Comparing Genomic Estimates With Pedigree Expectations

Why Compare Genomic Estimates With Pedigrees?

A recorded pedigree describes expected genetic sharing across possible transmissions of DNA. A genomic estimate summarizes sharing observed in the particular DNA samples that were analyzed. Those quantities are related, but they are not identical: even when a pedigree is correct, Mendelian transmission makes realized sharing vary among pairs with the same recorded relationship. Comparing the two is useful for checking whether the data broadly agree with the recorded relationships and for identifying pairs that merit closer review.

The comparison is meaningful only when both quantities use the same scale. A kinship coefficient is not the same as a relationship coefficient, and software may report still other measures. Nor does a large difference alone identify its cause. Pedigree mistakes, ordinary sampling variation, and technical or population-related artifacts can all produce discrepancies.

Pedigree Expectations and Genomic Estimates

Definition: The kinship coefficient \(\phi_{ij}\) for individuals \(i\) and \(j\) is the probability that one allele copy chosen at random from each individual is identical by descent (IBD), meaning the copies descend from the same ancestral allele. A pedigree expectation is the kinship coefficient implied by the recorded pedigree, averaged over possible inheritance paths under that pedigree. A genomic estimate, written here as \(\hat{\phi}_{ij}\), estimates kinship from observed genetic data.

For the familiar relationship values below, assume that pedigree founders are unrelated and not inbred, and that there is no unmodeled shared ancestry. If the pedigree records inbreeding or related founders, expectations should be calculated from that pedigree rather than read from this simplified list.

Recorded relationship	Pedigree expectation for kinship	Relationship coefficient, \(2\phi\)
Parent and child	1/4 = 0.25	0.50
Full siblings	1/4 = 0.25	0.50
Half siblings, grandparent and grandchild, or avuncular pair	1/8 = 0.125	0.25
First cousins	1/16 = 0.0625	0.125
Unrelated founders under the assumptions above	0	0

The relationship coefficient is twice the kinship coefficient under this convention. Some reported genomic measures use this doubled scale; others, including some IBD-based outputs, have their own definitions. Before making a comparison, identify precisely what the estimate represents and convert it if needed. Do not compare a kinship estimate of 0.25 with a relationship expectation of 0.50 and call the difference a discrepancy.

For a nonfounder \(i\) whose parents are \(a\) and \(b\), the pedigree kinship with a different individual \(j\) is the average of the parents’ kinships with \(j\). The diagonal kinship includes the individual’s inbreeding coefficient \(F_i\):

\phi_{ij}=\frac{\phi_{aj}+\phi_{bj}}{2}\quad (i\ne j), \qquad \phi_{ii}=\frac{1+F_i}{2}.

These recursions show how expectations can be obtained from recorded parentage, including pedigrees more complicated than the standard relationship list. A pedigree-based value is an expectation, not a guarantee that every genomic estimate for that relationship will equal the listed number.

Worked Example: Check the Scale Before Comparing

Suppose a hypothetical parent–child pair has a genomic kinship estimate of \(\hat{\phi}_{ij}=0.252\). The pedigree expectation for kinship is 0.25. Compare them on the kinship scale, then express both on the relationship-coefficient scale.

Kinship-scale residual: define residual as estimate minus pedigree expectation. Thus \(e_{ij}=0.252-0.25=0.002\). The estimate is 0.002 kinship units above expectation.

Relationship-coefficient scale: multiply both values by 2. The estimate is \(2(0.252)=0.504\), and the expectation is \(2(0.25)=0.500\). The difference is \(0.504-0.500=0.004\), exactly twice the kinship-scale residual.

The two comparisons tell the same story because the scales were converted consistently. The small residual is not evidence by itself that the pedigree is correct; its interpretation also depends on the uncertainty and the estimator’s behavior.

Residuals and Relationship-Specific Summaries

Definition: For a pair with pedigree-expected kinship \(\phi_{ij}^{\mathrm{ped}}\), the residual is \(e_{ij}=\hat{\phi}_{ij}-\phi_{ij}^{\mathrm{ped}}\). A positive residual means the estimate is above expectation; a negative residual means it is below expectation. For a set of \(m\) pairs of the same recorded relationship, the mean residual summarizes signed bias, while the root mean squared residual (RMSE) summarizes the typical magnitude of discrepancies, with larger discrepancies receiving more weight.

