Choosing Among Nonparametric Linkage Statistics

Why the Choice of Sharing Statistic Matters

Nonparametric linkage methods ask whether affected relatives share marker information more than expected under no linkage, without specifying a disease-allele frequency or penetrance model. But “excess sharing” can mean different things. A statistic may target the average amount shared across relative pairs, the frequency of particular family sharing configurations, or a weighted score that gives large sharing patterns extra influence. The choice affects which pedigrees contribute most and which alternatives are easiest to detect.

These methods are not interchangeable labels for one calculation. A test based on a mean can miss a distinctive pattern if positive and negative deviations offset. A statistic that retains more of the family configuration can be more sensitive to a structured alternative, but its behaviour may depend more strongly on pedigree size and the dependence among relatives. A marker-based affected-pedigree-member score can use observed allele matches, while IBD-based statistics require estimates of which alleles were inherited from the same ancestor.

For IBD-based methods, let \(S_{fij}\) denote the number of alleles shared identical by descent by relatives \(i\) and \(j\) in family \(f\), or its conditional expectation given the observed marker data. For a sib pair, \(S_{fij}\) can be 0, 1, or 2, and its no-linkage expectation is 1. For other relationships, the null expectation must reflect that relationship; it is not generally 1. When marker data are incomplete, an estimated IBD sharing value is uncertain, and that uncertainty should be carried into the statistic or its null calculation.

Mean-Sharing and Sharing-Class Statistics

A mean-sharing statistic summarizes excess IBD sharing over pairs or families. One general form is

where \(\mu_{0,fij}\) is the expected IBD sharing for that pair under no linkage and \(w_{fij}\) is a prespecified weight. The weights may give each pair equal influence or adjust the contribution of pairs within a larger pedigree. A positive value indicates excess sharing in the weighted sum. Because it is an aggregate, the statistic primarily targets a shift in average sharing.

A sharing-class statistic instead assigns family sharing configurations to classes and examines how often each class occurs. The classes might distinguish, for example, families with no affected-pair IBD2 sharing from families with at least one such pair. A class definition can retain aspects of the within-family configuration that a single pairwise mean discards. There is no single universal set of classes: the classes must be defined for the pedigree type and sharing information being analysed.

Whittemore–Halpern NPL Scores

Whittemore–Halpern nonparametric linkage (NPL) scores standardize an affected-relative sharing score against its no-linkage distribution. In a common formulation, the unstandardized score is a weighted sum of sharing among affected individuals. Write this score as \(T_{\mathrm{WH}}\); its precise construction and weights depend on the chosen NPL formulation and pedigree information. Its standardized form is

Here \(E_0\) and \(\operatorname{Var}_0\) are the mean and variance under no linkage, accounting for the pedigree structure and the sharing statistic. A positive score means that the observed weighted sharing exceeds its null mean. Standardization puts the result on a scale measured in null standard deviations; it does not by itself guarantee that a normal approximation gives an accurate p-value, especially with few families or complicated pedigrees.

Kong–Cox Allele-Sharing LOD Scores

Kong–Cox methods express evidence for excess allele sharing as a likelihood-ratio score. Rather than specifying a disease penetrance model, they use an exponential tilt of the no-linkage distribution of a family-level sharing score. Let \(T_f\) be the score for family \(f\), and let \(M_{0f}(\delta)=E_0[\exp(\delta T_f)]\), the null moment-generating function. For a tilt parameter \(\delta\geq 0\), the tilted distribution assigns a family score value \(t\) a likelihood ratio proportional to \(\exp(\delta t)/M_{0f}(\delta)\). For independent families, the log likelihood ratio is

The tilt parameter controls how strongly the alternative favours larger sharing scores. The null distribution of each family score still has to reflect its pedigree and marker information. This method therefore changes how evidence is summarized, not the need to define the score or obtain its correct null distribution.

A short derivative explains the role of the tilt. Differentiating the log likelihood ratio gives

At \(\delta=0\), the tilted expectation equals the null expectation. The initial slope is therefore the observed total score minus its null expected total. If that slope is positive, a small positive tilt increases the likelihood ratio. The maximization allows the data to determine the strength of the tilt, subject to the one-sided restriction \(\delta\geq0\).

Affected-Pedigree-Member Statistics and Marker Data

Affected-pedigree-member (APM) statistics focus on marker allele sharing among affected people in pedigrees. A key distinction is that marker genotypes directly show identity by state (IBS)—alleles with the same observed marker label—whereas IBD means copies inherited from the same ancestral allele. IBS matches can occur by chance, so an APM score uses a scoring rule, often involving marker allele frequencies, to assess how informative the observed matches are. APM statistics can be useful when the analysis is naturally expressed through affected members’ marker alleles rather than estimated pairwise IBD counts.

For example, one simple frequency-adjusted pair score compares the number of matching allele pairs with the number expected from independent draws from marker allele frequencies. If the allele frequencies are \(p_c\), the chance-match probability for two independent alleles is \(q=\sum_c p_c^2\). A simple centered score for genotypes \(G_i,G_j\) is

where \(I(a=b)\) is 1 for a match and 0 otherwise. This formula illustrates a frequency-adjusted IBS score; it is not the only APM scoring rule. In an actual pedigree analysis, the score and its null expectation must account for allele frequencies, pedigree relationships, and the way affected members were selected. A marker match is not automatically evidence of IBD or linkage.

Choosing a Statistic for the Data

The choice should reflect the information available and the sharing pattern the analysis aims to detect. The methods differ in how they represent evidence, not in whether the underlying scientific question concerns linkage.

Large pedigrees contribute many related pairs, but those pairs are not independent observations. A method that gives substantial weight to a large family may therefore be sensitive to pedigree structure, while a method that reduces each family to a class or a single score may emphasize different information. The relevant question is not simply which statistic uses the most pairs; it is whether its weighting and null distribution match the families and marker data available.

Common Pitfall: Comparing Raw Scores as Though They Were Equivalent

• Wrong reasoning: “The Kong–Cox LOD is numerically larger than the NPL score, so it provides stronger evidence.”
• Correction: These statistics have different scales and constructions. Compare evidence using appropriate null calibration, not the raw numerical sizes of unlike statistics.
• Wrong reasoning: “A shared marker allele proves the affected relatives inherited the same allele.”
• Correction: IBS is an observed match; IBD is shared ancestry. An IBS-based score must account for chance matching and pedigree structure.
• Wrong reasoning: “A nonparametric statistic has no assumptions.”
• Correction: The statistic still depends on correct relationships, suitable marker information, the specified family score, and a valid no-linkage null distribution. Ascertainment and family dependence must also be handled in the analysis.