Worked Analysis of an Admixed Cohort

From Genotypes to an Admixed-Cohort Analysis

An ancestry analysis is not a single calculation. Its results depend on which samples and genetic markers are retained, how population structure is inspected, which reference samples represent the possible sources, and how the final estimates respond to reasonable alternatives. These choices matter especially when the cohort and available reference data have uneven geographic representation.

The worked analysis below uses hypothetical data throughout. It illustrates a practical sequence, not a universal set of filtering thresholds or a claim that broad geographic groups are discrete biological populations. The analysis distinguishes two roles for the methods: principal component analysis (PCA) describes major patterns of genetic variation, while a reference-based model estimates proportions under an explicitly chosen set of source populations.

Definition: A global ancestry proportion is a model-based summary of an individual's genome-wide ancestry relative to specified reference sources. For two sources, write the proportion from source A for individual \(i\) as \(q_i\), and the proportion from source B as \(1-q_i\), where \(0\leq q_i\leq1\). The estimate is conditional on the reference samples, markers, and model; it is not a direct measurement of every segment of the genome.

Step 1: Filter Samples and Variants

Begin by checking the genotype data rather than interpreting structure immediately. Sample-level checks can identify individuals with substantial missing genotype data or inconsistent records. Variant-level checks can remove poorly measured markers, markers with very low frequency for the planned analysis, or markers that cannot be reliably harmonized across datasets. The precise thresholds depend on study design, platform, and analytic goal; they should be specified before examining results where possible.

For PCA, centering and scaling genotypes and pruning correlated markers are standard preparations. Correlated markers can give one genomic region disproportionate influence on the components. The earlier material on centering, scaling, and LD pruning before PCA gives the detailed calculations. In a worked cohort analysis, report the number of samples and markers at each stage so that filtering is auditable.

Hardy–Weinberg checks require particular care in an admixed sample. A pooled sample may depart from Hardy–Weinberg proportions because it combines groups with different allele frequencies, even if the groups separately satisfy the model. Thus, a pooled departure is not by itself proof of a genotyping error; use suitable comparison groups and interpret the check in the context of structure.

Worked Example: A Sequential Genotype-Filtering Summary

Suppose a hypothetical study begins with 1,240 participants and 600,000 autosomal variants. The prespecified sample filter removes 40 participants whose genotype missingness exceeds the study's 3% threshold. The retained cohort is \(1{,}240-40=1{,}200\) participants. Assume no additional sample is removed at this stage.

Apply sequential variant filters to the 600,000 variants. First remove 24,000 with missingness above 2%, leaving \(600{,}000-24{,}000=576{,}000\). Next remove 18,000 of the remaining variants for falling below the chosen minor-allele-frequency threshold, leaving \(576{,}000-18{,}000=558{,}000\). Finally remove 8,000 remaining variants that cannot be harmonized reliably with the reference data, leaving \(558{,}000-8{,}000=550{,}000\).

After centering, scaling, and LD pruning, suppose 42,000 markers remain for PCA. This last number is not obtained by subtracting an assumed fixed percentage: pruning depends on the observed correlation pattern. The analysis report should distinguish the 550,000 variants passing the stated quality filters from the 42,000 markers used for this PCA. It should also state that the filtering thresholds were analytic choices, not universal standards.

Step 2: Inspect Structure With PCA

PCA summarizes major directions of variation in the genotypes supplied to it. The coordinates of an individual on a principal component (PC) describe position along a direction of genetic variation in that analysis. They are not ancestry proportions, and a PC axis does not acquire a geographic interpretation solely because a cohort or reference group has been assigned a place name. Interpretation depends on the samples, markers, and analysis used to construct the axes.

When sample sizes differ greatly among geographic sources, the largest group can exert substantial influence on the components and on how finely its internal variation is represented. Unequal sampling can therefore affect the apparent structure. A balanced reference subset is one useful diagnostic, but it is not automatically the uniquely correct analysis: reducing a group also discards information. Compare results with the full available panel and document the sampling scheme.

Worked Example: Reading PCA Under Uneven Representation

Suppose the 1,200-person hypothetical cohort includes 720 participants from a coastal region, 360 from an inland region, and 120 from a highland region. Their shares are \(720/1{,}200=0.60\), \(360/1{,}200=0.30\), and \(120/1{,}200=0.10\). The coastal group therefore accounts for 60% of the cohort.

