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Relatedness and Kinship Estimation · Tutorial 170 of 989

Missing Genotypes And Pairwise Denominators

Compare marker-denominator choices, quantify the uncertainty caused by uneven genotype overlap, and diagnose relatedness estimates that may reflect missing-data patterns.

Intermediate 11 min read

What You'll Learn

  • Define complete-case and pairwise-available marker sets for a genomic relationship estimate
  • Calculate pairwise estimates using the number of jointly observed markers as the denominator
  • Explain how overlap and linkage disequilibrium affect estimator precision
  • Diagnose how sample-specific missingness can alter the marker mix and apparent relatedness
  • Identify when pairwise estimates may not form a positive semidefinite relationship matrix
  • Choose checks that make pairwise relatedness estimates more comparable

Why the Denominator Matters

A genomic relationship estimate summarizes genotype similarity across markers. In real data, some genotypes are missing. The calculation must then specify which markers contribute to each estimate and how many observed markers form its denominator. Using one denominator for every pair can discard usable data; using a different denominator for each pair can make estimates less directly comparable.

The central distinction is between a complete-case marker set, containing markers observed in every sample being analyzed, and a pairwise-available marker set, containing markers observed in both members of a particular pair. Neither choice is automatically best. The appropriate calculation depends on the missingness pattern and on whether the resulting estimates will be compared or used in a matrix-based analysis.

Definition: Let \(g_{ik}\in\{0,1,2\}\) be the observed count of a reference allele for sample \(i\) at marker \(k\), and let \(p_k\) be the allele frequency used for that marker. Define the standardized genotype \(z_{ik}=(g_{ik}-2p_k)/\sqrt{2p_k(1-p_k)}\), when \(0<p_k<1\). The marker contribution to the standardized genomic relationship estimate for samples \(i\) and \(j\) is \(Y_{ijk}=z_{ik}z_{jk}\). A missing genotype has no \(z_{ik}\) value and contributes no product.

For a fixed marker panel of \(M\) markers, a standard off-diagonal genomic relationship estimate is the average of these products. Write \(\mathcal C\) for markers with observed genotypes in every sample, and \(\mathcal O_{ij}\) for markers observed in both samples \(i\) and \(j\). Then the two denominator choices are:

$$ \widehat K_{ij}^{\mathrm{CC}}= \frac{\sum_{k\in\mathcal C}Y_{ijk}}{|\mathcal C|} \qquad\text{and}\qquad \widehat K_{ij}^{\mathrm{pair}}= \frac{\sum_{k\in\mathcal O_{ij}}Y_{ijk}}{|\mathcal O_{ij}|}. $$

Here “CC” denotes complete-case markers and “pair” denotes pairwise-available markers. These formulas assume the same allele-frequency inputs and marker scaling are used in both calculations. The denominator is the number of markers actually contributing to that estimate—not the number of markers on the original panel. An estimate with no jointly observed markers is undefined.

Worked Examples of the Two Denominators

Worked Example: Complete-Case and Pairwise Estimates

Consider three hypothetical samples, A, B, and C, and six markers. A and B have observed genotypes at all six markers. C is missing at markers 4, 5, and 6. Suppose the standardized genotype products \(Y_{AB,k}\) are \(0.4,\ 0.2,\ -0.2,\ 0,\ 0.6,\ -0.4\), respectively. The allele-frequency inputs and genotype coding are held fixed.

For a complete-case calculation across all three samples, only markers 1, 2, and 3 qualify. Their products sum to \(0.4+0.2-0.2=0.4\), so:

$$ \widehat K_{AB}^{\mathrm{CC}}=\frac{0.4+0.2-0.2}{3} =\frac{0.4}{3}=0.1333. $$

For the pairwise calculation, A and B share observed genotypes at all six markers. Their products sum to \(0.4+0.2-0.2+0+0.6-0.4=0.6\), giving:

$$ \widehat K_{AB}^{\mathrm{pair}}=\frac{0.6}{6}=0.1000. $$

The estimates differ because they average different marker sets, not because either sum was divided incorrectly. If marker contributions vary, changing the set can change the observed average. Pairwise availability preserves four contributions that the cohort-wide complete-case rule discards.

