Jointly Representing Ancestry and Kinship

Two Views of the Same Samples

Genomic analyses often need to describe two features of the same set of people. Principal components (PCs) summarize broad patterns of genetic variation: samples with similar coordinates across leading PCs tend to have similar profiles along those variation axes. A kinship matrix, in contrast, records pairwise genetic dependence, with one entry for each pair of samples. These views are related, but they answer different questions. A PCA plot is a low-dimensional map; a kinship matrix is a sample-by-sample collection of pairwise estimates.

The distinction matters in genomic comparisons. Two people can occupy nearby positions in a broad ancestry pattern without being close relatives. Conversely, close relatives can have similar genotypes for reasons that are not captured by a plot of only the first two PCs. A useful joint representation therefore preserves both kinds of information and makes it possible to inspect them together without treating one as a substitute for the other.

Definitions and Assumptions

Let \(n\) samples be typed at \(m\) genetic markers. For illustration, suppose genotypes are diploid allele dosages \(g_{ij}\in\{0,1,2\}\), where \(g_{ij}\) is the dosage for sample \(i\) at marker \(j\). If \(p_j\) is the allele frequency used for marker \(j\), a commonly used standardized value is

This scaling centers dosage at its expected value under the allele frequency and scales it by its binomial standard deviation. A common genomic relationship matrix is \(K=ZZ^\mathsf{T}/m\), where \(Z\) is the \(n\times m\) matrix of standardized genotypes. Other estimators and marker weightings are used, so a reported matrix must be interpreted using its actual definition. In particular, a kinship coefficient and a genomic relationship matrix entry are not automatically on the same numerical scale. Under the conventional pedigree definition, the additive relationship for a pair is \(2\phi\), where \(\phi\) is their kinship coefficient; genomic estimators may use that or another stated scaling.

PCA of standardized genotypes identifies orthogonal directions that explain variation across samples. A sample’s score on a PC is its coordinate along that direction. The first few PCs give a compact view of leading variation, but do not preserve every feature of the genotype data. The kinship matrix retains pairwise entries for all sample pairs, including information that is not visible in a two-dimensional plot.

Joint interpretation assumes that the PCA results and matrix refer to the same samples and compatible genotype data, and that sample identifiers are correctly aligned. The marker set, allele-frequency choices, scaling, and any sample exclusions should be documented. If these differ across the two calculations, the outputs may still be compared, but they are not two summaries of precisely the same input representation. The genotype data should also have passed the quality checks appropriate to the analysis; artifacts in genotypes can affect both summaries.

Why the Views Are Related but Not Interchangeable

When PCA is computed from the same standardized genotype matrix used to form \(K=ZZ^\mathsf{T}/m\), the relationship is exact. Let \(K u_k=\lambda_k u_k\), where \(u_k\) is a unit-length eigenvector and \(\lambda_k\) its eigenvalue. The eigenvectors give sample directions and the eigenvalues quantify variation along those directions. If \(U\) contains all eigenvectors and \(\Lambda\) is the diagonal matrix of eigenvalues, then

Thus each matrix entry is assembled from contributions across all eigen-directions. Keeping only the first \(q\) PCs gives the approximation \(K_{i\ell}\approx\sum_{k=1}^{q}\lambda_k u_{ik}u_{\ell k}\), not generally the full entry. If PCs are calculated from a different marker set, weighting, or scaling than the matrix, even this exact decomposition need not apply to the reported pair.

This also explains why a PC scatterplot cannot be read as a kinship matrix. The plot shows selected coordinates, often only two, while a matrix entry depends on the combined contributions from all directions in the decomposition. And neither PC distance nor visual proximity is itself a kinship coefficient.

Building a Joint Sample-Level Display

A practical representation links each sample’s PC coordinates to the corresponding row and column of the matrix. The linkage can be a sample table beside a PC scatterplot and a kinship heatmap, or an interactive display in which selecting a plotted sample highlights its matrix row and column. Whichever format is used, sample identifiers and matrix ordering must be explicit.

A Short Matrix Decomposition

The relationship between a full matrix and all PCs follows directly from the spectral decomposition. For a symmetric matrix \(K\), suppose \(K=U\Lambda U^\mathsf{T}\), with orthonormal columns in \(U\). Its \(i,\ell\) entry is the dot product of row \(i\) and row \(\ell\) of \(U\), weighted by the eigenvalues:

For example, take the hypothetical two-sample matrix \(K=\begin{pmatrix}1&0.2\\0.2&1\end{pmatrix}\). Its unit eigenvectors are \(u_1=(1,1)^\mathsf{T}/\sqrt{2}\) and \(u_2=(1,-1)^\mathsf{T}/\sqrt{2}\), with eigenvalues \(1.2\) and \(0.8\). The off-diagonal entry reconstructed from both directions is \(1.2(1/\sqrt{2})(1/\sqrt{2})+0.8(1/\sqrt{2})(-1/\sqrt{2})=0.6-0.4=0.2\). Keeping only the first direction would instead give \(0.6\). This is why a truncated PC representation does not, in general, reproduce the full matrix’s pairwise values.

Diagnostics, Alternatives, and Limitations

Several checks help prevent a visually persuasive but incorrect joint interpretation:

• Check sample identity and order. A row-order mismatch can attach a valid matrix row to the wrong PC point. Verify identifiers, not just the number of samples.
• Check the matrix definition and scale. A kinship coefficient, an additive relationship estimate, and a standardized genomic matrix entry should not be compared as if their numerical values were interchangeable.
• Check what the PCs represent. A leading axis describes variation in the analyzed genotypes; it does not, by itself, assign a sample to a named ancestry group or establish a relationship.
• Check whether the views share inputs. If the matrix and PCA use different markers, frequency estimates, or samples, explain that difference. Apparent disagreement may reflect construction choices rather than a biological contradiction.
• Check the display for lost information. A two-PC plot suppresses later directions, while a heatmap can hide fine structure if its color range is poorly chosen. State the scales and inspect the underlying values when needed.

A PCA plot alone is efficient for viewing broad structure, but cannot display all pairwise dependence. A kinship heatmap alone shows pairwise patterns, but can be difficult to scan for broad gradients across many samples and does not show each sample’s location on selected PCs. A linked plot and matrix, or a table that preserves both, is useful when both questions matter. A full matrix is also more information than some analyses need; for a large sample, report or display selected pairs or a well-defined summary while retaining the scale and selection rule.

The two summaries are often computed from overlapping genotype information. They should therefore not be presented as independent confirmations of a finding. Joint representation is a way to compare complementary summaries, not a guarantee that either estimate is unbiased or that population structure and close relatedness have been fully separated. The interpretation depends on marker selection, allele-frequency choices, genotype quality, and the population represented in the analysis.

Common Pitfall