IBD Sharing Versus IBS Matching

Why Matching Is Not the Same as Sharing by Descent

When two affected relatives have the same marker allele, it is tempting to say that they share that allele. But “share” can mean two different things. The relatives may have allele copies with the same observed state, or they may have inherited copies descended from the same ancestral copy. Only the second meaning is identity by descent (IBD). This distinction matters in linkage analysis: the evidence of interest is whether affected relatives share ancestral chromosomal copies, not merely whether their genotype labels match.

For example, suppose two siblings both have genotype A/B at a marker. The matching genotypes tell us that each sibling carries an allele with state A and an allele with state B. They do not, by themselves, tell us whether the A copies came down the same parental chromosome, whether the B copies did, or whether neither did. The observed genotypes describe allele states; IBD describes the origins of the inherited copies.

Definition: Two allele copies are identical by descent (IBD) if they are copies of the same ancestral allele at the locus. They are identical by state (IBS) if they have the same allele state at the marker, whether or not they descend from the same ancestral copy. In the absence of mutation or other changes to the allele state, IBD copies are also IBS. The converse is not guaranteed.

Counting IBD and Describing IBS

For a full-sib pair at a locus, each sibling inherits one allele copy from each parent. The pair’s IBD state is the number of those parental copies that the siblings inherited from the same parental homologues. Write \(K\) for this number: \(K=0\) means they share no parental copy IBD, \(K=1\) means they share one, and \(K=2\) means they share both. The value of \(K\) refers to inheritance origin, not to how many allele labels look alike.

IBS describes the observed marker states. One possible summary for two diploid genotypes is the number of matching allele copies, counting repeated states with their multiplicities. If \(c_{ia}\) is the number of copies of allele state \(a\) in sibling \(i\)'s genotype, define

M_{\mathrm{IBS}}=\sum_a \min(c_{1a},c_{2a}).

This count ranges from 0 to 2. For example, A/B compared with A/B has \(M_{\mathrm{IBS}}=2\), while A/B compared with A/A has \(M_{\mathrm{IBS}}=1\). Other analyses may use a simpler indicator of whether the pair matches at least one allele, so the precise IBS summary should be stated when it is used. Neither summary, by itself, gives \(K\).

Worked Example: Identical Genotypes, Different IBD States

Consider two hypothetical parents, each with genotype A/B. Their two affected children both have genotype A/B. Calculate the allele-copy match and compare two possible transmission histories.

Observed IBS match. Each child has one A and one B, so \(c_{1A}=c_{2A}=1\) and \(c_{1B}=c_{2B}=1\). Therefore,

M_{\mathrm{IBS}}=\min(1,1)+\min(1,1)=1+1=2.

Possible history 1: IBD state 0. The first child inherits the father's A and the mother's B. The second inherits the father's B and the mother's A. Neither sibling received the same paternal homologue as the other, and neither received the same maternal homologue. Thus \(K=0\), even though both genotypes are A/B.

Possible history 2: IBD state 2. Both children inherit the father's A and the mother's B. They received the same paternal and maternal homologues, so \(K=2\). The observed genotypes are again A/B and A/B, with \(M_{\mathrm{IBS}}=2\).

The same IBS summary is therefore compatible with very different IBD states. Genotype matching alone cannot choose between these histories.

Why IBD Implies IBS, but Not the Reverse

Under the usual assumption that marker alleles do not mutate between the relevant ancestors and the observed relatives, descent preserves the allele state. If two relatives carry copies descended from the same ancestral copy, both copies have that copy’s marker state. So IBD implies IBS under this assumption. The reverse implication fails because different ancestral copies can have the same state.

This asymmetry can be expressed as a one-way implication:

\text{IBD} \Longrightarrow \text{IBS} \qquad\text{(assuming no change in allele state),}

but observing IBS does not establish IBD. The counterexample in the previous example supplies a direct justification: the siblings can match at both allele states while \(K=0\), because they inherited A and B from opposite parental homologues. A shared label can arise through descent, or because separate copies happen to have the same state.

Marker allele frequencies help explain why this distinction is practically important. When an allele state is common, two relatives can match at that state by chance even if they did not inherit the same copy. The chance of an IBS match depends on the allele frequencies and the matching definition; it is not itself a count of shared ancestral copies. A rare matching state may be more informative about shared origin than a common one, but it still does not turn a state match into proof of IBD.

Worked Example: A Single Matching State Does Not Identify Its Origin

Suppose the hypothetical parents each have genotype A/B, and two affected children each have genotype A/B. As before, the observed count is

M_{\mathrm{IBS}}=\min(1,1)+\min(1,1)=2.