A complete-case marker set is common across all pairs, which makes the marker basis consistent. Pairwise sets can be larger, but their composition and size may vary from pair to pair. A useful report therefore describes the denominator rule and gives the overlap count, rather than presenting the estimate alone.

Worked Example: Unequal Overlap and Precision

Suppose two hypothetical sample pairs are unrelated, and for this illustration assume their standardized products are independent across markers, have mean zero, and have variance 1. Pair A–B has 900 jointly observed markers; pair C–D has 225. Under these simplifying assumptions, the standard error of an average of \(m\) products is \(1/\sqrt m\).

For A–B, the standard error is \(1/\sqrt{900}=0.03333\). For C–D, it is \(1/\sqrt{225}=0.06667\). Thus the estimate based on 225 independent markers has twice the standard error of one based on 900 markers.

Suppose the observed estimates are \(0.040\) for A–B and \(0.080\) for C–D. Approximate 95% intervals using the normal approximation are:

$$ 0.040\pm1.96(0.03333)=(-0.0253,\ 0.1053) $$
$$ 0.080\pm1.96(0.06667)=(-0.0507,\ 0.2107). $$

These are illustrative uncertainty intervals under the stated independent-product model, not universal intervals for genomic relationship estimates. The assumptions used for this calculation are explicit: independent marker products, variance 1, and a sufficiently large marker count for a normal approximation. The wider interval for C–D shows why an estimate with fewer observed markers should not be treated as equally precise merely because it is calculated on the same numerical scale.

How Missingness Affects Precision

With independent marker contributions of common variance \(\sigma_Y^2\), the variance of their average over \(m\) markers is \(\sigma_Y^2/m\), and its standard error is \(\sigma_Y/\sqrt m\). This explains the basic precision cost of a small pairwise denominator. In genomic data, however, nearby markers may be correlated through linkage disequilibrium. The raw number of observed markers can then overstate the amount of independent information.

If the observed markers are correlated, precision depends on their correlation pattern as well as their count. An effective marker count is a way of expressing how much independent information the correlated set carries; it is generally smaller than the raw count when correlations are substantial. The previous material on linkage disequilibrium and effective marker counts is relevant here: two pairs with the same number of observed markers can have different precision if their observed sets have different correlation structures.

The simple standard-error formula also assumes that the marker products have a common mean and variance. If those assumptions do not hold, the formula is only a rough guide. In practice, report the number of jointly observed markers, inspect the distribution of pairwise overlap, and use an uncertainty method appropriate to the estimator and marker dependence. A smaller denominator signals less information, but it does not by itself determine the exact standard error.

When Pairwise Availability Can Mislead

Pairwise deletion is safest when genotype missingness is unrelated to the genotype products being averaged and the observed marker sets represent comparable information. If certain samples tend to be missing at particular markers, different pairs can be evaluated on systematically different parts of the panel. A pairwise estimate may then reflect both genetic similarity and marker selection.

The basic mechanism can be stated directly. Let \(R_{ijk}=1\) when both genotypes for pair \(i,j\) are observed at marker \(k\), and \(R_{ijk}=0\) otherwise. The estimate is an average of \(Y_{ijk}\) only where \(R_{ijk}=1\). If inclusion depends on the value of \(Y_{ijk}\), the observed average can differ systematically from the average over the intended panel. Even when missingness is not literally determined by a pair’s product, technical or sample-specific patterns can select different marker subsets. Differences between estimates may then be mistaken for differences in relatedness.

Worked Example: A Pair-Specific Marker Pattern

Consider a hypothetical ten-marker panel for pair A–B. Suppose the products that would be obtained if all genotypes were observed are \(0.5\) at markers 1–4, \(-0.5\) at markers 5–8, and \(0\) at markers 9–10. Across the full panel, the products sum to \(4(0.5)+4(-0.5)+2(0)=0\), for a mean of \(0/10=0\).

