Common Errors in Parametric Linkage Interpretation

Why Careful Interpretation Matters

A parametric linkage result is conditional on a specified inheritance model, pedigree data, and genetic map. Even when the calculations are correct, a conclusion can go wrong if the recombination fraction is treated as unrestricted, phase is guessed incorrectly, family-specific maxima are added as though they came from one common model, or a LOD score is read as a probability that a causal variant has been found.

The central quantity is the LOD score: the base-10 logarithm of a likelihood ratio comparing a specified linkage model with the unlinked model. The score summarizes relative support in the observed data under those models; it does not by itself settle whether the model assumptions hold or identify the biological cause of a disease.

Definition: In a two-point linkage analysis, the recombination fraction $\theta$ is the probability that a transmitted gamete is recombinant between the disease locus and a marker, under the specified model. Its allowed range is $0\leq\theta\leq 0.5$. The value $\theta=0$ represents no recombination in the model, while $\theta=0.5$ represents independent assortment at the two loci. For a fixed $\theta$, the LOD score is $$ Z(\theta)=\log_{10}\!\left(\frac{L(\theta)}{L(0.5)}\right), $$ where $L(\theta)$ is the likelihood of the observed pedigree data under that value of $\theta$, with the remaining model specification held fixed.

A recombination fraction cannot be negative or exceed one half: values outside that range do not describe the two-point linkage parameter. The upper boundary is the unlinked reference in this definition of the LOD score. Consequently, if the largest likelihood over the allowed range occurs at $\theta=0.5$, then the maximum LOD score is zero. A LOD score evaluated at some other fixed value can be negative; that means the data have lower likelihood there than under $\theta=0.5$, not that the estimated recombination fraction is negative or that there is a special “negative linkage” state.

Boundary Estimates Are Allowed, but Need Careful Interpretation

A maximum-likelihood estimate can occur at a boundary. In a simple example with $n$ independent, informative meioses and known phase, let $r$ be the number classified as recombinant. If each meiosis has recombinant probability $\theta$, the likelihood, apart from a factor not depending on $\theta$, is $$ L(\theta)\propto \theta^r(1-\theta)^{n-r},\qquad 0\leq\theta\leq 0.5. $$ For $0<r<n$, differentiating the log likelihood gives an unconstrained maximum at $r/n$; the constrained estimate is that value if it is no greater than $0.5$, and otherwise the maximum is at $0.5$. If $r=0$, the likelihood decreases as $\theta$ increases, so its maximum is at zero. This simple binomial calculation illustrates the boundaries; a full pedigree likelihood can be more complex.

Worked Example: A Maximum at Zero

Suppose, hypothetically, that a simplified analysis has 10 independent informative meioses, all classified as nonrecombinant. Assume phase is known, the classifications are error-free, and the meioses have a common recombination fraction. Then $r=0$, and the likelihood is $L(\theta)\propto(1-\theta)^{10}$. Because $1-\theta$ decreases over the allowed range, the maximum occurs at $\hat{\theta}=0$.

The likelihood ratio at that estimate, relative to $\theta=0.5$, is $$ \frac{L(0)}{L(0.5)} =\frac{1^{10}}{(0.5)^{10}} =1024. $$ Thus the maximum LOD score in this simplified calculation is $$ Z(0)=\log_{10}(1024)=3.010. $$

This boundary estimate means that zero recombination gives the largest likelihood among the allowed values for these data and this model. It does not establish that the loci are physically at exactly the same position: a finite set of meioses can contain no observed recombinants even when the underlying recombination probability is positive.

The distinction between an estimated boundary and a known biological value matters particularly with sparse or uninformative data. A maximum at $\theta=0$ is not a measurement of zero base pairs, and a maximum at $\theta=0.5$ does not tell us whether the loci are on different chromosomes or far apart on the same chromosome. In either case, interpret the estimate within the model and the information in the pedigree.

Phase Determines Which Transmissions Count as Recombinant

Phase specifies which alleles lie together on each of an individual's two homologous chromosomes. For a heterozygous disease locus with alleles $D$ and $d$, and a heterozygous marker with alleles $M$ and $m$, coupling phase is $DM/dm$; repulsion phase is $Dm/dM$. A transmitted allele pair is called recombinant or nonrecombinant relative to that phase. The same observed pair can therefore have opposite classifications under the two phase assignments.

Worked Example: One Transmission, Two Phase Assignments

Consider a hypothetical heterozygous parent whose transmitted disease allele is known to be $D$, and whose transmitted marker allele is $m$. Suppose the analysis compares two possible phases and fixes $\theta=0.1$. Assume the transmitted pair is observed without error and use the single informative transmission likelihood.