Now compare two possible histories that both produce those data. In the first, child 1 inherits paternal A and maternal B, while child 2 inherits paternal B and maternal A. The children match at A and at B, but share neither parental homologue: \(K=0\). In the second, both inherit paternal A and maternal B. They match at A and B and share both parental homologues: \(K=2\).

The arithmetic for \(M_{\mathrm{IBS}}\) is identical in both histories, but the IBD count differs by two. Thus even a complete genotype match is not a direct measure of the number of inherited copies shared by descent. The fact that both children are affected does not change what their observed marker genotypes establish about origin.

Reading Parental Origins in a Small Example

If parental genotypes and transmissions are informative, an example can illustrate what origin information contributes. This is different from trying to infer origin from sibling genotype matching alone. For now, the purpose is to see why the distinction matters, not to set out a general procedure for inferring IBD from incomplete relative genotypes.

Worked Example: One Shared Parental Homologue

In a hypothetical family, the father has genotype A/B and the mother has genotype C/D. Two affected siblings have genotypes A/C and A/D. Assume no mutation at the marker and that the stated parental transmissions are known.

IBS count. The siblings both have one A, and there is no other matching state. Hence \(c_{1A}=c_{2A}=1\), while every other state contributes zero:

M_{\mathrm{IBS}}=\min(1,1)=1.

IBD state. Both siblings inherited A from the father, so they share the paternal A-bearing homologue. One inherited maternal C and the other maternal D, so they did not share the same maternal homologue. They therefore share exactly one parental copy IBD: \(K=1\).

For comparison, if the siblings instead had genotypes A/C and B/D, and the transmissions were father-to-first A, mother-to-first C, father-to-second B, and mother-to-second D, then they would share no parental homologue: \(K=0\). The IBS count would be

M_{\mathrm{IBS}}=\min(1,0)+\min(0,1)+\min(1,0)+\min(0,1)=0.

In this example, the parental transmissions—not the mere appearance of the sibling genotypes—establish the IBD state. The contrast also shows that the two summaries answer different questions: \(K\) counts shared parental copies, while \(M_{\mathrm{IBS}}\) counts matching states in the observed genotypes.

Connection to Affected-Relative Linkage

In nonparametric linkage analysis, affected relatives are informative because, under linkage, they may inherit a disease-associated chromosomal region from a shared ancestor more often than expected under the relevant null model. The sharing quantity of interest is IBD. In the affected-sib setting, excess-sharing tests compare observed sharing against the expected IBD sharing under the null; they do not define a pair as IBD-shared simply because its marker genotypes match.

IBS genotypes may contribute information about IBD when considered with pedigree structure, parental data, and marker properties. But an IBS match is an observation, not an inheritance history. Treating every match as an IBD event can overstate sharing, particularly when marker alleles are common. Conversely, missing or ambiguous genotype information can make the IBD state uncertain even when some allele matches are visible. The distinction is therefore essential both for interpreting an observed pair and for understanding what a linkage statistic is intended to summarize.

The earlier material on the sib-pair IBD null distribution gives the standard null probabilities for the three IBD states under its stated assumptions. Those probabilities concern \(K\), not the IBS match count \(M_{\mathrm{IBS}}\). Replacing one with the other changes the quantity being compared with the null and can invalidate the intended interpretation of an excess-sharing result.

Common Mistake

Wrong reasoning: “The two affected siblings both have A/B, so they share two alleles IBD.”

Correction: They share two allele states, so \(M_{\mathrm{IBS}}=2\). But their IBD state could be 0, 1, or 2 depending on which parental homologues they inherited. Without information about origin, identical genotype labels do not determine \(K\). In the absence of mutation, any genuinely shared IBD copy must have the same state in both siblings, but the observed state match alone does not prove that the copies are IBD.

A related mistake is to assume that an IBS match is worthless. It can be informative when combined with pedigree and marker information; it simply is not interchangeable with IBD. The careful statement is: “The relatives match at this marker state,” unless their shared ancestral origin has been established or appropriately inferred.

Key takeaway: IBS describes whether observed allele states match; IBD describes whether allele copies descend from the same ancestral copy. In affected-relative linkage, excess sharing concerns IBD. Matching marker alleles alone can neither establish nor count that sharing.

Check Your Understanding

Answer each question by distinguishing observed allele states from inherited origins.

Two siblings both have genotype C/D. What is their allele-copy IBS count under the definition in this tutorial? Does that determine their IBD state?
State the condition under which IBD implies IBS, and explain why IBS does not necessarily imply IBD.
For genotypes A/B and A/A, calculate \(M_{\mathrm{IBS}}\). What does that count say, and what does it not establish?
Two siblings inherit the same paternal homologue but different maternal homologues at a locus. What is their IBD state \(K\)?
Why can a common marker allele produce matching genotypes without establishing that affected relatives share that allele IBD?