What a Multipoint LOD Profile Shows
A multipoint linkage analysis evaluates evidence for linkage at a series of possible chromosome positions, using information from several markers together. Its results are often displayed as a curve: the horizontal axis gives map position, usually in centimorgans (cM), and the vertical axis gives the LOD score. Reading this curve involves two different questions. How strongly does the data support linkage? And how narrowly does the profile localize the position? Peak height addresses the first question; the shape and width around the peak help address the second.
A profile is conditional on the analysis model. For interpretation, assume the pedigree structure and disease model have been specified, including the inheritance and penetrance assumptions, and that the genetic map and other inputs used by the analysis are suitable. The same model and data must be used consistently along the chromosome. A peak is therefore a feature of the data under that model, not a model-free discovery of a disease gene.
In a parametric analysis, the likelihoods depend on the specified inheritance and disease model and on the pedigree data. As the assumed disease-locus position moves, the recombination relationships between that position and the typed markers change. The profile summarizes how well each position fits relative to the no-linkage reference. It does not directly display the probability that the locus is at a particular position.
Reading the Peak Height
A positive LOD score means that, under the specified model, the likelihood at that position is greater than the no-linkage likelihood. A score of zero means the likelihoods are equal. A negative score means the linkage likelihood is smaller than the reference likelihood. Because the score is a base-10 logarithm, a difference of one LOD unit corresponds to a tenfold difference in likelihood ratio.
For example, \(Z(x)=3\) means \(L(x)/L_0=10^3=1000\): the observed data are 1,000 times as likely under that linkage model at \(x\) as under the no-linkage reference model. This is a likelihood comparison, not a posterior probability that linkage is true. It does not account by itself for prior odds, uncertainty in the model assumptions, or the consequences of searching many positions.
A LOD score of 3 has traditionally been used as a benchmark for strong evidence of linkage, and a score near or below \(-2\) has often been treated as evidence against linkage at the tested position. These are conventions, not universal decision rules. The appropriate interpretation depends on the analysis design, the assumptions, and whether a threshold has been calibrated for the relevant search. In particular, one should not treat a threshold for a single prespecified position as automatically valid for a genome-wide scan.
Worked Example: Finding the Highest Scoring Position
Suppose a hypothetical parametric multipoint analysis gives the following profile. The positions are reported in cM on one chromosome, and each score is calculated using the same specified model and no-linkage reference.
| Position (cM) | LOD score |
|---|---|
| 0 | 0.2 |
| 5 | 1.1 |
| 10 | 2.4 |
| 15 | 3.2 |
| 20 | 2.7 |
| 25 | 1.4 |
The largest listed score is \(3.2\), at \(15\) cM. Thus, among the positions evaluated, the profile peaks at 15 cM. Its likelihood ratio at the peak is $$ 10^{3.2}=10^3\times 10^{0.2}\approx 1000\times 1.585=1585. $$
Under the specified model, the data at this position are about 1,585 times as likely as under the no-linkage reference. This is strong evidence by the traditional LOD-3 benchmark, but the peak does not identify a causal variant. Nor does a score reported at a grid point prove that the exact underlying maximum is at precisely 15 cM: the estimate is limited by the positions evaluated and the resolution of the profile.
Peak Width and a One-LOD Support Interval
Peak height and peak width describe different features. A high, narrow peak indicates a large maximum likelihood ratio concentrated in a relatively small map region. A high, broad peak indicates strong evidence against the no-linkage reference over a wider range of positions, but less precise localization from the profile. A lower peak can still be narrow; narrowness alone does not imply strong evidence for linkage.
A convenient descriptive summary is the one-LOD support interval: the set of positions whose LOD scores are no more than one unit below the maximum. If \(Z_{\max}\) is the maximum score, its threshold is \(Z_{\max}-1\). The endpoints are where the profile crosses that threshold, when such crossings occur on both sides of the peak. This interval summarizes the shape of the likelihood profile. It is not automatically a confidence interval with a specified coverage probability.
For a plotted or tabulated profile, an endpoint may fall between evaluated positions. Linear interpolation on the displayed LOD scale can give a useful approximate endpoint, but it is only an approximation to the crossing in the underlying profile. Report the method and avoid implying a finer resolution than the grid and map support.
Worked Example: Comparing Peak Height and Width
Consider two hypothetical profiles evaluated at positions 0, 5, 10, 15, 20, 25, and 30 cM. Profile A has scores \(0.4, 1.8, 3.6, 4.2, 3.5, 1.7, 0.5\); Profile B has scores \(0.1, 1.5, 3.0, 3.9, 3.0, 1.5, 0.1\). Assume both use the same type of no-linkage reference and are based on the same model specification.
Profile A peaks at 15 cM with \(Z_{\max}=4.2\), so its one-LOD threshold is \(4.2-1=3.2\). On the left, the threshold lies between 5 cM (score 1.8) and 10 cM (score 3.6). Linear interpolation gives $$ 5+\frac{3.2-1.8}{3.6-1.8}(10-5) =5+\frac{1.4}{1.8}\times5 \approx 8.89\ \text{cM}. $$ On the right, it lies between 20 cM (score 3.5) and 25 cM (score 1.7): $$ 20+\frac{3.5-3.2}{3.5-1.7}(25-20) =20+\frac{0.3}{1.8}\times5 \approx 20.83\ \text{cM}. $$ The approximate one-LOD support interval for A is 8.89 to 20.83 cM, about 11.94 cM wide.