Now suppose A and B both have observed genotypes at markers 1–4 and 9–10, but one or both genotypes are missing at markers 5–8. Their pairwise estimate uses six markers:

$$ \widehat K_{AB}^{\mathrm{pair}}= \frac{4(0.5)+2(0)}{6} =\frac{2}{6}=0.3333. $$

The apparent positive estimate comes from the selected marker subset. For comparison, suppose markers 9 and 10 are also the only markers complete across every sample in a cohort, and A–B have product zero at both. The complete-case estimate is then \((0+0)/2=0\). The pairwise and complete-case values differ because the pairwise set includes four positive products that the cohort-wide complete-case set excludes.

This construction illustrates a possible selection effect; it does not imply that ordinary random missingness will routinely produce a large bias. Under a model where missingness is independent of products and marker contributions are exchangeable with a common mean, pairwise averaging does not systematically favor positive products. The example instead motivates checking whether particular samples or technical batches have distinctive missingness patterns, and whether the loci that remain for a pair are representative of the intended panel.

Assumptions, Comparability, and Diagnostics

A pairwise estimate has a clear interpretation as an average over its observed marker set. Interpreting it as a comparable estimate of the same underlying relationship across all pairs requires more. At minimum, use consistent genotype coding, allele-frequency inputs, marker filters, and scaling; ensure that enough markers are shared; and consider whether the observed sets are representative. Missingness associated with genotype, allele count, or technical quality can undermine that representation.

A practical diagnostic sequence is:

1
Summarize overlap.
For each pair, count jointly observed markers. Inspect the range and distribution of those counts, not only the cohort-wide missingness rate.
2
Inspect sample patterns.
Compare per-sample missingness and identify pairs whose overlap is unusually small or concentrated in particular marker regions or technical batches.
3
Compare like with like.
Where possible, recalculate a set of comparisons using a shared marker set or a consistent overlap threshold. Check whether conclusions change under this restriction.
4
Match the output to its use.
For pairwise screening, retain and report pair-specific denominators. For methods requiring a relationship matrix, check whether pair-specific marker sets compromise matrix properties.

One consequence of pair-specific averaging is that the full matrix of pairwise estimates need not be positive semidefinite. A matrix formed from a common set of standardized genotype vectors has the Gram-matrix structure that guarantees positive semidefiniteness. If each entry is computed using a different subset, that common-vector construction may no longer hold. This matters for downstream analyses that require a valid relationship matrix; it does not mean every pairwise estimate is individually invalid.

Common Mistakes

  • Dividing by the original marker count. Missing genotypes contribute no observed product. Dividing the observed sum by all markers implicitly treats missing contributions as zero and can pull estimates toward zero.
  • Treating all pairwise estimates as equally precise. A pair with fewer jointly observed, or more correlated, markers generally carries less information. Report overlap and assess uncertainty.
  • Calling pairwise deletion unbiased without conditions. The method avoids discarding every marker missing in any sample, but it does not protect against missingness that selects particular genotype products or marker subsets.
  • Interpreting differing estimates as differing biology immediately. First check whether the pairs were evaluated on comparable markers and whether their overlap differs substantially.
Key takeaway: The pairwise denominator is the number of markers observed in both members of the pair. Pairwise availability can preserve information, but uneven overlap changes precision, and sample-specific marker selection can change the estimate itself. Interpret estimates alongside their denominators and missingness patterns.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following questions.

  1. For a pairwise estimate, what determines the denominator, and why should a missing product not be treated as zero?
  2. Under independent marker products with variance 1, what is the standard error of an average over 400 markers? How does it compare with an average over 100 markers?
  3. Why can two pairs with equal numbers of observed markers still have different precision?
  4. Give one way a sample-specific missingness pattern could make a pairwise estimate unrepresentative of the intended marker panel.
  5. Why might a matrix of pairwise estimates calculated on different marker subsets fail to be positive semidefinite?

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