Uncertainty In Pairwise Relatedness Estimates

Why a Relatedness Estimate Has Uncertainty

A pairwise relatedness estimate is calculated from observed genetic markers, not from every possible genetic realization of the pair. If a different set of markers or allele copies had been observed, the estimate could differ. Its uncertainty therefore depends on how variable the marker-level contributions are and on how much independent information those markers provide.

A useful starting point is to treat an estimate as an average of contributions across markers. When contributions are independent and have a common variance, averaging more markers reduces sampling variance. When allele frequencies differ across markers, their contributions need not have the same variance: a marker with an allele frequency near zero or one can carry less stable information for a frequency-standardized calculation.

The exact variance depends on the estimator and its assumptions. The derivation below uses a simple allele-copy model to make the frequency dependence explicit. It is an illustration of the sampling-variance calculation, not a claim that every software estimator has this exact variance. In practice, uncertainty calculations should match the estimator, its inputs, and the marker-sampling assumptions.

A Frequency-Sensitive Allele-Copy Contribution

Consider one allele copy from each of two individuals at a marker. Let \(A\) and \(B\) equal 1 when the allele copy carries the allele of interest and 0 otherwise. Suppose both copies have allele frequency \(p\), with \(0<p<1\). Define their standardized product as

In this model, the target \(\theta\) is the correlation between the two allele-copy indicators:

Thus \(E[(A-p)(B-p)]=\theta p(1-p)\), so \(E(Y)=\theta\). The standardized product is centered and scaled so that its expectation targets the same \(\theta\) at each marker, even when marker frequencies differ.

The variance of \(Y\) depends on both \(p\) and \(\theta\). The following short derivation makes that dependence explicit. Write \(a=p(1-p)\). For a Bernoulli variable, \((A-p)^2=a+(1-2p)(A-p)\), and the same identity holds for \(B\). Taking expectations of the product of these squared terms gives

Dividing by \(a^2\) gives \(E(Y^2)=1+\theta(1-2p)^2/[p(1-p)]\). Since \(E(Y)=\theta\), subtracting the squared mean yields

This result is for the stated allele-copy model. It shows that, for a nonzero positive \(\theta\), the per-marker variance can become large when \(p\) is close to 0 or 1. At \(p=0.5\), the frequency-dependent term is zero. When \(\theta=0\), the variance in this model is 1 for every \(p\). These details matter: it is not correct to say that allele frequency always changes variance in the same way for every target or every estimator.

Combining Contributions Across Markers

Suppose the markers are independent under the sampling model, each contribution has expectation \(\theta\), and marker frequencies \(p_l\) are treated as known. For an equally weighted estimate, the variance of the average is the sum of the contribution variances divided by \(M^2\). The standard error is the square root:

If all markers have the same variance \(v\), this simplifies to \(\operatorname{Var}(\hat{\theta})=v/M\) and \(SE(\hat{\theta})=\sqrt{v/M}\). Thus, under independence and equal per-marker information, multiplying the marker count by four halves the standard error. This square-root relationship is not a guarantee that any particular collection of four times as many markers will have four times the information: the marker contributions must also satisfy the assumed independence and variance conditions.

Confidence Intervals and Their Interpretation

When the estimate is approximately normally distributed around \(\theta\), an approximate 95% confidence interval is \(\hat{\theta}\pm1.96SE(\hat{\theta})\). This approximation is more plausible when many independent contributions are averaged and no small number of contributions dominates the total variance. It can be poor for a small marker count, strongly skewed contributions, or estimates constrained to a limited range.

The interval procedure has a repeated-sampling interpretation: if the assumptions hold and the same method is used across repeated samples, approximately 95% of intervals constructed this way cover the fixed target \(\theta\). It is not a statement that there is a 95% probability that the fixed target lies in this particular interval.

Comparing Two Estimates

To compare two pairwise estimates, focus on their difference and its standard error. Let \(\hat{\theta}_1\) and \(\hat{\theta}_2\) estimate targets \(\theta_1\) and \(\theta_2\). The variance of their difference is

If the estimates are independent, the covariance is zero, so the standard error of the difference is the square root of the sum of their variances. If they are dependent, that shortcut is not justified. For example, two pairs that share an individual may have correlated estimates; a comparison then needs a covariance estimate or a method that accounts for their dependence.

Assumptions and Common Pitfalls

A standard error is only as useful as the model behind it. In the variance derivation and examples, the allele frequency is treated as known, contributions have the stated expectation, and markers are independent. The resulting uncertainty does not automatically include error from estimating allele frequencies, missing genotypes, genotype calling, or violations of the model. Those sources require additional methods or assumptions.

The independent-marker calculation also cannot be used unchanged when marker contributions are correlated. In general, the variance of an average includes covariance terms:

A calculation that omits nonzero covariance may understate or overstate uncertainty, depending on its sign and size. Likewise, when comparing two estimates, overlapping confidence intervals are not themselves a direct test of the difference; the relevant calculation uses the uncertainty of the difference, including covariance when necessary.

• Mistake: assuming more markers always reduce uncertainty by exactly \(1/\sqrt{M}\). That rate follows for independent contributions with comparable variance. Unequal variances or correlated contributions change the calculation.
• Mistake: treating allele frequency as irrelevant after standardization. Standardization can equalize the expected contribution, but it does not ensure equal variances. The derived formula shows frequency dependence for positive \(\theta\).
• Mistake: comparing two point estimates without their uncertainty. A larger estimate is not automatically meaningfully different. Calculate the standard error of the difference and interpret an interval or test in context.
• Mistake: treating a confidence interval as a probability statement about a fixed target. The 95% describes the long-run coverage of the procedure under its assumptions.