From pedigrees to a linkage result
A linkage scan asks whether affected relatives share more chromosomal material identical by descent (IBD) than expected under the no-linkage null. A scan does not directly identify a causal variant: it evaluates evidence that a disease-associated locus may lie near a genomic region where affected relatives share inherited chromosome segments. To make that evidence interpretable, the analysis must connect pedigree information, multipoint IBD estimates, a specified sharing statistic, and a null calibration that matches the search that was actually conducted.
This worked tutorial uses affected sib pairs to keep the arithmetic visible. The same logic applies to other affected-pedigree-member NPL statistics, but their sharing summaries and dependence structures differ. In particular, pairs from the same extended pedigree are not automatically independent observations.
The probabilities are estimates of sharing conditional on observed marker information, not proof that a disease allele is present at that position. Multipoint analysis uses information across markers, together with the pedigree and genetic map, to estimate sharing along the chromosome. Marker informativeness and missingness affect how precisely those probabilities can be estimated.
A sharing score at one position
For a simple worked statistic, let \(q_i=E(K_i\mid G)\) be the multipoint expected IBD count for affected sib pair \(i\), and let \(T=\sum_{i=1}^{m}q_i\) be the total sharing score across \(m\) pairs. Larger \(T\) means greater estimated sharing. Under the no-linkage null, the mean and standard deviation of \(T\) depend on the pedigree structure, marker data, and analysis procedure. We therefore obtain them from a null distribution appropriate to the analysis rather than assume that all pair scores are independent.
For clarity, the first example uses four independent pedigrees, each contributing exactly one affected sib pair. Its null distribution is summarized by a mean and standard deviation obtained from valid null replicates. In a real analysis with multiple related pairs per pedigree, the replicates must preserve the relevant within-pedigree dependence.
Worked Example: Turning IBD Probabilities Into an NPL Score
Suppose four independent hypothetical pedigrees each contribute one affected sib pair. At a candidate position, multipoint analysis gives the following probabilities. Each row sums to one.
| Pair | \(P(K=0\mid G)\) | \(P(K=1\mid G)\) | \(P(K=2\mid G)\) | \(q_i=E(K\mid G)\) |
|---|---|---|---|---|
| A | 0.10 | 0.50 | 0.40 | 1.30 |
| B | 0.05 | 0.45 | 0.50 | 1.45 |
| C | 0.20 | 0.60 | 0.20 | 1.00 |
| D | 0.10 | 0.40 | 0.50 | 1.40 |
For pair A, the expected count is \(q_A=0(0.10)+1(0.50)+2(0.40)=1.30\). The same calculation gives \(q_B=0(0.05)+1(0.45)+2(0.50)=1.45\), \(q_C=0(0.20)+1(0.60)+2(0.20)=1.00\), and \(q_D=0(0.10)+1(0.40)+2(0.50)=1.40\). Thus the observed total sharing score is
Suppose the valid null replicates for this position have mean \(4.00\) and standard deviation \(0.90\) for \(T\). The standardized score is \(Z=(T-4.00)/0.90=1.28\). The observed score is 1.28 null standard deviations above the null mean. That is a descriptive standardization, not a p-value and not evidence by itself that the result is significant.
Why the null calibration matters
A standardized statistic is only as useful as its null distribution. In the simple model where each pair is independent and has the ordinary sib-pair null distribution \(P(K=0)=1/4\), \(P(K=1)=1/2\), and \(P(K=2)=1/4\), the expected count is \(1\) and its variance is \(1/2\). For \(m\) independent pairs, the total has null mean \(m\) and variance \(m/2\). That calculation is a useful reference, but it is not a general substitute for calibration: real scans use marker data, may include related pairs, and may apply ascertainment-aware procedures.
A practical calibration simulates or permutes data under the no-linkage null and repeats the analysis. Each replicate must use the same pedigree structure, marker map and missingness pattern, and the same statistic. The null procedure must also respect the family-ascertainment design; an invalid permutation can produce a null distribution that does not represent the data-generating process. For a genome-wide claim, each replicate must cover the complete genome-wide search and record its maximum statistic.
The choice of NPL statistic also matters. A statistic that weights pairs equally and one that weights them according to pedigree information need not yield identical rankings or null distributions. The statistic, its direction, and its calibration should be specified before interpreting a peak. A larger score is evidence for excess sharing only when the statistic is defined so that larger values represent more sharing.
Comparing positions and calibrating the full scan
A scan evaluates a statistic at many genomic positions. The largest observed value is selected because it is the maximum, so its null distribution is not the same as the null distribution at one position chosen in advance. A single-position p-value answers a pointwise question. A region-wide p-value accounts for a search over a stated region. A genome-wide p-value accounts for selecting the maximum over the complete genome-wide search. These are distinct claims.
