Method-of-Moments Estimation of Genomic Relatedness

From Standardized Genotypes to Relatedness

A genomic relationship matrix summarizes how similar individuals are across a set of genetic variants. Its off-diagonal entries are averages of standardized genotype cross-products. The method-of-moments perspective gives those averages a statistical interpretation: under a specified allele-frequency and inheritance model, their expectation is a pairwise additive relationship parameter. This interpretation also makes clear what the estimate depends on, and why its precision is limited by the number and dependence of the loci used.

The earlier material on genomic relationship matrices introduced the construction. Here the focus is its moment derivation and sampling variability. The key assumptions are that genotypes are coded consistently, the allele frequencies used for standardization are appropriate for the model, and the loci provide the dependence structure accounted for in the uncertainty calculation.

This is a method-of-moments estimator because it equates an observed sample moment—the average standardized cross-product—to its model expectation. It is not a count of shared alleles at every locus, nor does one pairwise estimate have to fall in a particular pedigree category. Finite samples can produce estimates below zero or above one.

Why the Cross-Product Estimates Relatedness

State the model before interpreting the average. For each locus, suppose the two genotypes have the same allele-frequency baseline \(p_l\), with \(E(X_{il})=E(X_{jl})=2p_l\) and \(\operatorname{Var}(X_{il})=\operatorname{Var}(X_{jl})=2p_l(1-p_l)\). Define \(A_{ij}\) as their additive relationship parameter under the model, so that \(\operatorname{Cov}(X_{il},X_{jl})=2p_l(1-p_l)A_{ij}\). For a non-inbred individual, the corresponding diagonal parameter is \(A_{ii}=1\); in the presence of inbreeding it is \(A_{ii}=1+F_i\), where \(F_i\) is the inbreeding coefficient. The off-diagonal parameter is twice the kinship coefficient: \(A_{ij}=2\phi_{ij}\).

The covariance assumption expresses the link between shared inheritance and correlated allele dosage. It is a model statement, not a guarantee for arbitrary samples: ancestry differences or a mismatched frequency baseline can also induce genotype covariance. Under the stated model, standardization gives the moment directly.

Averaging over loci preserves the expectation when the same relationship parameter applies across the analyzed loci: \(E(\hat A_{ij})=A_{ij}\). Thus the estimator is unbiased under this model, conditional on the frequencies used and the selected loci. If the target is kinship rather than additive relationship, the corresponding estimate is \(\hat\phi_{ij}=\hat A_{ij}/2\) for a distinct pair. These two scales should not be confused.

Sampling Variability Across Loci

An estimate varies because the observed cross-product at each locus varies around its expectation. Let \(Y_l=Z_{il}Z_{jl}\), so \(\hat A_{ij}=M^{-1}\sum_lY_l\). If loci are independent and have a common variance, then \(\operatorname{Var}(\hat A_{ij})=\operatorname{Var}(Y_l)/M\). More generally, when loci are independent but variances differ, the variance is \(M^{-2}\sum_l\operatorname{Var}(Y_l)\). Independence is an assumption about the analyzed loci, not a consequence of standardizing genotypes.

A useful approximation follows if the standardized genotype pair at each locus is jointly normal, with means zero, variances one, and correlation \(A_{ij}\). For a bivariate normal pair, the fourth-moment identity gives \(E(Z_{il}^2Z_{jl}^2)=1+2A_{ij}^2\). Subtracting the squared product mean yields \(\operatorname{Var}(Y_l)=1+A_{ij}^2\). Hence, for \(M\) independent loci,

Genotypes are discrete, so joint normality is an approximation, especially at a small number of loci or at very low allele frequencies. A practical alternative is to estimate the across-locus standard deviation of the observed products and divide it by \(\sqrt{M}\), provided the loci are sufficiently independent. This empirical calculation reflects variation in the actual panel, but it does not fix bias caused by a wrong model or an unsuitable frequency baseline.

Dependence, Diagnostics, and Failure Modes

Linkage disequilibrium can make nearby loci dependent. In that case, the variance includes covariance terms:

Positive covariance generally increases uncertainty relative to the independent-locus calculation. Standardizing each locus does not remove this dependence. A useful diagnostic is to compare uncertainty estimates under different reasonable, pre-specified LD-pruning or block-based approaches and check whether conclusions depend heavily on the chosen set of loci. The number of genotyped variants alone can overstate the amount of independent information.

Several checks help distinguish noisy estimates from a problematic analysis. Confirm that allele coding is aligned across individuals, that the same reference frequency and standardization are used for both members of a pair, and that loci with \(p_l=0\) or \(p_l=1\) are excluded because the denominator is zero. Inspect the distribution of locus-wise products: a few extreme values can dominate an average, and a very small panel can yield unstable estimates. Unexpectedly high relationships across many pairs may signal shared ancestry structure or a frequency-model mismatch rather than close pedigree relationships. Estimates outside familiar pedigree values are not, by themselves, proof of an error; the estimator measures a noisy genome-wide moment.

What the Estimate Does—and Does Not—Say

The estimator is attractive because it is simple, fast, and directly tied to an expectation. It can summarize genomic similarity for many pairs without assigning each pair to a pre-specified family category. Its interpretation is nevertheless conditional: the relationship scale depends on the allele-frequency reference, the genetic model, and the loci included. The next material on allele-frequency choices develops one important consequence of that dependence; here, the practical point is to state the baseline and avoid treating the estimate as an absolute, model-free quantity.

A moment estimate should therefore not be converted mechanically into a label such as “siblings” or “unrelated.” Different pairs can have similar estimates, and finite-genome variation, dependence, ancestry, and model assumptions affect the observed value. The standard error quantifies sampling variation only to the extent that its assumptions are credible; it does not account automatically for systematic bias.