Multipoint Linkage Likelihoods for Pedigrees

Why Multipoint Likelihoods Use Hidden Inheritance Paths

A pedigree rarely reveals every allele transmitted at every locus. Marker genotypes provide indirect evidence about those transmissions, while disease phenotypes provide evidence about which disease alleles were inherited. A multipoint linkage likelihood combines both kinds of evidence across a mapped set of loci. Its central operation is to sum over latent inheritance paths: each possible path receives a probability from the genetic map and a likelihood from the observed data.

The candidate disease-locus position matters because it determines the recombination probabilities connecting the disease locus to the markers. A path that fits the marker genotypes well may fit the disease phenotypes poorly, or vice versa. The likelihood weighs these contributions together rather than selecting one guessed path.

Definition: An inheritance path specifies, for every meiosis in the pedigree and every locus in the ordered map, which of the transmitting parent's two homologous chromosomes was transmitted. A multipoint pedigree likelihood at candidate disease position \(t\) is the probability of the observed marker and disease data under the specified pedigree model and map, after summing over unobserved inheritance paths and, when not fixed, founder haplotypes.

States, Transitions, and Data Contributions

Suppose the map contains marker loci \(M_1,\ldots,M_K\) in order and a disease locus \(Q\) at candidate position \(t\). Insert \(Q\) into the ordered map. For each meiosis, record at each locus whether the transmitted homolog is 0 or 1. The collection of these indicators across all meioses is the inheritance vector \(v_j\) at locus \(j\). If there are \(m\) meioses, \(v_j\) has \(m\) entries and can take \(2^m\) values.

Let \(\theta_j\) be the recombination fraction between adjacent loci \(j\) and \(j+1\). For one meiosis, the probability of a switch in transmitted homolog across that interval is \(\theta_j\), and the probability of no switch is \(1-\theta_j\). Under the usual model of independent recombination across meioses and intervals, the transition probability for the entire inheritance vector is

P(v_{j+1}\mid v_j) =\prod_{r=1}^{m} \theta_j^{I(v_{jr}\ne v_{j+1,r})} (1-\theta_j)^{I(v_{jr}=v_{j+1,r})},

where \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. The map or mapping function supplies the recombination fractions. The independence-across-intervals assumption is appropriate for a no-interference model; if a different recombination model is specified, its transition probabilities must be used instead.

Conditional on founder haplotypes and an inheritance vector, transmitted alleles determine the pedigree members' genotypes. Marker observations then contribute probabilities through the marker genotype model, and disease observations contribute probabilities through the penetrance model. These are probabilities of the observed data conditional on the genotypes; they are not probabilities that a particular inheritance path is true.

Likelihood construction: Let \(F\) denote the founder haplotypes, \(Y\) all observed marker data, and \(D\) all observed disease phenotypes. A general likelihood at position \(t\) is

L(t)= \sum_F P(F) \sum_{v_1,\ldots,v_J} P(v_1)\prod_{j=1}^{J-1}P(v_{j+1}\mid v_j) \;P(Y\mid F,v_1,\ldots,v_J) \;P(D\mid F,v_1,\ldots,v_J,t).

Here the ordered loci include the disease locus, \(J\) is their number, and the model determines the initial inheritance-state probabilities \(P(v_1)\). The likelihood terms for marker and phenotype data may themselves factor across individuals or loci when the model's conditional-independence assumptions justify that factorization.

In a simple model with conditionally independent marker observations and phenotypes, the two data terms can be written as products of their genotype-specific probabilities. If founder haplotypes are treated as known, the outer sum over \(F\) is omitted. If they are unknown, their prior probabilities must be included; omitting that uncertainty can change the likelihood.

Worked Examples of the Components

Worked Example: Probability of a Transmission Pattern Across Two Intervals

Hypothetical setup. In one meiosis, consider three ordered loci \(A,B,C\), with recombination fractions \(\theta_{AB}=0.10\) and \(\theta_{BC}=0.20\). At \(A\), the transmitted homolog is 0. Find the probability that the path is 0 at \(A\), switches to 1 at \(B\), and remains 1 at \(C\).

