Why Diagnose Information at a Linkage Peak?
An NPL statistic summarizes evidence of allele sharing among affected relatives, but the amount of information behind that summary can vary across chromosome positions and families. A marker may be highly polymorphic yet provide little information about which alleles relatives inherited. Missing genotypes, uninformative parental genotypes, or ambiguous inheritance can also leave IBD uncertain. These issues matter when deciding how much confidence to place in the observed pattern of support.
The goal here is diagnostic: describe how well the available marker data resolve IBD and whether the families contributing to an NPL result are similarly informative. These summaries help distinguish a broadly supported signal from one arising in a small, unusually informative subset. They do not replace the calibrated NPL test or its significance assessment.
IBD Posterior Uncertainty and Information
For a pair of full siblings at a locus, let \(K\) be the number of alleles they share identical by descent (IBD), so \(K\) can equal 0, 1, or 2. Before using marker data, the usual Mendelian probabilities are \(P(K=0)=0.25\), \(P(K=1)=0.50\), and \(P(K=2)=0.25\). These give \(E(K)=1\) and \(\operatorname{Var}(K)=0.50\). After observing marker genotypes, inference is represented by posterior probabilities \(P(K=k\mid G)\), where \(G\) denotes the observed genotypes and relevant pedigree data.
For a sibling pair, one convenient realized diagnostic is \(I=1-\operatorname{Var}(K\mid G)/0.50\). A value near 1 means the observed genotypes have greatly reduced uncertainty relative to the prior; a value near 0 means they have reduced little. This is a diagnostic, not a test statistic or a probability that the inferred IBD state is correct. For one particular genotype outcome, \(I\) can be negative if the posterior variance exceeds the prior variance. Do not silently truncate such a value; investigate the pedigree, model, or data and report how the summary was defined.
Information can also be summarized for more complex pedigrees using inheritance vectors: the vector records the parental alleles transmitted to pedigree members. In that setting, the relevant prior variance or uncertainty measure must match the inheritance quantity used by the NPL analysis. Comparisons across families are meaningful only when the summaries use compatible definitions and reference distributions.
Worked Example: Posterior IBD Information for a Sibling Pair
Suppose a hypothetical pair of affected siblings has posterior probabilities \(P(K=0\mid G)=0.80\), \(P(K=1\mid G)=0.20\), and \(P(K=2\mid G)=0\) at a marker. Calculate the posterior mean, variance, and variance-based information diagnostic.
The posterior mean is \(E(K\mid G)=0(0.80)+1(0.20)+2(0)=0.20\) shared alleles. The second moment is \(E(K^2\mid G)=0^2(0.80)+1^2(0.20)+2^2(0)=0.20\). Therefore, \(\operatorname{Var}(K\mid G)=0.20-(0.20)^2=0.16\) shared-allele-squared.
Using the sibling prior variance of 0.50, \(I=1-0.16/0.50=0.68\). The observed genotypes reduce posterior variance by 68% relative to this prior-variance benchmark. This says the IBD state is more resolved than it would be without these data; it does not say that there is a 68% probability of a particular state or that linkage is established.
What Makes a Marker Informative?
Marker informativeness is not determined by the number of alleles alone. A marker with several common alleles can still be uninformative for a particular family if the parental genotypes do not reveal which allele was transmitted. Conversely, a simpler genotype pattern may be quite informative when it distinguishes the parental origins of the alleles carried by relatives. Useful ingredients include parental and offspring genotypes, allele frequencies, the pedigree structure, and—when using multiple markers—the recombination model connecting positions.
A principled information summary averages uncertainty over possible marker data under a specified model. For a sibling pair, the law of total variance gives a useful justification:
The identity follows by writing \(K-E(K)=[K-E(K\mid G)]+[E(K\mid G)-E(K)]\), squaring, and taking expectations. The cross-product has expectation zero because \(E[K-E(K\mid G)\mid G]=0\). Thus, averaging over marker data, reduced posterior variance corresponds to variation in posterior expected sharing across possible data outcomes. An expected variance-based information fraction is \(1-E\{\operatorname{Var}(K\mid G)\}/\operatorname{Var}(K)\), which equals \(\operatorname{Var}\{E(K\mid G)\}/\operatorname{Var}(K)\) and lies between 0 and 1. The realized diagnostic for a single family need not lie in that range.
At a single marker, inference uses the observed genotypes and a model for allele frequencies and Mendelian transmission. For multipoint inference, neighboring markers and recombination probabilities also inform the inheritance state. Marker data must not be treated as independent evidence when linkage between markers is part of the model. Genotyping error, allele-frequency misspecification, and pedigree errors can all distort posterior probabilities; high apparent precision under a faulty model is not reliable information.
Worked Example: Missing Genotypes Leave Sharing Unresolved
Consider a hypothetical affected sibling pair at a marker. One sibling has a recorded genotype, but the other sibling’s genotype and both parents’ genotypes are missing. Suppose, for this illustration, the recorded data do not distinguish among the three IBD states, so the posterior probabilities remain \(0.25, 0.50, 0.25\) for \(K=0,1,2\). Calculate the information diagnostic and explain the result.
The posterior mean is \(0(0.25)+1(0.50)+2(0.25)=1\). The second moment is \(0^2(0.25)+1^2(0.50)+2^2(0.25)=1.50\), so the posterior variance is \(1.50-1^2=0.50\). Therefore, \(I=1-0.50/0.50=0\).