A PCA of the cohort and reference samples places the three reference groups at different locations on PC1 and PC2. The cohort participants occupy a broad range between and around those locations. This pattern is consistent with genetic variation shared across the sampled groups, but the plot alone does not supply a percentage of ancestry for any individual. In particular, a coordinate such as PC1 = 0.4 is not a 40% ancestry estimate: component scores depend on the fitted axes and their scaling.

For a diagnostic comparison, randomly select 120 cohort participants from each region, giving a balanced subset of 360. Each region then has a share of \(120/360=1/3\). Refit PCA using the same marker-processing rules and compare the broad arrangement of samples with the full-cohort result. If the arrangement changes materially, report that dependence on sample composition. If the balanced plot looks clearer, that is not evidence that the full cohort is wrong; it shows that the visual summary is sensitive to representation. Keep track of whether each plot uses newly fitted axes or projects samples onto fixed axes, since these are different comparisons.

Step 3: Choose and Check Reference Panels

An ancestry model needs reference samples whose allele frequencies are used to represent its possible sources. Selection should be guided by the scientific question and by the available sampling: consider geographic coverage within each proposed source, sample quality, marker overlap, and whether the reference groups are appropriate for the cohort. A panel labelled with a broad geographic name may not represent all the genetic variation found across that region.

Unequal reference-panel sizes matter too. A large panel can estimate its group's allele frequencies more precisely than a small panel, but it can also dominate an analysis if the model or PCA construction gives sample counts direct influence. Balancing the number of references, using appropriate model weights, and comparing the result with the unbalanced panel are possible checks. These choices answer somewhat different questions, so the analysis should state what each panel represents rather than presenting one as universally definitive.

Before estimating proportions, align the variants and alleles across cohort and references, and check that the same allele is being counted in every dataset. Otherwise an allele-frequency contrast can be reversed by inconsistent coding. The reference-panel choice and data harmonization should be fixed and documented before interpreting individual estimates.

Step 4: Estimate Global Ancestry Proportions

A simple two-source model makes the dependence of an estimate on reference frequencies explicit. At a biallelic locus \(j\), let \(p_{Aj}\) and \(p_{Bj}\) be the frequencies of the counted allele in reference sources A and B. Under a two-source mixture with proportion \(q_i\) from A, the expected allele frequency for individual \(i\)'s ancestry mixture at that locus is

p_{ij}=q_i p_{Aj}+(1-q_i)p_{Bj}.

This is a mixture model, not a statement that a person's genome is literally a single random draw from one of two homogeneous groups. In the simple dosage-based illustration below, the observed allele frequency is the average allele dosage divided by two. A least-squares estimate chooses \(q_i\) to make the observed frequencies close to the mixture frequencies across loci. This illustrative estimator assumes the two references adequately represent the specified sources, the loci are consistently coded, and the markers provide usable information about the source contrast. With many markers, a fitted model can use genotype likelihoods and account for sampling variation; its precise assumptions depend on the method.

Estimation formula: If \(y_j\) is the observed counted-allele frequency at locus \(j\), the equal-weight least-squares estimate for two sources is the value of \(q\) in the interval from 0 to 1 that minimizes \(\sum_j [y_j-\{p_{Bj}+q(p_{Aj}-p_{Bj})\}]^2\). When the unconstrained solution lies in that interval, it is

\hat q = \frac{\sum_j (p_{Aj}-p_{Bj})(y_j-p_{Bj})} {\sum_j (p_{Aj}-p_{Bj})^2}.

Worked Example: Estimating a Two-Source Proportion

Consider three hypothetical, consistently coded markers. The reference frequencies for source A are 0.80, 0.20, and 0.60; those for source B are 0.20, 0.70, and 0.30. In a hypothetical admixed sample, the observed allele dosages averaged over individuals are 1.50, 0.70, and 1.10 copies per person. Dividing each average dosage by two gives observed allele frequencies \(y_1=0.75\), \(y_2=0.35\), and \(y_3=0.55\).

The source contrasts \(p_{Aj}-p_{Bj}\) are \(0.60\), \(-0.50\), and \(0.30\). The numerator of the estimate is \(0.60(0.75-0.20)+(-0.50)(0.35-0.70)+0.30(0.55-0.30)=0.330+0.175+0.075=0.580\). The denominator is \(0.60^2+(-0.50)^2+0.30^2=0.360+0.250+0.090=0.700\). Therefore \(\hat q=0.580/0.700=0.829\), to three significant figures. This is within the allowed interval, so no boundary adjustment is needed. The estimated proportion from source B is \(1-0.829=0.171\).

