Why Describe Relatedness With States?
A kinship coefficient summarizes how likely two randomly chosen alleles, one from each person, are identical by descent. That summary is useful, but it does not say whether a pair typically shares no alleles, one allele, or two alleles IBD at a genomic position. These possibilities differ among relative pairs, even when their kinship coefficients are equal. For example, parent–offspring pairs and full siblings have the same kinship coefficient under the standard pedigree assumptions, but their IBD-state probabilities differ.
Here, an IBD (identical by descent) match means that two allele copies descend from the same ancestral allele copy. This is different from being identical by state: two copies can carry the same allele value without sharing a recent ancestral copy. IBD states are pedigree-based descriptions of descent, not simply observations that two genotypes look alike.
The non-inbreeding condition matters. It means that the two homologous alleles within either individual are not themselves IBD. Under this condition, the IBD matches between the individuals form a one-to-one matching: one allele in either person cannot be matched to both alleles in the other. The familiar three-state description and the kinship relationship derived below rely on that condition.
State probabilities describe the distribution of inheritance at a randomly selected autosomal locus under a specified pedigree. Along an actual chromosome, inheritance can change across genomic positions because of recombination. Thus, the probabilities describe the pedigree-based distribution across possible transmissions, or the expected genomic proportions over many positions—not a claim that every position has the same state.
From IBD States to Kinship
The kinship coefficient \(\phi\) for two individuals is the probability that one allele chosen at random from the first person and one chosen at random from the second are IBD. Assume both individuals are non-inbred and that the IBD matches are one-to-one. Each person’s allele is selected with probability \(1/2\), independently of the other person’s selection.
There is no IBD match, so the probability that the two selected alleles are IBD is 0.
There is one matching pair among the four possible cross-person allele choices. The probability of selecting that pair is \(1/2 \times 1/2=1/4\).
There are two matching pairs among the four possible cross-person allele choices. The probability of selecting an IBD pair is \(2/4=1/2\).
This is a weighted average of the conditional probabilities for the three states. It is valid for the non-inbred pair described above. If either person is inbred, the labels \(K=0,1,2\) by themselves may not specify the pattern of matches among all four allele copies, so this calculation need not hold.
Worked Pedigree Patterns
The standard examples below assume autosomal inheritance, Mendelian transmission, no mutation, and founders who are mutually unrelated and non-inbred. These assumptions rule out additional IBD sharing through the founders and make the state descriptions straightforward.
Worked Example: Parent and Offspring
Consider a non-inbred parent and their biological child. At an autosomal locus, the child inherits one of the parent’s two allele copies. That inherited copy is IBD with the corresponding allele copy in the parent. The child’s other allele comes from the other parent, assumed unrelated to the first.
Consequently, the pair always has exactly one IBD match: \(k_0=0\), \(k_1=1\), and \(k_2=0\). The probabilities sum to one, as required.
Using the non-inbred-pair formula, \(\phi=k_1/4+k_2/2=1/4+0=0.25\). Thus, the kinship coefficient is 0.25. The relatedness coefficient is \(r=2\phi=0.50\). This does not mean the pair shares exactly half of all their alleles at every genomic position; it summarizes the pedigree-based probability of an IBD match for a randomly chosen cross-person allele pair.
Worked Example: Full Siblings
For two non-inbred full siblings, consider the allele each inherits from their father and the allele each inherits from their mother. From either parent, the siblings independently receive one of that parent’s two homologous copies, each with probability \(1/2\). The chance they receive the same parental copy is therefore \(1/2\). Under the stated assumptions, the paternal and maternal transmission comparisons are independent.
Let \(K\) count how many of these two parental origins produce a shared IBD copy. It has the distribution of the sum of two independent Bernoulli variables, each with success probability \(1/2\). Thus, \(P(K=0)=(1/2)^2=1/4\), \(P(K=1)=2(1/2)(1/2)=1/2\), and \(P(K=2)=(1/2)^2=1/4\). Hence \(k_0=0.25\), \(k_1=0.50\), and \(k_2=0.25\).
The kinship coefficient is \(\phi=(0.50/4)+(0.25/2)=0.125+0.125=0.25\), and \(r=2\phi=0.50\). Full siblings and parent–offspring pairs therefore have the same \(\phi\) and \(r\), but different state distributions: siblings can have \(K=0\) or \(K=2\), whereas a parent–offspring pair has \(K=1\) under these assumptions.
Worked Example: First Cousins
Consider first cousins whose parents are full siblings, with the cousins’ other parents unrelated to the shared grandparents and to one another. Each cousin has only one allele inherited from the parent in the shared-grandparent branch; the other allele comes from an unrelated parent. Therefore, the cousins can share at most one IBD allele, so \(k_2=0\).
For either cousin, the allele inherited through that branch has probability \(1/4\) of tracing to each of the four allele copies in the two shared grandparents: the transmission must pass through the cousin’s parent and then through one of the grandparent’s two homologues. The two cousins’ transmissions along their separate branches are independent. For any one specified grandparent allele copy, the probability both cousins inherit a copy descending from it is \((1/4)(1/4)=1/16\). These four events are mutually exclusive, since each cousin has just one allele from the related-parent branch. Summing gives \(P(K=1)=4(1/16)=1/4\).
