Weighting Relative Pairs In NPL Analysis

Why Weighting Matters in NPL Analysis

A pedigree with many affected relatives contains more affected-relative pairs than a small pedigree. If every pair is given the same weight, a large pedigree can therefore contribute many times to the overall sharing statistic. That may be appropriate when the aim is to use every pairwise observation, but pairs from the same pedigree are generally dependent: they share people and inheritance paths. Pair count alone does not tell us how much independent information a family contributes.

Weighting makes explicit what is being allowed to drive the evidence for linkage. Equal-pair weighting gives each pair the same direct coefficient. Pedigree-level weighting limits the influence of a pedigree’s number of pairs. Information-based weighting accounts for how precisely the family’s sharing evidence is measured under the null. These are choices about the statistic, not changes to the underlying observed genotypes or IBD probabilities.

For a full-sibling pair, the null expectation of the proportion of alleles shared IBD is commonly \(1/2\). Other relationships can have different expectations, so \(1/2\) should not be applied indiscriminately. With uncertain marker-based IBD, the pair’s \(\pi_{kij}\) is an expected proportion, not necessarily an observed value of exactly 0, \(1/2\), or 1.

Three Ways to Weight Sharing

Write \(m_k\) for the number of affected-relative pairs included from pedigree \(k\). A simple pair-level score for that pedigree is \(T_k=\sum_{i<j} d_{kij}\). The total equal-pair score is \(T_{\mathrm{pair}}=\sum_k T_k\): each included pair has coefficient 1. Larger pedigrees can have larger scores partly because they contribute more pairs.

One simple pedigree-level alternative is to average the pair deviations within each pedigree and then give each pedigree the same coefficient. Its score is \(T_{\mathrm{ped}}=\sum_k T_k/m_k\). This prevents a pedigree’s contribution from growing automatically in proportion to its pair count. It is not the only possible pedigree-level convention: one might instead form a family-specific standardized score and combine those scores. The weighting rule must therefore be stated, rather than inferred from the phrase “pedigree-level weighting.”

Information-based weighting uses the precision of the family’s sharing evidence, rather than its raw pair count, to determine its contribution. In a score formulation, let \(U_k\) be the null-centered score contribution for family \(k\), and let \(I_k\) be its null information, the variance of that score under the usual score-test formulation. Under independent pedigrees, the combined score statistic is

This expression is useful when the family scores concern a common linkage departure and their null variances have been correctly calculated. It weights standardized family scores \(Z_k=U_k/\sqrt{I_k}\) in proportion to \(\sqrt{I_k}\), since \(U_k=\sqrt{I_k}Z_k\). A pedigree with more information can contribute more, but information is not assumed to be proportional to the number of pairs.

Information Is Not the Same as Pair Count

A family’s information depends on the variability of its score under the null, which in turn depends on pedigree structure, dependence among pairs, and how well the marker data resolve IBD sharing. A pair with uncertain IBD may contribute less precise evidence than a pair whose sharing state is well resolved. Likewise, several pairs that reuse the same affected relatives can provide less independent evidence than the same number of pairs drawn from separate, independent pedigrees.

The variance calculation makes this dependence explicit. For weights \(w_{kij}\), define a weighted family score \(T_k=\sum_{i<j}w_{kij}d_{kij}\). Its null variance is

The second sum includes covariance between distinct pairs in the same pedigree. If pedigrees are independent under the null, the variance of their total score is the sum of their family variances. If they are not independent, covariance between pedigrees must also be included. Given the chosen weights, the standardized statistic is the null-centered weighted score divided by its null standard deviation:

This standardization puts a score in units of its null standard deviation. It does not make two differently weighted statistics identical: they can use the same data but emphasize different patterns of sharing. Nor does standardization alone establish a particular reference distribution or guarantee a valid p-value; calibration must match the study design and null model.

A Short Derivation: Why Covariance Changes the Standard Error

For two pair deviations \(d_1\) and \(d_2\) in one pedigree, the family score is \(T=d_1+d_2\). Expanding its variance gives

The final equality follows by expanding \(E_0[(T-E_0T)^2]\) and collecting the two squared terms and the two cross-products. If the pair deviations are positively correlated, the cross-product increases the variance. Treating them as independent would omit this term and usually understate uncertainty. Negative covariance can instead reduce the variance, so the direction and size of the correction depend on the pedigree and statistic.

Assumptions and Interpretation

A weighting comparison is meaningful only if the sharing deviations and their null distribution are defined consistently. The analysis needs an appropriate no-linkage expectation for each relationship, a valid treatment of uncertain IBD from the marker data, and a variance that reflects dependence among pairs. For an information-based combination of family scores, the formula above also assumes independent pedigrees and compatible score contributions under a common linkage departure. If these conditions do not hold, the variance or calibration must be modified.

Weighting does not remove the need to account for the study design. Affected families may have been ascertained through their disease status, and the null distribution must be appropriate to that ascertainment. Likewise, comparing statistics with different weighting rules requires care: a larger standardized value under one rule does not mean that the rule is universally more powerful or that the underlying family data changed.

Common Mistake: Counting Pairs as Independent Evidence

• Wrong reasoning: “This pedigree has six affected pairs, so it contains six independent units of evidence.”
• Correction: Pairs can share relatives and inheritance paths. Use a null variance that includes their covariance; do not infer information from pair count alone.
• Wrong reasoning: “Equal-pair weighting and equal-pedigree weighting are just different ways of writing the same statistic.”
• Correction: The first gives a pedigree more direct weight when it has more pairs; the second can cap or normalize that influence. As the first worked example shows, their weighted averages can differ.
• Wrong reasoning: “A highly informative family should always receive weight proportional to its number of pairs.”
• Correction: Information depends on precision and dependence as well as count. State the weighting rule and calculate the corresponding null variance.