Affected-Pedigree-Member Sharing Statistics

Why Compare Sharing With a Pedigree-Specific Expectation?

If affected relatives share marker alleles more often than expected, that can be evidence that a marker is linked to a disease locus: relatives who inherit the same chromosomal segment may also inherit the same marker alleles. An affected-pedigree-member (APM) statistic summarizes this sharing across affected relatives. Its central comparison is not simply “how much sharing was observed?” but “how much sharing was observed relative to what this marker and this pedigree would produce under the null model?”

Two features matter from the outset. First, two allele copies can match by state (IBS) without having been inherited from the same ancestral copy; marker genotypes alone do not make those concepts interchangeable. Second, relatives are not independent observations. Their inheritance patterns are connected by the pedigree, so an appropriate null expectation and variance must account for that structure.

A Weighted Pairwise Sharing Score

One common way to construct an APM statistic begins with a score for each pair of affected relatives. Let \(p_a\) denote the population frequency of marker allele \(a\). For illustration, use the inverse-square-root weight \(w(p_a)=1/\sqrt{p_a}\): a match involving a less frequent allele receives a larger contribution. Other APM implementations can use different weighting functions, so the weight must be stated when a statistic is reported.

Suppose relatives \(i\) and \(j\) each have two observed marker-allele copies. Define their IBS score as the sum over the four cross-person copy comparisons, adding \(w(p_a)\) for each pair of copies that both carry allele \(a\). If an allele occurs twice in one person and once in the other, it contributes twice. This copy-pair convention makes the score explicit and avoids ambiguity about whether “sharing” means one match or two.

Here \(G_{ir}\) is copy \(r\) in person \(i\), and \(\mathbf{1}(\cdot)\) is 1 when its condition is true and 0 otherwise. \(T\) is the total pairwise sharing score for the affected members being analyzed. The same pair can contribute several matching copy pairs, and a person in a larger affected pedigree can participate in many pairs.

Expected Sharing Depends on Frequencies

A match is more likely for a common allele than for a rare one. The null expectation therefore has to account for allele frequencies; otherwise, a marker with many common alleles could appear to show excess sharing simply because matches are frequent at that marker.

For a simple baseline, suppose the two copies being compared are independent draws from the marker population. The probability that both copies are allele \(a\) is \(p_a^2\). Since a match on \(a\) contributes \(w(p_a)\), the expected weighted contribution from that comparison is \(p_a^2w(p_a)\). There are four cross-person copy comparisons in \(S_{ij}\), so the expected pair score for two independent genotypes is \(4\sum_a p_a^2w(p_a)\). This baseline is useful for understanding the frequency effect, but it is not the correct null expectation for related pedigree members.

Building the Null Model Around the Pedigree

For a pedigree, the null model must preserve the relationships among its members. A standard approach specifies the marker allele frequencies in the founders and the Mendelian transmission process through the pedigree. Under the no-linkage null, disease status does not select marker inheritance patterns through linkage to the disease locus; nevertheless, the pedigree still induces dependence among marker genotypes. The expected score and its variance must be computed under those pedigree inheritance rules, not by treating every relative pair as an independent pair of population draws.

In practice, the null distribution can be obtained by enumerating possible founder genotypes and transmissions when feasible, or by simulation from the specified pedigree model. For every null replicate, calculate the same \(T\) from the resulting genotypes. The null mean is \(E_0(T)\), and the null variance is \(\operatorname{Var}_0(T)=E_0[(T-E_0(T))^2]\). The subscript 0 indicates that these quantities are calculated under the no-linkage null.

Why Pedigree Structure Changes the Answer

If two affected siblings share a marker allele, their match is not surprising in the same way as a match between unrelated people. Siblings have a chance of inheriting the same parental allele copy, and multiple sibling pairs in a larger family are dependent because they reuse the same individuals and parental transmissions. Consequently, the number of affected pairs is not the number of independent observations. Increasing the number of affected relatives can increase \(T\), but it can also change the null mean and variance.

The marker itself matters as well. With inverse-square-root weighting, a match on an allele with frequency 0.10 contributes \(1/\sqrt{0.10}\approx3.162\), while a match on an allele with frequency 0.90 contributes \(1/\sqrt{0.90}\approx1.054\). Rare-allele matches therefore count more under this weighting rule. But a rare allele also has a different probability of appearing in the pedigree, so its contribution must be evaluated against a null distribution using the same allele frequencies. A large weighted score is not, by itself, evidence of linkage.

The no-linkage model also determines what is conditioned on. The pedigree, observed affected-member set, marker frequencies, missing genotypes, and any ascertainment assumptions should be represented appropriately in the null calculation. Changing those choices can change the distribution of \(T\). Thus a reported APM result is interpretable only alongside the scoring rule and the null model used to calibrate it.

A Short Justification for the Null Mean

The frequency contribution to the independent-copy baseline follows directly by summing over mutually exclusive matching alleles. For one comparison, the event “both copies are \(a\)” has probability \(p_a^2\), and its score contribution is \(w(p_a)\). The expectation of that comparison is therefore the sum of the score times the probability over all alleles:

Adding the expectations for four comparisons gives \(4\sum_a p_a^2w(p_a)\) by linearity of expectation; independence among the four comparisons is not required for that mean calculation. For a pedigree, the same logic applies once the relevant joint probabilities are calculated under its inheritance model. Variance is more demanding: it includes covariances among score components, so it cannot generally be found by adding pairwise variances as if relatives were independent.

Common Mistakes and Practical Interpretation

• Using the unrelated-person baseline for relatives. The formula \(4\sum_a p_a^2w(p_a)\) assumes independent allele-copy draws. Use a pedigree-specific null for siblings and other relatives.
• Treating every pair as independent. Pairs in a family overlap in people and inherited alleles. Their dependence affects the variance of the total score.
• Confusing IBS with IBD. Equal observed marker alleles are IBS. They can be shared by descent, but genotype matching alone does not prove that they are.
• Ignoring the weighting rule. A score using \(1/\sqrt p\) is not interchangeable with an unweighted count or a statistic using a different weight.
• Reading a standardized score as a p-value. \(Z\) gives a distance from the null mean in null standard-deviation units. A p-value requires a calibrated null distribution and a stated tail direction.
• Interpreting excess sharing as causation. Evidence of linkage concerns co-inheritance in the studied pedigrees; it does not show that the marker allele causes disease.