Worked Example: An Empirical Genome-Wide P-Value
Suppose a hypothetical study scans all autosomes and records a maximum NPL statistic of 3.08. It generates \(B=9{,}999\) valid null replicates; each replicate repeats the complete autosomal scan and saves its genome-wide maximum. In 621 replicates, the maximum is at least 3.08. Using the common plus-one estimate,
The estimated genome-wide p-value is 0.0622. It is the fraction, with the plus-one adjustment, of full-genome null scans whose maximum is at least as large as the observed maximum. At a prespecified 5% genome-wide significance level, this result does not meet the threshold. It does not establish that the null is true; it says the scan did not provide sufficient evidence to reject the no-linkage null at that stated level under the calibration procedure.
If the same 621 exceedances had instead come from null scans restricted to one 40-cM interval, the resulting value would be a region-wide p-value for that interval, not a genome-wide p-value. The scope of the search in every null replicate must match the scope named in the conclusion.
Positions within a scan are correlated because nearby marker positions share inheritance information. Consequently, one should not treat each position as an independent test or infer a formal difference between two peaks merely by comparing their NPL values. A maximum-statistic calibration handles the scan-level selection directly, provided the null scans repeat the same search and analysis choices.
Worked Example: Comparing Candidate Positions
Consider a separate hypothetical scan with three prespecified positions in a region. The same NPL statistic is used at each position, and the values below are already standardized using the appropriate position-specific null distributions.
| Position | NPL statistic |
|---|---|
| 18 cM | 2.41 |
| 24 cM | 2.76 |
| 30 cM | 2.62 |
The largest observed statistic among these positions is 2.76 at 24 cM. This identifies the strongest observed sharing signal in the displayed set, but it does not show that 24 cM is significantly better than 18 or 30 cM. The positions are correlated, and no uncertainty calculation for their difference has been supplied. Nor does the table establish genome-wide significance: it only gives three regional values. To make a genome-wide claim, the full-genome maximum must be compared with maxima from full-genome null scans.
Reporting location and resolution
A linkage peak has a location estimate and a resolution limit. The maximum-scoring position is the observed peak location on the analysis grid; it is not necessarily the exact position of a causal locus. Resolution depends on factors such as marker density and informativeness, recombination, pedigree information, and the shape of the sharing profile. A broad or flat peak supports a less precise location than a sharply localized peak, but the grid spacing alone does not establish the true localization accuracy.
A useful report states the statistic and its direction, the peak position and observed value, the calibration scope and p-value, and the interval used to describe the peak’s resolution. It should say how that interval was defined—for example, by a prespecified drop from the maximum or by a confidence procedure—rather than call it a confidence interval without justification. The interval describes localization under that method; it does not guarantee that the causal variant lies within it.
Worked Example: Writing a Cautious Peak Summary
Suppose a hypothetical genome-wide scan has its maximum at 24 cM on chromosome 7, with NPL statistic 3.08 and empirical genome-wide p-value 0.0622 from 9,999 full-genome null scans. A prespecified peak interval, defined by the study’s stated drop-from-maximum rule, extends from 18 to 30 cM. The marker grid in this region is 2 cM.
A careful summary is: “The strongest observed affected-relative sharing signal occurred at 24 cM on chromosome 7 (NPL=3.08). Its empirical genome-wide p-value was 0.0622, so it did not meet the 5% genome-wide threshold. Under the stated peak-interval rule, the reported localization interval was 18–30 cM; the 2-cM grid should not be interpreted as 2-cM localization accuracy.”
This wording separates the observed peak from the interval and separates both from the genome-wide evidence. It does not identify a causal gene, claim a precise locus, or describe the interval as a confidence interval unless the method supports that interpretation.
Common mistake: treating a regional maximum as genome-wide evidence
A common error is to scan a short chromosome segment, compare its largest NPL statistic with null maxima from that same segment, and report the resulting p-value as genome-wide. The calculation may be correct for the segment, but the label is not: it does not account for having searched other chromosomes and regions. The correction is to either report it as a regional p-value with the interval stated, or calibrate the observed full-genome maximum against null replicates that each searched the full genome.
A second mistake is to interpret a high NPL value as proof of causation. NPL evidence concerns excess allele sharing among affected relatives under a linkage model; it does not by itself establish a causal variant, estimate a variant’s effect, or rule out alternative explanations. Conclusions should remain limited to the statistic, null hypothesis, and search that were actually analyzed.
Check Your Understanding
Answer each question using the definitions and distinctions developed above.
- If an affected sib pair has \(P(K=0\mid G)=0.25\), \(P(K=1\mid G)=0.50\), and \(P(K=2\mid G)=0.25\), calculate its expected IBD sharing count.
- Why does standardizing a sharing score to have null mean zero and variance one not, by itself, determine its p-value?
- A maximum statistic is compared with null maxima from scans of one chromosome only. What scope of p-value can this calibration support?
- Why must null replicates record the maximum over the complete genome when the observed result is described as genome-wide?
- What does a reported peak interval say, and why should it not automatically be called a confidence interval?