relatives in the pedigree.
- Do not confuse a pedigree connection with a valid loop contribution. A shared name alone does not establish that an additional route should be summed.
- State the assumptions. Founder relatedness, ancestor inbreeding, and other shared routes can change which contributions belong in the calculation.
Check Your Understanding
Use the generation distances and loop rules to answer these questions.
- Two parents share a noninbred common ancestor who is two generations from one parent and three from the other. What is the contribution of this loop?
- Why does an ancestor's inbreeding coefficient appear as the factor \(1+F_A\) in the loop formula?
- Two full siblings have a child. If their parents are unrelated, noninbred founders, how many valid loops connect the siblings, and what is the child's inbreeding coefficient?
- In the shared-parent example, why is a route through one of \(A\)'s parents not an additional valid loop between \(P\) and \(Q\)?
- What should you do if overlapping pedigree routes make it unclear whether summing loop contributions would duplicate an inheritance route?