Profile B peaks at 15 cM with \(Z_{\max}=3.9\), giving a one-LOD threshold of \(2.9\). Its left crossing is between 5 and 10 cM: $$ 5+\frac{2.9-1.5}{3.0-1.5}\times5 \approx 9.67\ \text{cM}. $$ Its right crossing is between 20 and 25 cM: $$ 20+\frac{3.0-2.9}{3.0-1.5}\times5 \approx 20.33\ \text{cM}. $$ The approximate interval is 9.67 to 20.33 cM, about 10.66 cM wide. Profile A has the higher peak, while Profile B has the slightly narrower one-LOD interval. Thus A gives the stronger peak likelihood ratio, but width and height should not be treated as interchangeable measures.
From a Profile to a Linked Region
A profile is usually interpreted as a whole, rather than by selecting one point and ignoring its neighbors. A broad elevated region can support linkage across multiple nearby positions, while a sharp isolated maximum may be sensitive to the data or assumptions. A local maximum is simply higher than nearby evaluated scores; it is not necessarily the global maximum or persuasive evidence against no linkage. Always check the score scale and the reference model before describing a peak as evidence for linkage.
The peak location is the position that maximizes the likelihood ratio over the positions considered. It is an estimate of where the disease locus may lie under the model, not a direct observation of the locus. Map positions are genetic distances, not necessarily physical distances in base pairs. The relationship between the two can vary along a chromosome, so a cM interval should not be converted to a physical interval without the relevant map information.
Worked Example: A Candidate Variant Within a Supported Region
Suppose a hypothetical profile has scores of 2.5 at 8 cM, 3.7 at 12 cM, 3.1 at 16 cM, and 1.8 at 20 cM. The maximum is \(3.7\) at 12 cM, so the one-LOD threshold is \(2.7\). Linear interpolation gives a left endpoint of $$ 8+\frac{2.7-2.5}{3.7-2.5}(12-8) =8+\frac{0.2}{1.2}\times4 \approx 8.67\ \text{cM}, $$ and a right endpoint of $$ 16+\frac{3.1-2.7}{3.1-1.8}(20-16) =16+\frac{0.4}{1.3}\times4 \approx 17.23\ \text{cM}. $$
A hypothetical candidate variant mapped to 14 cM lies within this approximate one-LOD interval. If the profile is linearly interpolated between 12 and 16 cM, its score at 14 cM is about \(3.4\), or \(0.3\) LOD units below the maximum. The likelihood ratio there is \(10^{3.4}\approx2512\), compared with \(10^{3.7}\approx5012\) at the peak; the ratio of these likelihood ratios is \(10^{-0.3}\approx0.501\). This describes the relative fit at the two positions under the linkage model.
Being inside the interval does not establish that the candidate variant causes the disease. The profile supports a region for linkage under the analysis assumptions; it does not by itself distinguish a causal variant from nearby linked markers or other variants in the region. A causal claim requires additional evidence beyond the location and shape of this linkage profile.
Why the Profile Depends on the Analysis
The curve is an output of a specified analysis, not an intrinsic property of the chromosome. Changes to the assumed disease model, map, or pedigree likelihood can change both the height and location of a peak. The interpretation should therefore state that the evidence is conditional on the assumptions used. If there is reason to doubt those assumptions, a high or narrow peak does not remove that concern.
The amount of localization also depends on how much information the data provide across the region. The markers, their positions, and the observed family data determine how sharply the likelihood changes as the candidate position moves. A broad profile may reflect limited ability to distinguish nearby positions; it does not mean that every position in the interval is equally likely. A narrow profile is likewise not proof that the causal change has been pinpointed.
When a profile is displayed on a finite grid, the reported maximum is the highest value among the evaluated points. A finer grid may refine the estimated peak coordinate, but it cannot create information absent from the pedigree data. If an interpolation or smoothing method is used, it should be understood as a way of displaying or approximating the profile, not as new evidence.
A Short Justification for the LOD Scale
The logarithmic scale makes likelihood ratios easier to compare. If \(Z(x)=\log_{10}(L(x)/L_0)\), exponentiating both sides with base 10 recovers the ratio:
Now compare a position \(x\) with the maximum position \(x_{\max}\), using the same reference likelihood. Subtracting their LOD scores cancels that reference:
Therefore, if \(Z(x)=Z_{\max}-1\), then $$ \frac{L(x)}{L(x_{\max})}=10^{-1}=0.1. $$ This is the basis for describing the one-LOD interval as the region where the likelihood is at least one tenth of its maximum, subject to the same model and reference. It does not turn that region into a probability interval: likelihood ratios alone are not posterior probabilities or confidence levels.
Common Mistakes
- Calling the peak a causal variant. The maximum identifies the best-supported position in the linkage profile under the assumed model. It does not show that a variant at that coordinate causes disease.
- Using height as a measure of precision. A higher peak means a larger maximum likelihood ratio against the reference. Precision is better described by the profile’s shape and width, with the limitations of a support interval stated.
- Calling a one-LOD interval a 95% confidence interval. The one-LOD rule is a descriptive likelihood-based summary. It has no automatic 95% coverage interpretation.
- Reading a LOD score as a probability. A score of 3 means a likelihood ratio of 1,000 under the stated comparison, not a 99.9% probability that linkage is true.
- Reporting more positional precision than the data support. A grid maximum at 15 cM is not necessarily an exact location. State the evaluated resolution and identify interpolated endpoints as approximate.
Check Your Understanding
Use the profile definitions and interpretation rules above to answer these questions.
- If a profile has a maximum LOD score of 2.5, what likelihood ratio does that correspond to at the maximum relative to the no-linkage reference?
- Two profiles have the same peak location, but one has a higher peak and the other a narrower one-LOD interval. What does each feature tell you?
- Why is a one-LOD support interval not automatically a 95% confidence interval?
- A candidate variant lies at the profile’s peak coordinate. What conclusion about causality is justified by the profile alone?
- Why should the coordinate of a grid maximum not be reported as an exact biological location?