It follows that \(k_1=0.25\), \(k_2=0\), and \(k_0=1-0.25=0.75\). The kinship coefficient is \(\phi=(0.25/4)+0=0.0625\), and \(r=0.125\). This state pattern differs from that of half siblings, although both pairs have \(k_0=0.50\), \(k_1=0.50\), \(k_2=0\) under the same founder assumptions. State probabilities do not uniquely identify every pedigree category.
When Three States Are Not Enough
The one-to-one matching assumption excludes inbreeding within either person. If an individual’s two homologous alleles are IBD to each other, identity relationships among the four copies can be more complicated than a count of cross-person matched pairs records. In that setting, a pair can have \(K=2\), yet the two matches need not be distinct one-to-one pairs. The conditional probability that randomly selected alleles are IBD is then not necessarily \(1/2\).
Worked Example: Why Inbreeding Changes the Calculation
Suppose, as a hypothetical extreme, that each of two individuals has two allele copies, and all four copies descend from the same ancestral allele copy. Counting two cross-person matches gives \(K=2\). But every possible choice of one allele from each person is IBD, so the conditional probability of an IBD match is \(4/4=1\), not \(2/4=1/2\).
The three-state count alone has lost information about the identity relationships among copies within each person. Applying \(\phi=k_1/4+k_2/2\) to this situation would give the wrong answer. Inbred pairs require a richer description of the identity partitions among the four copies; the three \(k\)-probabilities alone are insufficient.
Interpreting and Checking State Probabilities
State probabilities answer a different question from a single kinship coefficient: they describe how the number of IBD matches is distributed. They are particularly informative when relationship categories have equal kinship but different transmission patterns. However, they are not a universal pedigree fingerprint. Half siblings, grandparent–grandchild pairs, and avuncular pairs have the same standard state probabilities, \(k_0=0.50\), \(k_1=0.50\), \(k_2=0\), under non-inbred, unrelated-founder assumptions. Their pedigrees differ even though these three probabilities agree.
Several checks help detect errors in a calculation or interpretation:
- Check the probability total. The three states are exhaustive and mutually exclusive, so \(k_0+k_1+k_2=1\), with each probability between 0 and 1.
- Check the pedigree assumptions. Inbreeding, related founders, or a pedigree loop can create identity patterns not represented by the simple one-to-one states.
- Check what “sharing” means. Matching allele values in observed genotypes are not automatically IBD; they may be identical by state without a shared ancestral copy.
- Check whether the conclusion over-identifies the pedigree. Equal state probabilities or equal kinship coefficients do not prove that two pairs have the same relationship category.
A state-based pedigree description is a model of inheritance, not a direct empirical classification from one marker. Real genomic data can deviate from an idealized pedigree expectation through finite genomic length, recombination, genotyping error, and population-level allele-frequency patterns. Later genomic analyses use observed data to estimate relatedness; the definitions here clarify the inheritance quantities those analyses aim to represent.
Comparison and Common Pitfalls
| Description | What it summarizes | Main limitation |
|---|---|---|
| IBD-state probabilities \(k_0,k_1,k_2\) | Probabilities of zero, one, or two one-to-one IBD matches for a non-inbred pair | Insufficient by themselves for inbred individuals |
| Kinship coefficient \(\phi\) | Probability that one randomly chosen allele from each person is IBD | Compresses different state patterns to one number |
| Pedigree relationship category | The stated genealogical connection between two people | Different categories can have the same kinship or state probabilities |
A common mistake is to say, “The pair has \(K=2\), so a random cross-person allele choice is IBD with probability \(1/2\),” without checking whether the individuals are non-inbred and the matches are one-to-one. That reasoning is valid only under those conditions. For an inbred individual, the two homologues can share ancestry with each other, so the cross-person identity pattern may not consist of two distinct matching pairs. The correction is to state the assumptions first and use the three-state kinship formula only when they hold; otherwise, represent the fuller identity structure.
The practical value of the states is therefore twofold: they distinguish inheritance patterns that a scalar kinship coefficient merges, and they make the assumptions behind that coefficient visible. They do not replace pedigree information, identify every relationship uniquely, or guarantee that observed allele matches are IBD.
Check Your Understanding
Answer each question using the non-inbred, unrelated-founder assumptions unless the question says otherwise. Solutions are included for checking your reasoning.
- A pair has \(k_0=0.40\), \(k_1=0.40\), and \(k_2=0.20\). What is its kinship coefficient under the standard assumptions? Solution: \(\phi=k_1/4+k_2/2=0.40/4+0.20/2=0.10+0.10=0.20\).
- Why do parent–offspring pairs and full siblings have equal kinship coefficients but different IBD-state probabilities? Solution: Both have \(\phi=0.25\), but a parent–offspring pair always has one IBD match, while full siblings have probabilities \(1/4,1/2,1/4\) for zero, one, and two matches.
- For a pair with \(k_0=0.50\), \(k_1=0.50\), and \(k_2=0\), calculate \(r=2\phi\). Solution: \(\phi=0.50/4=0.125\), so \(r=2(0.125)=0.25\).
- Can the state probabilities \(0.50,0.50,0\) distinguish half siblings from grandparent–grandchild pairs under the standard assumptions? Solution: No. Those categories can share the same three state probabilities, so the states do not uniquely determine the pedigree.
- Why is \(\phi=k_1/4+k_2/2\) not generally justified for an inbred pair? Solution: Inbreeding can make allele copies within a person IBD, so cross-person matches may not form distinct one-to-one pairs. The conditional probability of an IBD match given \(K=2\) need not be \(1/2\); a richer identity description is needed.