The required event is a switch across \(A\) to \(B\), followed by no switch across \(B\) to \(C\). Under independent recombination across these intervals,

P(0,1,1)=\theta_{AB}(1-\theta_{BC}) =(0.10)(1-0.20) =(0.10)(0.80) =0.0800.

Thus this particular path has probability 0.0800 conditional on the starting homolog at \(A\). It is not the probability of all paths ending at homolog 1 at \(C\): other switch patterns would also need to be included for that event. For a pedigree with multiple meioses, multiply the corresponding transition probabilities across meioses as well, under the stated independence assumption.

Worked Example: Summing Two Latent Paths

Hypothetical setup. Suppose a simplified pedigree model has two possible inheritance paths, \(A\) and \(B\), each with prior probability 0.50. The probability of the observed marker data given path \(A\) is 0.80, and given path \(B\) it is 0.20. The probability of the observed phenotype given path \(A\) is 0.05, and given path \(B\) it is 0.80. Assume marker and phenotype observations are conditionally independent given the path. Find the joint likelihood of the observed marker and phenotype data.

For each path, multiply its prior path probability by both conditional data probabilities, then add the path contributions:

\begin{aligned} L &=P(A)P(Y\mid A)P(D\mid A) +P(B)P(Y\mid B)P(D\mid B)\\ &=(0.50)(0.80)(0.05)+(0.50)(0.20)(0.80)\\ &=0.0200+0.0800\\ &=0.100. \end{aligned}

The joint likelihood is 0.100 under this model. The marker-data probability alone is \(P(Y)=(0.50)(0.80)+(0.50)(0.20)=0.500\), so the conditional probability of the phenotype data given the marker data is \(0.100/0.500=0.200\). The phenotype likelihood is not obtained by choosing whichever path best matches the phenotype: both possible paths contribute, weighted by how probable they are and how well they explain the marker observations.

Worked Example: Placing a Candidate Disease Locus Between Markers

Hypothetical setup. Consider one meiosis and two perfectly informative marker observations. They show that homolog 0 was transmitted at both flanking markers, \(M_L\) and \(M_R\). A candidate disease locus \(Q\) lies between them, with recombination fractions \(\theta_{LQ}=0.05\) and \(\theta_{QR}=0.10\). Assume the prior probability of homolog 0 at \(M_L\) is 0.50. For illustration, the phenotype probability is 0.10 if homolog 0 is transmitted at \(Q\) and 0.70 if homolog 1 is transmitted there. Calculate the joint probability of the two observed marker inheritance states and the phenotype, summing over the unobserved state at \(Q\).

There are two possible states at \(Q\). If the state is 0, there is no switch on either side of \(Q\), so its contribution is

$$ (0.50)(1-0.05)(1-0.10)(0.10) =(0.50)(0.95)(0.90)(0.10) =0.04275. $$

If the state at \(Q\) is 1, there is a switch from \(M_L\) to \(Q\) and another switch from \(Q\) to \(M_R\). Its contribution is

$$ (0.50)(0.05)(0.10)(0.70)=0.00175. $$

Summing the mutually exclusive latent states gives

$$ L(t)=0.04275+0.00175=0.04450. $$

This calculation uses the stated map fractions, the assumed starting-state probability, and the specified phenotype probabilities. In an actual pedigree likelihood, marker and phenotype probabilities would be computed from all relevant relatives' genotypes and the penetrance model, rather than assigned directly to a single transmission state.

Why the Sum-Over-Paths Formula Holds

The likelihood is a marginal probability: the inheritance path and possibly founder haplotypes are unobserved, so their probabilities must be summed out. The justification is the law of total probability. For each possible \(F\) and complete path \(v=(v_1,\ldots,v_J)\), the joint probability of the observed data and that latent configuration is its prior probability multiplied by the conditional probability of the observations.