\bar{e}=\frac{1}{m}\sum_{k=1}^{m} e_k, \qquad \mathrm{RMSE}=\sqrt{\frac{1}{m}\sum_{k=1}^{m}e_k^2}.

Summarizing separately by relationship type is important. A genomic estimate around 0.12 has a different interpretation for a half-sibling pair, whose expectation is 0.125, than for first cousins, whose expectation is 0.0625. A single overall mean can conceal overestimation for one class and underestimation for another. Useful summaries therefore include the number of pairs, mean residual, RMSE, and often a plot of estimates against their corresponding expectations.

Worked Example: Summarize Full-Sibling Residuals

In a hypothetical data set, three recorded full-sibling pairs have kinship estimates 0.23, 0.27, and 0.19. Under the stated founder assumptions, each pair has pedigree expectation 0.25. Calculate the residuals, mean residual, and RMSE.

The residuals are \(0.23-0.25=-0.02\), \(0.27-0.25=0.02\), and \(0.19-0.25=-0.06\) kinship units. The mean residual is \[ \bar{e}=\frac{-0.02+0.02-0.06}{3}=-0.02. \] So the estimates average 0.02 kinship units below the pedigree expectation in this small set.

The RMSE is \[ \mathrm{RMSE}=\sqrt{\frac{(-0.02)^2+(0.02)^2+(-0.06)^2}{3}} =\sqrt{\frac{0.0004+0.0004+0.0036}{3}} =\sqrt{0.0014667}\approx 0.0383. \] Thus the typical discrepancy as measured by RMSE is about 0.0383 kinship units. The mean residual describes direction; the RMSE describes size without allowing positive and negative discrepancies to cancel.

Three pairs are too few to establish a systematic problem. These summaries describe these hypothetical pairs, not a universal performance level for genomic estimation.

Uncertainty: When Is a Residual Surprising?

A residual should be interpreted relative to an appropriate uncertainty scale. Genomic estimates vary because only a finite set of genomic regions is observed and because related individuals inherit different segments through Mendelian transmission. If a standardized residual is used, its denominator must represent uncertainty for comparison to the pedigree expectation, not merely a software-reported standard error that conditions on the realized genotypes.

When a suitable total standard deviation \(s_{ij}^{\mathrm{comp}}\) is available, a descriptive standardized residual is \(z_{ij}=e_{ij}/s_{ij}^{\mathrm{comp}}\). The scale should account for relevant genomic sampling uncertainty and, for a pedigree-level comparison, variation in realized sharing under that relationship. A large absolute value flags a pair for investigation; it is not automatically a calibrated hypothesis test or proof of an incorrect pedigree. Calibration depends on the estimator, marker data, and assumptions used to obtain the uncertainty.

Worked Example: Standardize a Discrepancy Carefully

Consider a hypothetical first-cousin pair. The pedigree expectation is 0.0625, the genomic estimate is 0.13, and an uncertainty analysis supplies a total comparison standard deviation of 0.02 kinship units. Assume this uncertainty measure is appropriate for comparing the estimate with the pedigree expectation.

The residual is \[ e_{ij}=0.13-0.0625=0.0675. \] The standardized residual is \[ z_{ij}=\frac{0.0675}{0.02}=3.375. \] The estimate is 0.0675 kinship units above expectation, or 3.375 comparison standard deviations above it. This is a substantial discrepancy relative to the supplied scale and warrants checking the pair and its data.

The calculation does not establish that the pair is not composed of first cousins. A pedigree error, unusual realized inheritance, or a systematic technical effect could contribute. Nor should a probability be attached to \(z_{ij}\) without a justified reference distribution and calibration.

A Short Justification: Why Residuals Center on Zero

Under a correct pedigree, the pedigree expectation averages over possible Mendelian transmissions. If the genomic estimator targets realized kinship without systematic bias, averaging its estimate over those transmissions and over the relevant sampling process gives the pedigree expectation. In symbols, under these assumptions,

E(\hat{\phi}_{ij}\mid\text{recorded pedigree})=\phi_{ij}^{\mathrm{ped}}, \qquad E(e_{ij}\mid\text{recorded pedigree}) =E(\hat{\phi}_{ij}-\phi_{ij}^{\mathrm{ped}}\mid\text{recorded pedigree})=0.

