Why Map and Inheritance Assumptions Matter
A parametric linkage analysis asks whether observed disease and marker data are more likely under linkage than under no linkage, given a specified model. That model includes assumptions not only about disease allele frequency and penetrance, but also about the order and spacing of markers and how disease status is inherited. If those assumptions are wrong, the calculated likelihood can change: a peak may weaken, shift, or appear under one specification but not another.
A high LOD score is therefore evidence conditional on the specified model; it does not establish that the map or inheritance model is correct. The practical question is not simply whether one analysis produces a peak, but whether the main conclusion persists across scientifically plausible specifications.
Where the Assumptions Enter
For two loci, the recombination fraction \(\theta\) is the probability that a meiosis transmits a recombinant rather than a nonrecombinant combination of alleles. No linkage corresponds to \(\theta=0.5\). In multipoint analysis, the marker order determines which loci are adjacent, and the interval distances determine the recombination probabilities between adjacent loci. Those probabilities govern the likelihood of inheritance paths along the map.
A mapping function translates map distance into a recombination fraction. For example, under the Haldane mapping function, which assumes no crossover interference, a distance \(d\) in Morgans corresponds to \(r=(1-e^{-2d})/2\). A different mapping function can give a different recombination fraction for the same map distance. Thus, an incorrect distance or mapping assumption changes the probabilities assigned to inheritance paths, even if the observed genotypes are unchanged.
The inheritance model also affects the likelihood. It specifies which disease genotypes are possible or probable, and how likely each genotype is to produce the observed phenotype. In a pedigree with uncertain genotypes or phase, the likelihood sums over possible underlying states, weighting each by its probability under the model. Changing the model can therefore change which paths or genotypes contribute most to the evidence for linkage.
Within one specified model \(M\), a LOD score at a position or recombination fraction compares the likelihood under linkage with the likelihood under no linkage:
Both likelihoods in this ratio use the same model \(M\). A positive LOD means the data are more likely under the specified linked value than under \(\theta=0.5\) for that model. Comparing LOD scores from different inheritance models requires care: each score has its own model-specific likelihood and no-linkage denominator, so their difference is not automatically a formal test or a direct measure of which model is true.
Worked Examples of Specification Changes
Worked Example: A Recombination-Distance Assumption Changes the LOD
Suppose 20 independent, informative meioses have known phase and no genotyping error; 2 are recombinant and 18 are nonrecombinant. For this illustration, treat the recombination fraction \(r\) as fixed by the map. The likelihood is \(r^R(1-r)^{N-R}\), where \(R\) is the recombinant count and \(N\) is the number of meioses. Compare two map-based values, \(r=0.05\) and \(r=0.20\), with the unlinked value \(r=0.50\).
At \(r=0.05\), the LOD is
At \(r=0.20\), it is
Both specified linked values give positive LOD scores, but the score is lower at \(r=0.20\). The data show relatively few recombinants, so the larger assumed recombination fraction assigns less probability to this pattern. This example isolates the effect of a fixed map-based value; in a full multipoint analysis, the map also affects probabilities across a sequence of markers.
Worked Example: A Wrong Marker Order Implies Extra Transitions
Consider three markers, \(M_1,M_2,M_3\), in the true order \(M_1-M_2-M_3\). In two hypothetical informative meioses, suppose the transmitted founder-haplotype states at these markers are \(A,A,B\) and \(A,B,B\), respectively. Assume each adjacent interval has recombination fraction \(r=0.05\), transitions between intervals follow the two-state recombination model, and we condition on the state at \(M_1\). These assumptions make the illustration a calculation on known inheritance paths, not a full pedigree likelihood.
Under the true order, each path has one transition and one nontransition. The joint probability of the two paths, conditional on their starting states, is therefore
Now suppose the analysis uses the wrong order \(M_1-M_3-M_2\), retaining the same assumed interval recombination fractions. The paths in that order become \(A,B,A\) and \(A,B,B\). The first has two transitions and the second has one, so their joint probability is
The likelihood ratio for the true versus the wrong order in this simplified calculation is \(0.00225625/0.00011875=19\), or \(\log_{10}(19)\approx1.279\) on a base-10 log scale. The wrong order makes the observed paths look less probable because it implies more transitions. Real analyses use observed genotypes and sum over uncertain inheritance paths, so the size and even the direction of an order effect depend on the full data and model.