The calculation illustrates the model, not a reliable three-marker analysis. With only three loci, sampling variation or a poorly chosen marker could have a large effect. An actual analysis would use a much broader, appropriately pruned or otherwise modeled marker set and would report uncertainty and sensitivity to the references. The estimate is a genome-wide model summary relative to these two sources; it does not identify ancestry along particular chromosome segments.

Step 5: Check Sensitivity and Report the Analysis

A sensitivity check repeats the analysis under reasonable alternatives and compares the resulting summaries. For an unevenly represented cohort, useful alternatives include a geographically balanced reference subset, the full available panel, a panel with different within-region representation, and a different defensible set of markers. Keep other choices fixed when possible so that a difference can be associated with the choice being examined. If several choices change at once, it is harder to identify why results differ.

Compare not only a cohort-wide average but also the distribution of individual estimates and whether the same participants have notably different estimates under alternatives. State the reference groups and markers used, how representation was handled, and the size and direction of any changes. Stability across the checks indicates robustness to those particular alternatives; it does not prove that the references exhaust all relevant sources or that the model is correct.

Worked Example: Quantifying Reference-Panel Sensitivity

Suppose a hypothetical two-source analysis estimates a mean source-A proportion of 0.62 using the full reference panel. Repeating the same analysis with geographically balanced references gives 0.60, and using a second defensible panel gives 0.66. The balanced-panel change relative to the full-panel estimate is \(0.60-0.62=-0.02\), or a decrease of 2 percentage points. The second-panel change is \(0.66-0.62=0.04\), or an increase of 4 percentage points.

These are modest changes in the cohort mean, but the mean alone can conceal larger changes for some participants. The analyst should also compare individual estimates and their ranking or broad distribution across the runs, without assuming that rank agreement proves accuracy. If a prespecified marker-subset check yields a mean of 0.61, its difference from the full-panel result is \(0.61-0.62=-0.01\), or 1 percentage point lower. Together, these results suggest limited movement in the mean across these particular checks. They do not establish that 0.62 is the true ancestry proportion, because all runs may share the same limitations in the source definitions or model.

Why the Workflow Matters

The order of analysis helps separate data quality, structure, and model dependence. Filtering establishes which genotypes enter the analysis. PCA offers a view of major variation and can reveal that the cohort is not well summarized by a single homogeneous sample. Reference panels then define the sources against which proportions are estimated. Sensitivity checks ask whether conclusions are driven by reasonable changes in representation or marker choice.

A common mistake is to interpret a clean-looking PCA plot as proof that the ancestry model is valid, or to treat a model estimate as independent of its references. The correction is to treat PCA as a description of variation in the analyzed samples and ancestry proportions as estimates conditional on the selected model and references. Agreement between methods can be informative, but the methods do not become interchangeable: PC coordinates are not percentages, and a reference-based proportion is not established merely because it aligns with a PC plot.

Document filtering.
Report sample and variant counts, thresholds, and the marker set used for each analysis.

Inspect structure.
Use PCA to examine broad variation, noting the influence of unequal sample representation.

Define the references.
Describe geographic coverage and harmonization, and explain how panel sizes were handled.

Estimate and compare.
Report model-based proportions and compare them under defensible reference and marker alternatives.

Key takeaway: An admixed-cohort analysis is a chain of conditional choices. Clear filtering, cautious PCA interpretation, explicit reference-panel selection, and sensitivity checks make the ancestry estimates easier to evaluate and reproduce.

Check Your Understanding

Try these questions using the workflow and definitions above.

Why can a departure from Hardy–Weinberg proportions in a pooled admixed cohort arise even without a genotyping error?
A participant has PC1 = 0.4. Why is it incorrect to report this as 40% ancestry from a source population?
In the two-source model, what does \(q_i\) represent, and which choices condition the interpretation of its estimate?
A balanced-reference analysis changes a cohort mean from 0.62 to 0.60. What is the change in percentage points, and what does that sensitivity result not establish?
Why should the analysis report the number of variants used for PCA separately from the number passing initial quality filters?