Two Different Questions About Genetic Similarity
PCA and kinship estimation both summarize patterns in genome-wide genotypes, but they target different features. PCA represents major directions of genetic variation among the sampled individuals. Proximity in that representation can indicate similar broad ancestry, relative to the data and the PCs used. A pairwise kinship estimate instead targets the extent to which two individuals share alleles identical by descent (IBD), relative to a specified allele-frequency baseline. Recent relatives tend to have more IBD sharing than unrelated individuals from that baseline.
These quantities can be associated without being interchangeable. Individuals from the same population may have similar PC scores while having little recent familial sharing. Conversely, close relatives can have different PC scores, especially when ancestry varies within a family or when the displayed PCs capture ancestry differences. A PC distance is not a kinship coefficient, and a kinship estimate is not a measure of how close two points lie on a PCA plot.
The distance depends on which PCs are included and how their scores are scaled. PCA axis signs can reverse without changing pairwise Euclidean distances, but rescaling an axis changes those distances. Thus \(D_{ij}\) has no universal threshold that defines a particular ancestry match or relationship. The kinship coefficient has a different scale: under the conventional pedigree interpretation, expected values are about \(1/4\) for parent–offspring and full-sibling pairs, \(1/8\) for second-degree relatives, and \(1/16\) for first cousins. Estimates can depart from these expectations, and their interpretation depends on population structure and the allele-frequency baseline.
Why PCA and Kinship Can Disagree
PCA is constructed to represent variation across the genotypes in a chosen sample. If allele frequencies differ among ancestry groups, those differences can contribute strongly to the variation captured by early PCs. This is useful for describing broad genetic structure, but it means that PC proximity is relative to the sampled data: changing the samples, variants, or PCs can change the representation and the distances.
Kinship estimation asks a more conditional question: is there excess allele sharing consistent with recent familial inheritance, given a model or estimate of allele frequencies? Population structure matters here too. If the allele-frequency baseline is unsuitable, two people who share ancestry can appear more related than they are under the intended interpretation. The earlier material on robust kinship estimation with population structure addresses this concern. Even a well-calibrated kinship estimate and a carefully computed PC distance are not measuring the same thing.
A short covariance argument clarifies why ancestry can dominate a representation based on genotype variation. For a genotype variable \(G\) across a sampled population and an ancestry grouping \(A\), the law of total variance gives:
The first term is variation within ancestry groups; the second is variation among their mean genotypes. When group means differ, the between-group term contributes to total variation that PCA can represent. This is a statement about variation in the sample, not a claim that PCA has identified family relationships. Kinship, by contrast, concerns IBD sharing; it cannot be inferred solely from a large or small contribution of ancestry-group mean differences.
Worked Comparisons
Worked Example: Nearby in PCA, Not Close Relatives
In a hypothetical analysis, two individuals have scores \((0.10,-0.05)\) and \((0.12,-0.04)\) on the first two PCs. Their estimated kinship is \(0.003\). Calculate their distance in these two PCs and interpret the pair.
Using the specified Euclidean distance, \(D_{ij}=\sqrt{(z_{i1}-z_{j1})^2+(z_{i2}-z_{j2})^2}\). Substitution gives:
The scores are close in this two-PC representation. The kinship estimate, \(0.003\), is much smaller than the conventional pedigree expectation for close relatives. Taken together, these results are consistent with similar broad ancestry and little evidence of recent familial relatedness. They do not prove the individuals are unrelated: the estimate has uncertainty, and the distance is defined only for these PCs and this scaling.
Worked Example: A Larger PC Distance Within a Family
Suppose two hypothetical siblings from an admixed family have scores \((-0.20,0.10)\) and \((0.30,-0.10)\) on two PCs. Their kinship estimate is \(0.245\). Calculate the distance and explain why the results are not contradictory.
The distance is:
The estimated kinship, \(0.245\), is close to the pedigree expectation of \(1/4=0.25\) for full siblings. The PC distance is larger than in the first example, but its numerical size has no universal relationship cutoff. Siblings can inherit different combinations of parental ancestry-associated alleles, and the selected PCs may reflect those differences. The kinship estimate indicates substantial familial sharing under its assumptions; the scores indicate a difference in this particular representation. Before accepting either result, an analyst should check sample identity, genotype quality, PCA inputs, and the kinship estimator's population-structure handling.
