Uncertainty from Small Ancestry Reference Panels

Why Reference-Panel Size Matters

An ancestry estimate depends in part on how well the reference panels represent the allele frequencies of the source populations in the model. A small panel can give a noticeably imprecise estimate of those frequencies. That uncertainty can then affect the estimated ancestry proportions of an admixed sample, even when the ancestry model and the target genotypes are held fixed. Reporting only a point estimate can therefore make a limited panel appear more informative than it is.

This tutorial focuses on sampling uncertainty: variation in estimates that would arise if a different random set of individuals were sampled from the same source population. It does not measure every source of uncertainty. In particular, resampling cannot correct a panel that is systematically unrepresentative of its intended population. The population model, choice of panels, and target data also matter.

Allele-Frequency Uncertainty

Definition: Suppose \(m\) allele copies are sampled independently from a population with frequency \(p\) for a specified allele. If \(X\) is the number of copies of that allele, then \(X\sim\operatorname{Binomial}(m,p)\), and the sample allele-frequency estimate is \(\hat p=X/m\). Under this model, \(E(\hat p)=p\) and \(\operatorname{Var}(\hat p)=p(1-p)/m\).

For \(n\) diploid individuals, there are \(m=2n\) allele copies at a locus. Treating those copies as independent is appropriate under the simple binomial sampling model, such as random sampling from a homogeneous population with independent allele copies. Relatedness, inbreeding, or additional population structure can make copies or individuals dependent; in those settings, the binomial variance can understate uncertainty. An individual-level resampling scheme can preserve the two alleles and other measured data from each sampled person.

The plug-in standard error is \(\sqrt{\hat p(1-\hat p)/m}\). It is useful for seeing how precision changes with sample size, but a symmetric interval based on it can perform poorly when the frequency is near zero or one, especially with a small sample. A Wilson score interval is one standard alternative for a binomial proportion.

A Wilson Score Interval

For a two-sided confidence level of approximately 95%, let \(z=1.96\), the 97.5th percentile of the standard normal distribution. The Wilson interval is obtained by considering candidate values \(p\) that satisfy the score-test inequality \(m(\hat p-p)^2/[p(1-p)]\le z^2\). Solving this quadratic inequality for \(p\) gives the endpoints below.

\frac{\hat p+\frac{z^2}{2m}}{1+\frac{z^2}{m}} \ \pm\ \frac{z}{1+\frac{z^2}{m}} \sqrt{\frac{\hat p(1-\hat p)}{m}+\frac{z^2}{4m^2}}

The expression before the plus-or-minus sign is the interval center; the remaining term is its half-width. Unlike the simple symmetric plug-in interval, the Wilson interval does not generally center on \(\hat p\). Its endpoints remain within the possible range from zero to one.

Worked Example: An Allele-Frequency Interval from a Small Panel

In hypothetical reference-panel data, 15 copies of a specified allele are observed among 50 sampled allele copies. Estimate the allele frequency and construct a 95% Wilson interval. Assume the 50 copies are independent draws from a single population.

The estimate is \(\hat p=15/50=0.300\). With \(m=50\) and \(z=1.96\), \(z^2=3.8416\), so the denominator in the Wilson expression is \(1+3.8416/50=1.076832\). The center is \((0.300+3.8416/(2\cdot50))/1.076832=0.338416/1.076832\approx0.31427\).

The half-width is \(1.96\sqrt{0.300(0.700)/50+3.8416/(4\cdot50^2)}/1.076832\). Inside the square root, \(0.300(0.700)/50=0.004200\) and \(3.8416/10000=0.00038416\), giving a square root of approximately \(0.06771\). Thus the half-width is approximately \(0.12324\), and the endpoints are \(0.31427-0.12324=0.19103\) and \(0.31427+0.12324=0.43751\).

Rounded to three decimal places, the interval is \([0.191,0.438]\). Under the binomial sampling model, this procedure has approximately 95% coverage over repeated samples. It is not a statement that there is a 95% probability the fixed population frequency lies in this particular interval.

Panel Size and Precision

Under independent binomial sampling, the standard error of \(\hat p\) is approximately \(\sqrt{p(1-p)/m}\). The frequency estimate is therefore more precise, on average, with more sampled allele copies. If the number of copies is multiplied by four while the underlying frequency is unchanged, the standard error is divided by two. This is a statement about expected sampling variability; a particular larger sample can still produce a less convenient estimate or interval.