Worked Example: An Inheritance Assumption Changes Phenotype-Linked Evidence
Consider a hypothetical family with one child. A parent is heterozygous for a disease allele \(D\) and a marker allele \(A\), in coupling phase \(D\)-\(A\); the other parent contributes neither allele. The child is affected and carries \(A\) from the heterozygous parent. For a deliberately simple comparison, assume known parental genotypes and phase, no genotyping error, and evaluate the likelihood at \(\theta=0.10\) against \(\theta=0.50\).
Under a fully penetrant dominant model, the child must inherit \(D\) to be affected. The joint probability of inheriting \(D\) and marker allele \(A\) is \(0.5(1-\theta)\). Thus, for the observed affected child with \(A\), the likelihood ratio is
Under a recessive model, instead suppose the heterozygous parent is a carrier, the other parent is a noncarrier, and the phenocopy probability is \(0.01\). The child cannot have two disease alleles under those parental genotypes, so an affected child is a phenocopy in this simplified model. That phenotype is independent of which marker allele the carrier parent transmits. The probability of an affected child carrying \(A\) is \(0.01\times0.5\) at either value of \(\theta\), giving a likelihood ratio of 1 and a LOD of 0.
The contrast is a consequence of the assumed relationship between disease genotype and phenotype, not evidence that one inheritance model is correct. In a realistic pedigree analysis, uncertainty in genotypes, penetrance, and phase would need to be represented in the likelihood rather than fixed as in this small illustration.
Why Map Errors Can Move or Flatten a Peak
A multipoint LOD profile evaluates linkage evidence across positions on a map. If marker order is wrong, the path of marker inheritance used to calculate the probability of disease-locus inheritance can be wrong as well. A peak may then be displaced, weakened, or strengthened for reasons that reflect the assumed order rather than a stable localization signal. An incorrect interval distance changes transition probabilities: an interval treated as too short assigns too little probability to recombination, while one treated as too long assigns more. The resulting profile may become too sharp or too diffuse, and its maximum can move because different positions imply different combinations of interval probabilities.
These effects do not have a universal direction. For instance, a larger assumed recombination fraction penalizes a dataset with few recombinants in the fixed-\(r\) example above. But in a multipoint pedigree likelihood, many possible inheritance paths and disease-locus positions contribute. The net change depends on their combined probabilities, marker informativeness, and the observed family data.
A Practical Stability Assessment
A sensitivity analysis asks whether the substantive conclusion survives changes to plausible assumptions. It should vary specifications for a reason, rather than searching many alternatives until one produces the most attractive peak. A useful analysis first states the primary specification, then compares it with alternatives supported by map information or biological knowledge.
State marker order, interval distances and mapping function, along with the assumed inheritance model and other fixed parameters. This defines the conditions under which the primary LOD profile is interpreted.
Check credible alternative marker orders or distances, and inheritance models that are scientifically defensible for the disease and pedigree. Do not treat arbitrary alternatives as equally plausible simply because they can be computed.
For each specification, examine the peak’s strength and location and the shape of the profile. Keep each LOD tied to the model that produced it; a change across models is a sensitivity finding, not automatically a formal model comparison.
If a broad region remains supported but peak height varies, report that the regional signal is more stable than its strength. If location or evidence changes substantially, make that dependence explicit rather than presenting one peak as assumption-free.
A compact comparison table can make the assessment transparent. It should identify the changed assumption and record comparable features of each resulting profile. Where specifications differ in the phenotype model, avoid implying that LOD values alone rank those models on a common scale.
Common Mistake and Key Takeaway
- Wrong reasoning: “The largest LOD score identifies the correct marker order and inheritance model.” A LOD score measures linkage evidence conditional on the model used to calculate it; it does not by itself validate that model.
- Correction: Evaluate plausible specifications, note which features of the conclusion persist, and describe model-dependent changes. Do not interpret a model-specific score as proof that its assumptions are true.
Map and inheritance assumptions matter because they assign probabilities to the very transmission and phenotype patterns that generate linkage evidence. A credible conclusion is not necessarily identical under every conceivable model, but its dependence on reasonable alternatives should be visible. The strongest interpretation distinguishes robust features of the evidence from features that move when the specification changes.
Check Your Understanding
Answer each question using the model and data assumptions stated in the prompt.
- Why can using a larger fixed recombination fraction lower a LOD score when the observed meioses contain few recombinants?
- In the marker-order example, why does the assumed order \(M_1-M_3-M_2\) assign the first inheritance path two transitions?
- What does a positive LOD establish, and what does it not establish about the inheritance model?
- If a peak remains in roughly the same region across plausible map distances but its height changes, which feature is more stable: location or strength?
- Why should LOD scores from two different inheritance models not automatically be treated as a formal ranking of those models?