The second equality follows because the pedigree expectation is fixed conditional on the recorded pedigree, so it can be taken outside the expectation. This is a statement about repeated realizations under the model. It does not require every observed residual to be zero, and it does not imply that a small group’s mean residual will be exactly zero. An estimator with bias, an incorrect pedigree, or violated data assumptions can also shift residuals away from zero.

Diagnosing Patterns Rather Than Assigning Causes

A discrepancy is a diagnostic signal, not a diagnosis. Examine the size, direction, and pattern of residuals alongside sample and technical information. A single extreme pair may reflect a mistaken relationship record, a sample-label problem, or an unusual but possible realization. Many pairs shifted in the same direction are more suggestive of a systematic issue than of independent pedigree mistakes, although the pairs may not be statistically independent.

Verify definitions and scale.
Confirm that the estimate is kinship, or convert a relationship coefficient or other output to the matching quantity. Confirm the pedigree relationship and its assumptions.

Assess uncertainty.
Compare residuals with an appropriate uncertainty measure. Small residuals can be ordinary variation; a large standardized residual merits attention but is not conclusive alone.

Look for structure in the discrepancies.
Check whether residuals cluster by relationship class, individual, batch, missingness, marker set, or ancestry group. A cluster can help distinguish a local issue from a broader systematic effect.

Review plausible explanations.
Check pedigree records and sample identity, and consider Mendelian sampling, marker dependence, missing genotypes, allele-frequency choices, population structure, and technical artifacts.

Report evidence, not certainty.
State which pairs or classes disagree, by how much, and under which assumptions. Treat follow-up checks as necessary before changing pedigree records or drawing biological conclusions.

Pairs in a relationship-specific summary often share individuals, so their residuals are not necessarily independent. A group mean and RMSE remain descriptive summaries, but standard errors or formal tests that assume independent pairs may be misleading unless dependence is addressed. Likewise, ancestry differences or allele-frequency misspecification can affect many comparisons together; residuals should not be read as a direct measure of individual pedigree accuracy without considering these shared influences.

Common Mistakes

Comparing different scales. Treating a kinship expectation of 0.25 and a relationship coefficient of 0.50 as a mismatch is an arithmetic and definition error. Convert both to the same scale first.
Expecting exact agreement. Reasoning that every full-sibling estimate must equal 0.25 ignores Mendelian sampling and finite genomic information. The pedigree value is an expectation across possible transmissions.
Calling any large residual a pedigree error. A large difference can arise from a wrong record, unusual inheritance, or technical and modeling artifacts. Investigate competing explanations rather than assigning a cause from the residual alone.
Using the wrong uncertainty. A standard error conditional on observed genotypes may not describe variation around the pedigree expectation. Use a comparison uncertainty that includes the relevant sources of variation.
Pooling relationship classes indiscriminately. A pooled average can conceal opposite biases across relationship types. Summarize and inspect residuals by the relationship expectation being evaluated.

Key takeaway: Compare like with like: match the genomic measure to the pedigree kinship scale, calculate estimate-minus-expectation residuals, and summarize them by relationship type. Interpret discrepancies against appropriate uncertainty and diagnose their patterns without treating a residual as proof of its cause.

Check Your Understanding

Answer each question using the assumptions and definitions in this tutorial.

A pair has a pedigree kinship expectation of 0.125 and a genomic kinship estimate of 0.11. What is the residual, and what does its sign mean?
Why is it incorrect to expect every genomic estimate for a full-sibling pair to equal 0.25 exactly?
Three pairs have residuals of \(-0.03\), \(0.01\), and \(0.02\). Calculate their mean residual and explain what it summarizes.
A software output is a relationship coefficient on the \(2\phi\) scale. What conversion is needed before comparing it with a pedigree kinship expectation?
What additional uncertainty must be considered when using a standardized residual to assess agreement with a pedigree expectation?
Several pairs involving the same sequencing batch have positive residuals. Give two plausible explanations and explain why this pattern does not, by itself, establish either one.