Why Ask a Nonparametric Linkage Question?
A linkage analysis asks whether a chromosome region is inherited along with a disease in a family more often than would be expected if the region and disease were unlinked. Parametric linkage analysis answers this question using a specified model for disease inheritance. A nonparametric linkage question instead focuses on the observed sharing of marker-region alleles among relatives who have the disease.
This shift is useful when the mode of inheritance, disease-allele frequency, or penetrance is uncertain. It does not make the analysis assumption-free: the family relationships, observed affection status, marker information, and way families entered the study still matter. The key distinction is that the linkage alternative is framed as excess sharing among affected relatives, rather than as a fully specified model of how a disease allele produces affection.
The Null and the Linkage Alternative
For a candidate marker region, let \(\theta\) denote the recombination fraction between that region and a disease-susceptibility locus. Under the conventional linkage null, \(\theta=1/2\): inheritance at the two loci is unlinked. The nonparametric linkage alternative of interest is directional: affected relatives share the marker region IBD more than expected under that null. In this setting, the question is not whether a particular disease model fits better, but whether the observed family sharing points toward linkage.
A practical comparison requires a reference distribution or expected amount of sharing under no linkage for the pedigree being studied. The reference depends on relationships and pedigree structure; it is not one universal number for every pair or family. The tutorials on sib-pair null distributions and excess-sharing tests develop particular calculations. Here, the essential question is how to define the hypotheses and interpret the comparison.
A simple descriptive summary can make that comparison clear. Let \(Q_{\text{obs}}\) be an observed average IBD-sharing probability among a chosen set of affected relatives, and let \(Q_0\) be its expected value under the appropriate no-linkage reference for those relatives. Define the descriptive excess as
A positive \(\Delta\) describes more sharing than the no-linkage reference; a negative \(\Delta\) describes less. This difference alone is not a test statistic with a known null distribution, and it does not establish statistical significance. Formal inference also requires accounting for sampling variability, marker information, and the study design.
Worked Example: Interpreting a Positive Descriptive Excess
In a hypothetical family analysis, a marker-genotype procedure supplies an average IBD-sharing probability of \(Q_{\text{obs}}=0.67\) among the affected relatives. A pedigree-specific no-linkage calculation, supplied for this example, gives \(Q_0=0.42\). What does the difference say?
Use the descriptive comparison:
The observed average is 0.25 probability units above the no-linkage reference. In context, the affected relatives show more estimated IBD sharing at this region than the reference predicts. This is directionally consistent with the excess-sharing linkage alternative, but it is not by itself a p-value or proof of linkage. We have not calculated how variable the statistic would be under the null.
What “Nonparametric” Does—and Does Not—Mean
The term nonparametric refers to the way the disease inheritance model is avoided. The analysis need not specify whether susceptibility is dominant or recessive, assign a disease-allele frequency, or give penetrance probabilities for each genotype. Instead, it uses the affection status of relatives and their marker-region inheritance to ask whether affected relatives share more than expected without linkage.
This does not mean that no assumptions are made. The analysis still needs family relationships to be recorded correctly and marker data to provide useful information about inheritance. It also needs a valid no-linkage reference for the pedigree and a study design that does not create spurious sharing. For example, population structure or unrecognized relatedness can make relatives share alleles for reasons that are not evidence of linkage to the disease. Selection of families based on having multiple affected members is relevant too: the reference and inference must respect how families were ascertained.
Nor is the alternative a claim that a chromosome region causes the disease. Excess sharing can support linkage: it suggests that a susceptibility locus may be near the marker region and co-inherited with affection in these families. It does not, on its own, identify a causal variant, quantify an individual’s disease risk, or establish the biological mechanism.
Worked Example: Why the No-Linkage Reference Matters
Suppose two hypothetical marker regions are summarized using the same IBD-sharing measure and comparable affected relatives. At region A, the observed average sharing is 0.62 and its supplied no-linkage reference is 0.45. At region B, the observed average is 0.70 and its reference is 0.60. Which region has the larger descriptive excess?
For region A:
For region B:
Region B has the larger raw observed sharing value, but region A has the larger excess over its own no-linkage reference: \(0.17\) compared with \(0.10\). The relevant contrast is not simply which region has the highest observed sharing; it is how sharing compares with the appropriate null expectation. These descriptive differences alone do not determine which region has stronger statistical evidence, because uncertainty and the full null distributions have not been assessed.
A Short Justification for the Null Comparison
The logic of the no-linkage reference follows from independence of inheritance at unlinked loci. Consider a pedigree with a marker region and a disease-susceptibility locus. Under the no-linkage null, inheritance at the marker is independent of inheritance at the disease locus. In an idealized setting where affection depends on disease-locus inheritance and not on marker inheritance, selecting relatives because they are affected does not itself select a particular marker-sharing pattern.
To express that idea, let \(M\) denote a marker-sharing outcome and \(A\) the observed affection pattern. Conditional on the pedigree structure, the no-linkage assumptions imply:
Thus, under the null, the marker-sharing distribution for the affected relatives is the distribution specified by the no-linkage reference for that pedigree. Evidence for the targeted alternative arises when the observed sharing is unusually high relative to that reference. This argument depends on the stated conditions: if marker inheritance and affection are associated for reasons other than linkage, or if the pedigree or ascertainment is mishandled, the comparison may not have its intended interpretation.
Worked Example: Separating a Linkage Question From a Disease Model
In a hypothetical family, several relatives are affected, but the investigator cannot justify choosing a dominant or recessive disease model. A marker analysis supplies \(Q_{\text{obs}}=0.64\) for their average IBD sharing, and the appropriate no-linkage reference for this family structure is \(Q_0=0.46\). What can be concluded without choosing an inheritance model?
Calculate the descriptive excess:
The affected relatives have an average sharing estimate 0.18 probability units above the supplied no-linkage reference. This permits a model-free statement about the direction of the sharing evidence: it is consistent with excess sharing at the marker region. It does not require the investigator to assign disease-allele frequencies or penetrance values. However, it does not show that the excess is statistically unusual, and it does not determine whether the disease is dominant, recessive, or caused by a particular variant.
How to State the Question Precisely
A clear nonparametric linkage analysis begins by naming the affected relatives and marker region, stating that the reference is no linkage, and specifying that the alternative of interest is excess IBD sharing. The comparison should be interpreted in the context of the relevant pedigree structure and study design. Keeping these elements explicit prevents the descriptive phrase “the family shares more” from being mistaken for a completed statistical test.
Common Mistake
A common mistake is to reason, “The affected relatives share the marker region, so the marker must be linked to the disease.” Relatives share genetic material even when a marker is unlinked; the relevant evidence is excess sharing compared with the no-linkage expectation, not sharing by itself. Even a positive excess is only a direction of evidence until its variability and study design are accounted for. A second mistake is to call the analysis “assumption-free.” It avoids a specified disease-inheritance model, but still depends on valid family, marker, null-reference, and ascertainment assumptions.
Check Your Understanding
Answer each question by distinguishing the linkage question from a disease-inheritance model.
- State the no-linkage null in words and explain the role of the recombination fraction under that null.
- What feature of the affected relatives’ marker-region inheritance defines the targeted nonparametric linkage alternative?
- An observed IBD-sharing summary is 0.58 and its supplied no-linkage reference is 0.43. Calculate the descriptive excess and state what it does, and does not, establish.
- Why is a no-linkage reference for one pedigree not automatically appropriate for every other pedigree?
- Give one reason why “nonparametric” does not mean “assumption-free.”