Under the model, the path probability factors into an initial-state probability and adjacent-locus transition probabilities. If the marker and disease observations are conditionally independent given the genotypes induced by \(F\) and \(v\), their joint conditional probability is the product of the two data terms. Summing those joint probabilities over all mutually exclusive latent configurations gives the marginal likelihood:

\begin{aligned} P(Y,D) &=\sum_F\sum_v P(F,v,Y,D)\\ &=\sum_F P(F)\sum_v P(v) \,P(Y\mid F,v)\,P(D\mid F,v,t)\\ &=\sum_F P(F)\sum_{v_1,\ldots,v_J} P(v_1)\prod_{j=1}^{J-1}P(v_{j+1}\mid v_j) \,P(Y\mid F,v)\,P(D\mid F,v,t). \end{aligned}

The factorization is justified only if the conditional-independence assumptions hold. For example, a model that allows correlated genotyping errors cannot automatically factor the marker likelihood into independent per-observation terms. The sum itself remains the principle; the conditional probability model determines the terms being summed.

Comparing Linkage at Candidate Positions

A multipoint likelihood can be evaluated at each candidate disease-locus position by inserting that position into the map and using the corresponding recombination fractions. The marker-to-marker map remains fixed; what changes is the transition structure connecting the disease locus to the markers and hence the likelihood of the joint data.

A conventional multipoint LOD score compares the likelihood under linkage at position \(t\) with the likelihood under no linkage between the disease locus and the marker map:

Z(t)=\log_{10}\!\left(\frac{L_{\text{linked}}(t)} {L_{\text{unlinked}}}\right).

The unlinked model uses recombination fraction \(1/2\) between the disease locus and the marker transmissions, while retaining the other specified pedigree and marker assumptions. A positive LOD score means the observed data are more likely under the linked model than under this unlinked model; it is not the probability that linkage is true. The likelihood ratio depends on the models and their assumptions, including the penetrance and founder-haplotype models.

Common Mistakes and Practical Checks

Selecting one best-fitting path. A single path may explain the observations well, but the likelihood is the sum over all possible paths, each weighted by its probability and its data likelihood.
Adding probabilities across intervals instead of multiplying. For a specified sequence of independent interval events, multiply their probabilities. Sum only over alternative, mutually exclusive paths.
Using marker evidence twice. Marker observations enter their own likelihood contribution. If marker data have already been used to calculate a conditional distribution over paths, do not multiply by the same marker likelihood again without accounting for that conditioning.
Ignoring the candidate position. The disease locus must be placed in the ordered map, and its map-based transition probabilities must be updated when \(t\) changes.
Confusing an inheritance path with observed genotype data. A path is latent transmission information. Genotypes and phenotypes are observations whose probabilities are calculated conditional on that latent information and the model.

A useful calculation check is to verify that transition probabilities from any given state sum to 1 across all possible next states. Also check that each path contribution is nonnegative and that the final likelihood is a probability for the specified data under the model. These checks catch many indexing and arithmetic errors before a likelihood ratio is interpreted.

Key takeaway: A multipoint pedigree likelihood combines map-based inheritance transitions with marker and phenotype evidence conditional on latent genotypes, then sums over all possible inheritance paths and any uncertain founder haplotypes. The candidate disease position changes the transitions linking disease and marker loci.

Check Your Understanding

Answer each question by identifying the relevant model component or calculation.

For one meiosis across two adjacent intervals, what event does the product \(\theta_1(1-\theta_2)\) represent?
Why does a pedigree likelihood sum over inheritance paths instead of using only the most likely path?
In a multipoint likelihood, what determines the marker-data contribution conditional on a path and founder haplotypes?
When a candidate disease-locus position changes, which transition probabilities need to be reconsidered?
What is the unlinked disease-locus assumption in the conventional multipoint LOD likelihood ratio?