Method-Of-Moments Estimation From Allele Sharing

From Allele Counts to Pairwise Kinship

Two people can carry the same allele at a marker because they inherited copies from a shared ancestor or because the allele is common in the population. A pairwise kinship estimator uses the allele frequency to account for this background sharing. Its basic signal is the product of the two people’s centered allele counts: counts above the population expectation tend to occur together more often for related pairs.

Here, \(G_{il}\) is the number of copies of a specified allele carried by person \(i\) at biallelic locus \(l\), so \(G_{il}\) is 0, 1, or 2. Let \(p_l\) be that allele’s frequency in the reference population and let \(q_l=1-p_l\). The centered count is \(G_{il}-2p_l\), since the expected count under the population allele frequency is \(2p_l\). The earlier material on genotype coding and allele-frequency inputs establishes this count convention.

This distinction matters for pedigrees. For full siblings with unrelated, non-inbred parents, the maternal-to-maternal and paternal-to-paternal copy pairs each have IBD probability \(1/2\); the two opposite-parent pairs have probability 0. Their average is \((1/2+1/2+0+0)/4=1/4\), so the siblings’ kinship coefficient is \(1/4\). The probability can differ across fixed copy pairs even though their average defines \(\phi_{ij}\).

Why Centered Counts Carry a Kinship Signal

Consider one locus and temporarily omit the locus subscript. Write \(A_{ia}\) for the indicator that allele copy \(a\) of person \(i\) carries the specified allele, where \(a\) is 1 or 2. Then \(G_i=A_{i1}+A_{i2}\). Each allele-copy indicator has population mean \(p\). For a particular cross-person pair of copies \(a,b\), let \(f_{ab}\) be the probability that those two copies are IBD.

Under the population model used for this moment, an IBD pair shares the same ancestral allele, which carries the specified allele with probability \(p\). If the pair is not IBD, the two copies are independent draws with allele frequency \(p\). Therefore,

The covariance formula applies to that particular copy pair and uses its own IBD probability \(f_{ab}\). Now expand the centered genotype product. Since \(G_i-2p=(A_{i1}-p)+(A_{i2}-p)\), its product with the corresponding quantity for person \(j\) is the sum of four centered copy-pair products. Taking expectations adds their four covariances:

The final equality follows from the definition \(\phi_{ij}=(f_{11}+f_{12}+f_{21}+f_{22})/4\). In the full-sibling example above, the four covariances are \(p(1-p)/2\), \(0\), \(0\), and \(p(1-p)/2\). Their sum is \(p(1-p)\), which equals \(4\phi_{ij}p(1-p)\) when \(\phi_{ij}=1/4\). Thus the genotype-product identity is about the sum of the four copy-pair covariances, not a claim that each covariance equals \(\phi_{ij}p(1-p)\).

Deriving the Pooled Method-of-Moments Estimator

At locus \(l\), the centered product is \(C_{ijl}=(G_{il}-2p_l)(G_{jl}-2p_l)\). The result just derived gives \(E[C_{ijl}]=4\phi_{ij}p_lq_l\). Method of moments equates an observed moment to its model expectation. Summing over a set of \(m\) loci gives an observed total \(\sum_l C_{ijl}\) and an expected total \(4\phi_{ij}\sum_l p_lq_l\). Solving this equation for the unknown kinship coefficient gives the pooled estimator:

This moment equation requires allele-frequency inputs with \(p_lq_l>0\) for at least some included loci, so the denominator is positive. A locus fixed for either allele has \(p_lq_l=0\) and supplies no expected centered-product signal under this model. The formula pools loci by summing their raw centered products and their corresponding frequency variances; it does not give every locus the same scale regardless of frequency.

The key assumptions for interpreting this expectation are that the allele-frequency inputs describe the population from which the relevant ancestral alleles are drawn, and that non-IBD copies are independent draws from that population at the locus. The allele is coded consistently across people, and the genotype counts represent the two allele copies at a biallelic locus. Population differences or a mismatched frequency input can change the centering and hence the expected product. The moment identity itself does not require loci to be independent. Dependence between loci does, however, affect how much information a set of markers provides; the point estimate alone does not quantify its uncertainty.

The estimator targets \(\phi_{ij}\), the kinship coefficient as defined above. A different convention sometimes reports a relationship coefficient equal to \(2\phi_{ij}\). Keep the target explicit when comparing numbers: for example, a first-degree pair has expected kinship \(1/4\), corresponding to \(2\phi_{ij}=1/2\) under that alternative scale.

Worked Estimates for Related Pairs

For a parent and child, the kinship coefficient is \(1/4\): one of the parent’s two copies is transmitted to the child, and the average IBD probability across the four cross-person copy pairs is \(1/4\). Full siblings with unrelated, non-inbred parents also have expected kinship \(1/4\), as the copy-pair calculation above shows. These are pedigree expectations; a marker-based estimate varies with the observed genotypes.

What the Estimate Can and Cannot Say

For a related pair, the expected centered products are positive under the stated model, while for an unrelated pair with \(\phi_{ij}=0\), their expectation is zero. Individual products can nevertheless be negative. In the pooled estimator, positive and negative contributions can produce \(\widehat{\phi}_{ij}<0\), and sampling variation can also produce estimates above a familiar pedigree benchmark. These values are possible for an unconstrained moment estimator; they are not probabilities and need not lie between 0 and 1.

Allele frequency determines the expected scale because \(p_lq_l\) is largest at \(p_l=0.5\) and approaches zero as an allele becomes rare or nearly fixed. A centered product at a common-allele marker and the same-sized product at a rare-allele marker therefore do not have the same expected moment scale. The denominator accounts for that difference in the pooled equation. This is why substituting an arbitrary frequency, or centering at a value inconsistent with the population, can distort the estimate.

The estimate is a measure of pairwise IBD sharing under a population model; it does not show that one person’s genotype caused the other’s, and it does not by itself identify a specific pedigree. A useful first check is to confirm the target, allele coding, and frequency inputs before interpreting a numerical result.