Worked Example: Comparing Two Reference-Panel Sizes

Consider two hypothetical panels that both produce an estimated allele frequency of \(0.300\), but the first has 50 sampled allele copies and the second has 200. Compare their plug-in standard errors to assess the effect of panel size. Assume the binomial model is appropriate for both panels.

For the panel with 50 copies, the estimated standard error is \(\sqrt{0.300(1-0.300)/50}=\sqrt{0.004200}\approx0.06481\). For the panel with 200 copies, it is \(\sqrt{0.300(1-0.300)/200}=\sqrt{0.001050}\approx0.03240\).

The second panel has four times as many allele copies and about half the estimated standard error. Both standard errors are in allele-frequency units, which are dimensionless proportions. These calculations compare precision under the stated model; they do not establish that either panel represents its intended source population well.

How Frequency Uncertainty Affects an Ancestry Proportion

A simple two-source, one-locus model makes the propagation of reference-panel uncertainty explicit. Let \(p_A\) and \(p_B\) be the allele frequencies in two source populations, and let \(r\) be the allele frequency in a target sample. If the target is a mixture with ancestry proportion \(q\) from source A and \(1-q\) from source B, the model is \(r=q p_A+(1-q)p_B\). When \(p_A\ne p_B\), solving for \(q\) gives \(q=(r-p_B)/(p_A-p_B)\).

This equation is a simplified illustration, not a complete genome-wide ancestry analysis. If \(p_A\) and \(p_B\) are estimated from finite reference panels, then the resulting estimate of \(q\) is uncertain even if \(r\) is treated as fixed. When the two source frequencies are close, the denominator \(p_A-p_B\) is small, so frequency errors can have a large effect on the estimated proportion.

Approximation: If the two reference-panel frequency estimates are independent, their variances are \(p_A(1-p_A)/m_A\) and \(p_B(1-p_B)/m_B\), and \(r\) is held fixed, a first-order delta-method approximation is \(\operatorname{Var}(\hat q)\approx [q/(p_A-p_B)]^2\operatorname{Var}(\hat p_A)+[(1-q)/(p_A-p_B)]^2\operatorname{Var}(\hat p_B)\). This approximation uses the local slopes of the function defining \(q\); it does not include sampling uncertainty in \(r\).

Worked Example: Propagating Reference-Frequency Uncertainty

In hypothetical data, reference panel A has 70 copies of the allele among 100 sampled copies, and panel B has 30 among 100. A target sample has an allele frequency of \(r=0.540\). For this calculation, treat \(r\) as fixed and assume independent binomial sampling in the two panels. Estimate \(q\) and approximate its standard error from reference-panel sampling alone.

The reference estimates are \(\hat p_A=70/100=0.700\) and \(\hat p_B=30/100=0.300\). Substitution into the mixture equation gives \(\hat q=(0.540-0.300)/(0.700-0.300)=0.240/0.400=0.600\).

The estimated variances of the reference frequencies are \(\widehat{\operatorname{Var}}(\hat p_A)=0.700(0.300)/100=0.002100\) and \(\widehat{\operatorname{Var}}(\hat p_B)=0.300(0.700)/100=0.002100\). The slopes in the delta-method expression are \(\hat q/(0.700-0.300)=0.600/0.400=1.5\) for panel A and \((1-\hat q)/(0.700-0.300)=0.400/0.400=1.0\) for panel B. Therefore, \(\widehat{\operatorname{Var}}(\hat q)\approx1.5^2(0.002100)+1.0^2(0.002100)=0.004725+0.002100=0.006825\), and \(SE(\hat q)\approx\sqrt{0.006825}=0.08261\).

A rough normal approximation gives \(0.600\pm1.96(0.08261)\), or approximately \([0.438,0.762]\). This is an approximate interval for the ancestry proportion in the simplified model, based only on reference-panel uncertainty. It omits target-sample uncertainty, uses one locus, and can be unreliable near the boundaries or when the source frequencies are poorly separated.

Resampling Reference Panels

For a multi-locus ancestry model, analytical propagation may be inconvenient: estimates can depend on many allele frequencies, the fitting procedure may be nonlinear, and proportions must remain nonnegative and sum to one. A practical alternative is to resample reference individuals and refit the same ancestry model. Each resampled panel represents one plausible sample from the observed reference panel, conditional on that panel being representative of its source population.

Fit the model to the observed panels.
Record the ancestry-proportion estimate for the target sample and keep the loci, model settings, and data-processing choices fixed.