Under coupling phase $DM/dm$, the $D$-bearing chromosome carries $M$. A transmission of $D$ with $m$ is therefore recombinant, with likelihood contribution $\theta=0.1$. Under repulsion phase $Dm/dM$, $D$ is on the same chromosome as $m$, so the same transmission is nonrecombinant, with likelihood contribution $1-\theta=0.9$.

The likelihood ratio for repulsion versus coupling for this transmission is $$ \frac{0.9}{0.1}=9, $$ or $ \log_{10}(9)=0.9542 $ on the LOD scale. This comparison is between two phase assignments for this observation, not a linkage LOD score against $\theta=0.5$. The example shows why phase must be specified before classifying transmissions. When phase is uncertain, a linkage analysis must handle that uncertainty according to its model—for example, by summing likelihood contributions over possible phases with appropriate weights—rather than silently choosing whichever phase makes each transmission appear nonrecombinant.

Phase is not the same parameter as $\theta$. Changing phase changes the interpretation and likelihood of observed allele transmissions; changing $\theta$ changes the modeled probability of recombination. Treating phase as known when it is not, or using different convenient phases for different transmissions without a coherent likelihood model, can distort the linkage evidence.

Sum Scores at a Common Parameter Before Maximizing

When independent families share a common recombination fraction and the same specified model, their likelihoods multiply. Taking base-10 logarithms turns that product into a sum. Thus, at a fixed common value of $\theta$, family-specific LOD contributions can be added. If the target is the maximum combined LOD, first form that sum as a function of the common $\theta$, then maximize it over the allowed range.

Key distinction: With family $i$'s likelihood $L_i(\theta)$, the common-$\theta$ combined LOD is $$ Z_{\mathrm{combined}}(\theta) =\sum_i \log_{10}\!\left(\frac{L_i(\theta)}{L_i(0.5)}\right). $$ Its maximum is $\max_{0\leq\theta\leq0.5} Z_{\mathrm{combined}}(\theta)$. In general, this is not the same as adding the separately maximized family scores, $\sum_i\max_{\theta}Z_i(\theta)$, because those separate maxima may occur at different values of $\theta$.

Worked Example: Common-Parameter Sum Versus Separate Maxima

Suppose two hypothetical families each contribute six independent informative meioses under the simple known-phase model. Family A has one recombinant and five nonrecombinants; Family B has three recombinants and three nonrecombinants. Assume the families are independent and the analysis requires one common $\theta$.

For Family A, its likelihood is proportional to $\theta(1-\theta)^5$, so its maximum occurs at $\hat{\theta}_A=1/6$. Its maximized LOD is $$ Z_A(1/6) =\log_{10}\!\left( \frac{(1/6)(5/6)^5}{(1/2)^6} \right) \approx 0.632. $$ For Family B, the likelihood is proportional to $\theta^3(1-\theta)^3$, with its maximum at $\hat{\theta}_B=3/6=0.5$. Its maximized LOD is zero, because that maximum is at the unlinked reference. Adding these separate maxima gives $0.632$, but they do not describe a single common value of $\theta$.

For the common-$\theta$ analysis, combine the data first: there are 12 meioses, four recombinant and eight nonrecombinant. The likelihood is proportional to $\theta^4(1-\theta)^8$, maximized at $4/12=1/3$. The maximum combined LOD is $$ \begin{aligned} Z_{\mathrm{combined}}(1/3) &=\log_{10}\!\left( \frac{(1/3)^4(2/3)^8}{(1/2)^{12}} \right)\\ &=4\log_{10}(1/3)+8\log_{10}(2/3)-12\log_{10}(1/2)\\ &\approx 0.295. \end{aligned} $$ This differs from the sum of the separate maxima. The appropriate calculation depends on the scientific model: a common-$\theta$ model requires a common value, while a model allowing families to differ must represent that difference explicitly. Adding individually optimized scores without stating such a model does not do either job.

The same issue arises when reading a multipoint LOD profile. A score at a particular position is evaluated under that position's specified model and map; the peak is the largest score across the analyzed positions. It is important to know whether a reported number is a score at a fixed position, a maximum over position, or a sum across families at a shared position. These operations answer different questions.