The marker data have not reduced uncertainty relative to the sibling prior in this hypothetical case. This is not evidence that the siblings share no IBD alleles; their posterior mean is still 1, and all three states remain possible. It is evidence that these observations do not resolve which state occurred. With actual data, a recorded genotype can sometimes be informative even when other family genotypes are missing; its value depends on the allele frequencies and the family’s inheritance possibilities.
Handling Missingness and Summarizing Families
A missing genotype is not an additional allele and must not be assigned an arbitrary observed genotype. In likelihood-based inference, uncertainty about a missing genotype is handled by summing over its possible values. In general terms, the likelihood of the observed genotypes given an inheritance state is obtained by summing over compatible latent genotypes, weighted by their probabilities under the model. Missingness may therefore broaden the IBD posterior, though it does not always do so equally: missing a parent’s genotype can be especially consequential when that parent’s genotype would identify transmitted alleles.
Record missingness separately from posterior information. A family can have a high missing-genotype rate but retain useful IBD information from other relatives, or have nearly complete data that remain uninformative because the observed genotypes are ambiguous. Missingness patterns also deserve scrutiny. If genotyping failures are related to allele, phenotype, or pedigree structure, treating them as innocuous may be inappropriate. Genotyping errors and sample swaps can create misleadingly sharp or inconsistent posterior distributions.
For a pedigree-specific NPL assessment, it is useful to inspect at least three linked quantities: the family’s information summary, its missing-genotype pattern, and its contribution to the NPL statistic under the chosen method. For a larger pedigree, calculate information for the inheritance states or sharing quantity relevant to that NPL method rather than assuming that a sibling-pair formula applies unchanged. If the statistic combines multiple relative pairs within a family, distinguish a family-level contribution from the several pairwise quantities that may underlie it.
Worked Example: Finding Concentrated Support Across Families
Suppose a hypothetical scan reports four family-level contributions to a locus-specific NPL summary, alongside information diagnostics and missing-genotype rates. The entries below are illustrative outputs from one analysis, not values calculated from the variance formula above.
| Family | NPL contribution | Information diagnostic | Missing genotypes |
|---|---|---|---|
| A | 2.4 | 0.90 | 0% |
| B | 1.8 | 0.80 | 10% |
| C | 0.1 | 0.15 | 50% |
| D | -0.3 | 0.10 | 60% |
The contributions sum to \(2.4+1.8+0.1-0.3=4.0\). Families A and B contribute \(2.4+1.8=4.2\), while C and D together contribute \(0.1-0.3=-0.2\). Thus the first two families provide most of the positive support, and their information diagnostics are also higher. The positive contributions from A and B amount to \(4.2/(|2.4|+|1.8|+|0.1|+|{-0.3}|)=4.2/4.6\approx0.913\), or 91.3% of the sum of absolute family contributions. This is a descriptive concentration summary, not a formal measure of statistical significance.
The table suggests uneven information and support, but it does not establish that missingness caused the differences. The NPL contributions and information diagnostics summarize distinct features; they need not move together. A careful follow-up checks genotype quality, pedigree records, marker informativeness, and the analysis model before interpreting why families differ. The genome-wide significance assessment remains a separate part of the evidence.
Practical Diagnostic Workflow
Specify whether the diagnostic concerns a sibling-pair IBD count, a family sharing statistic, or an inheritance vector. Use the same target as the NPL analysis.
Review posterior IBD probabilities or an appropriate family-level uncertainty summary. Compare them with the correct prior inheritance distribution.
Tabulate missing genotypes and examine genotype quality, pedigree consistency, and model assumptions. Do not convert missing calls into observed alleles.
Place compatible information summaries beside family-level NPL contributions. Note whether support is broadly distributed or concentrated in a few pedigrees.
Describe the diagnostic and its reference distribution, identify important missingness or model concerns, and keep this assessment distinct from the calibrated test result.
Common Mistakes and Why the Diagnostics Matter
- Equating polymorphism with information. A marker with many alleles is not necessarily informative for a particular pedigree. Check whether the observed genotypes resolve transmissions and IBD states.
- Reading posterior probabilities as certainty. A posterior distribution describes uncertainty conditional on the data and model. A concentrated distribution can still be misleading if allele frequencies, pedigree structure, or genotype calls are wrong.
- Treating missing calls as a genotype. Missing data must be integrated over or otherwise handled by the stated method. Assigning a likely allele as though it had been observed can overstate information.
- Confusing a weakly informative family with a family showing no sharing. Low information means the data leave the sharing state uncertain; it does not imply low IBD sharing.
- Assuming information is evenly distributed. An aggregate NPL result can conceal that a few pedigrees dominate the observed support. Inspect family-level summaries without treating them as independent, separately calibrated tests.
These diagnostics help make a linkage result interpretable: they show whether the underlying inheritance data are resolving IBD and whether the evidence is shared across pedigrees. They do not determine a peak’s location precision, nor do they change the null distribution or significance threshold. Their value is in identifying where the evidence is strong, where it is weak, and which data or assumptions merit review.
Check Your Understanding
Use the definitions and examples above to test your interpretation.
- For a full-sibling pair, what does a posterior variance equal to the prior variance indicate about the variance-based information diagnostic?
- Why can a highly polymorphic marker still provide little IBD information in a particular pedigree?
- How should a missing genotype enter likelihood-based IBD inference, and why should it not be replaced by the most likely genotype?
- What is the difference between a family’s NPL contribution and its information diagnostic?
- If a few families have high information and large NPL contributions while most have low information, what can be concluded—and what cannot be concluded—from that pattern alone?