Worked Example: Similar PC Distance, Different Kinship Evidence
Consider two hypothetical pairs measured on the same two PCs and using the same scaling. Pair A has scores \((0.00,0.00)\) and \((0.10,0.00)\), with estimated kinship \(0.004\). Pair B has scores \((0.30,0.20)\) and \((0.40,0.20)\), with estimated kinship \(0.060\). Compare their distances and estimates.
For Pair A, \(D_A=\sqrt{(0.00-0.10)^2+(0.00-0.00)^2}=0.10\). For Pair B, \(D_B=\sqrt{(0.30-0.40)^2+(0.20-0.20)^2}=0.10\). The distances are identical because the pairs have the same coordinate difference.
The kinship estimates differ: \(0.004\) for Pair A and \(0.060\) for Pair B. The second estimate is near the pedigree expectation \(1/16=0.0625\) for first cousins, while the first is small relative to expectations for close relatives. This is illustrative, not a relationship classification rule: estimates have sampling error and depend on the frequency baseline and method. The comparison shows that equal distance in the chosen PCs does not imply equal kinship.
Assumptions, Diagnostics, and Alternatives
A meaningful comparison starts by stating what each number represents. For a PC distance, report the PCs used, whether scores were rescaled, and the sample and variant basis for the PCA. Distances from different PC sets or differently scaled scores should not be compared as though they had a common calibration. For kinship, identify the estimation method and allele-frequency assumptions, and consider whether population structure could bias the estimate. Interpret a value against the method's uncertainty and intended reference, not as an error-free pedigree label.
Useful checks include looking at the pair's genotype quality and missingness, confirming sample identities, and examining whether the PCA shows broad ancestry patterns that could affect kinship estimation. If a pair has an unexpectedly high kinship estimate, ask whether shared ancestry or allele-frequency misspecification could inflate it. If a putative close relative has unexpectedly low estimated kinship, check data quality, variant selection, and estimator assumptions. For an apparent disagreement between methods, inspect the underlying data and representation before deciding that one number must be wrong.
The main alternatives answer different questions. PC distances are useful for ranking similarity in a specified low-dimensional ancestry representation. Kinship estimates are more directly suited to screening for familial sharing, with interpretation conditional on their estimation assumptions. When the question is whether a pair shares long inherited genomic segments, segment-based IBD analysis can provide a complementary form of evidence; it is not the same calculation as PC distance or a single kinship coefficient. No one summary resolves every question, and agreement between two summaries does not make them equivalent.
A common mistake is to reason, “These two points are close on the PCA plot, so they must be close relatives.” The correction is that PCA proximity describes resemblance along the plotted axes, which may reflect shared broad ancestry; kinship estimates address excess IBD sharing under a frequency model. Another mistake is to treat a PC distance as an absolute quantity across analyses. Its value depends on the PCs and scaling, so state those choices before comparing distances.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two individuals have similar scores on the first two PCs. What can this establish, and what can it not establish about their relationship?
- In a chosen two-PC representation, a pair has coordinates \((0.2,0.1)\) and \((0.5,0.5)\). Calculate their Euclidean distance.
- Why might two full siblings have different PCA scores yet a kinship estimate near \(0.25\)?
- Why should a PC distance from one analysis not automatically be compared with a distance from another analysis?
- A pair has a small PC distance and estimated kinship \(0.07\). Give one cautious interpretation and one diagnostic question.
Solutions. (1) Similar scores establish proximity in the specified PCA representation; they do not establish recent familial relatedness. (2) \(D=\sqrt{(0.2-0.5)^2+(0.1-0.5)^2}=\sqrt{0.09+0.16}=\sqrt{0.25}=0.5000\), in the score units of that representation. (3) Siblings can inherit different combinations of ancestry-associated alleles, affecting selected PC scores, while still sharing substantial genomic material IBD; \(0.25\) is the conventional expected kinship for full siblings. (4) The selected PCs, sample, variants, and score scaling may differ, changing the distances. (5) The pair is close in the chosen PCA representation and the kinship estimate may indicate familial sharing, but neither result alone proves a pedigree relationship. Check whether the kinship method's allele-frequency assumptions and treatment of population structure are suitable.