What a Likelihood Ratio Does—and Does Not—Mean

Because $Z(\theta)$ is a base-10 logarithm, a LOD score $z$ corresponds to a likelihood ratio $10^z$. This ratio compares how well the observed data are supported by the numerator model and the unlinked reference model, with the assumptions used to calculate their likelihoods. It is not the probability that linkage is true, the probability that the null hypothesis is true, or posterior odds between hypotheses. Those probability statements require additional ingredients, such as prior probabilities, and are not supplied by the LOD score alone.

Worked Example: Translating a LOD Score into Relative Likelihood

Suppose, hypothetically, a specified linkage model at a marker position has a LOD score of $2.4$ relative to the unlinked model. The corresponding likelihood ratio is $$ 10^{2.4}\approx251.2. $$

The appropriate interpretation is that, under the specified models and assumptions, the observed data have about 251 times the likelihood under that linkage model as under the unlinked reference. It is not correct to say that there is a 99.6% probability of linkage, or that linkage has been proven. A likelihood ratio compares model support for these data; it does not assign probabilities to models.

The likelihood comparison is only as meaningful as its inputs. Pedigree structure, penetrance, allele-frequency assumptions, phase, genetic map, and treatment of missing genotypes can affect the likelihood. A large score under a misspecified model can be misleading, just as a small score under an appropriate model may reflect limited information. Interpret the reported evidence in the context of the assumptions and the analysis performed.

A Linkage Peak Does Not Identify a Causal Variant

Linkage evidence concerns co-segregation within families: the observed inheritance patterns are more compatible with a specified linked model than with the unlinked reference. A peak in a multipoint profile indicates a region where the modeled data provide their greatest relative support among the positions examined. It does not demonstrate that the marker itself causes disease, or that a particular variant in the region is causal.

A marker is used to track inheritance; it need not be the disease-causing change. A linked region may contain multiple variants, and the peak's location and strength depend on the available pedigree information and model assumptions. Identifying a causal variant requires evidence beyond the existence of a linkage peak. Keep the claim proportional to the result: linkage supports a chromosomal region under the stated analysis, not a specific molecular mechanism or causal allele.

Common Mistakes and a Practical Check

Allowing $\theta$ outside its range. A negative estimate or an estimate above $0.5$ is not a valid recombination fraction in this framework. Constrain estimation to $0\leq\theta\leq0.5$ and interpret boundary maxima as model-based estimates.
Calling every observed allele pair recombinant or nonrecombinant without phase. The classification depends on the parental haplotypes. State or model phase before using a transmission as recombination evidence.
Adding family-specific maximum LOD scores by default. Those maxima can correspond to different recombination fractions. For a common-$\theta$ model, sum the family LOD contributions at the same $\theta$, then maximize the sum.
Calling a likelihood ratio a probability. A ratio such as 251 to 1 is relative likelihood support for the observed data under two specified models, not a 251-to-1 posterior probability or certainty that linkage is real.
Calling the peak a causal variant. The peak supports relative linkage evidence for a region under the analysis assumptions; it does not identify which variant, if any, causes the phenotype.

Check the parameter and phase.
Confirm that $\theta$ is in its allowed range and that phase is known or handled explicitly before interpreting transmissions.

Identify what was combined.
Determine whether the score is evaluated at a fixed $\theta$, summed across families at a common value, or maximized over a parameter or genomic position.

Translate the score accurately.
If useful, convert $z$ to $10^z$ and describe it as a likelihood ratio under the specified models—not as a probability that linkage is true.

Limit the biological conclusion.
Report evidence for linkage to a region under the stated assumptions; do not claim that a peak alone proves a causal variant.

Key takeaway: Interpret parametric linkage results by keeping the recombination-fraction boundary, phase assignment, parameter over which scores are combined or maximized, and meaning of the likelihood ratio explicit. A linkage peak is evidence about co-segregation under a model, not proof of causation.

Check Your Understanding

Answer these questions by referring to the model and interpretation, not just the numerical result.

In the two-point LOD definition, why is $\theta=0.5$ the unlinked reference, and what does a maximum at $\theta=0.5$ imply about the maximum LOD?
A finite pedigree has no observed recombinants and its likelihood is maximized at $\theta=0$. What does this estimate say, and what does it not establish?
For a parent with haplotypes $DM/dm$, is transmission of $D$ with $m$ recombinant or nonrecombinant? How would the answer change under phase $Dm/dM$?
Two families have different maximizing values of $\theta$. Why is the sum of their separately maximized LOD scores not automatically the maximum LOD under a common-$\theta$ model?
A reported LOD score is 1.5. What likelihood ratio does it represent, and why is that not the probability that linkage is true?
What conclusion can a multipoint linkage peak support, and why does it not by itself establish a causal variant?