Resample individuals within each reference group.
Sample the same number of individuals as in that group, with replacement. Resampling individuals rather than treating two alleles from one person as unrelated preserves their pairing and any measured genotype-level information.

Refit and repeat.
Refit the ancestry model for each resampled set of panels. Use a consistent alignment of ancestry labels so that a proportion for one source is compared with that same source across runs.

Summarize the resampled estimates.
Use their spread as a measure of sensitivity to reference sampling. For example, percentile endpoints can be used for a bootstrap interval, while the full distribution can reveal skewness or boundary effects.

If the goal is to isolate uncertainty from the reference panels, keep the target sample fixed during these repetitions. If target sampling uncertainty is also part of the question, resample target individuals as well, and state that the resulting uncertainty includes both sources. In either case, resampling does not account automatically for uncertainty due to model choice, omitted source populations, or a mismatch between a reference panel and its intended population.

Worked Example: A Percentile Interval from Panel Resampling

Suppose a hypothetical ancestry analysis estimates the target's proportion from source A as \(0.580\) using the observed panels. Investigators resample individuals within each of the two reference groups, hold the target data fixed, and refit the model 2,000 times. After aligning labels, the 50th and 1,950th sorted estimates are \(0.420\) and \(0.750\), respectively.

For a 95% percentile interval with \(B=2{,}000\) resamples, the lower percentile is 0.025 and the upper percentile is 0.975. Using the nearest-rank convention, the corresponding ranks are \(\lceil0.025(2{,}000)\rceil=50\) and \(\lceil0.975(2{,}000)\rceil=1{,}950\). The interval is therefore \([0.420,0.750]\).

This interval describes the spread of fitted ancestry proportions generated by resampling the observed reference individuals, conditional on the fixed target data and the fitting procedure. It is substantially wider than a single point estimate might suggest. It is not a guarantee that the interval captures all uncertainty about ancestry, and the stated resampling procedure does not include target sampling variation.

Interpreting and Reporting the Uncertainty

A useful report identifies the source of uncertainty, the method used, and the assumptions that make the interpretation reasonable. For an allele-frequency interval, state the allele count and the number of sampled copies, along with the interval method. For an ancestry-proportion interval, state whether it came from an analytical approximation or reference-panel resampling, and whether target-sample uncertainty was included.

For multiple ancestry groups, each fitted vector of proportions should satisfy the model constraints: every component is between zero and one, and the components sum to one. Marginal intervals for separate components do not generally sum to one, because each interval is calculated separately. Do not combine marginal endpoints as though they formed one possible ancestry vector; examine the joint resampled vectors when the joint uncertainty matters.

Intervals describe sampling variability under their stated model and sampling design. They do not establish that a reference panel is representative, that the chosen ancestry model is correct, or that a population label has a simple biological meaning. Small panels make estimates less precise, but a large panel can still be systematically mismatched to its target population.

Common Pitfall

Wrong reasoning: “The estimated allele frequency is 0.300, so the reference frequency is known to be 0.300; any uncertainty in ancestry must come from the target sample.”
Correction: The estimate is based on a finite sample. Under binomial sampling, its estimated standard error is \(\sqrt{0.300(0.700)/m}\), which is nonzero for finite \(m\). Since ancestry estimates use reference frequencies, their uncertainty can include reference-panel sampling variation even when the target data are held fixed.
Related caution: A narrow resampling interval is not evidence against panel mismatch. Resampling observed individuals measures variability around the observed panel, not systematic differences between that panel and its intended population.

Key takeaway: Reference-panel sample size affects the precision of allele-frequency estimates, and those frequency estimates can affect ancestry proportions. Use intervals or resampling to quantify the uncertainty included by a clearly stated model and design; do not mistake that uncertainty measure for protection against model error or panel mismatch.

Check Your Understanding

Work through these questions using the assumptions stated in each problem.

A panel has 20 copies of an allele among 80 sampled allele copies. What is the sample allele-frequency estimate, and what binomial model would be used to form a score interval?
Under independent binomial sampling, what happens to the standard error of an allele-frequency estimate if the number of sampled copies is multiplied by four while the frequency is unchanged?
In the two-source mixture equation, why can a small difference between source allele frequencies make the estimated ancestry proportion sensitive to reference-panel sampling?
In a reference-panel bootstrap, why might investigators keep the target data fixed? What additional uncertainty would be included if they also resampled target individuals?
Explain why a narrow reference-panel bootstrap interval does not show that the reference panels are free of systematic mismatch.