Multipoint IBD Probabilities in Family QTL Models

Why Multipoint IBD Matters

At a quantitative-trait locus (QTL), relatives may share alleles inherited from a common ancestor. That sharing can induce covariance in their trait values. Marker genotypes near the QTL help infer which alleles relatives inherited, but usually do not reveal the QTL inheritance state with certainty. Markers on both sides of a QTL can be more informative than a single marker because recombination probabilities connect inheritance states across the genetic map.

The task here is to calculate the probability distribution of IBD sharing at a QTL given pedigree and marker data, then use its expected sharing to form the QTL covariance matrix. The key is to carry uncertainty forward: a pair that has some chance of sharing zero QTL alleles and some chance of sharing two does not generally have the same covariance as a pair known to share exactly one.

Inheritance States and Recombination Along a Map

An inheritance state records which parental homolog was transmitted to each relative at a locus. For two siblings in a nuclear family, a convenient state has four binary entries: the homolog inherited by each sibling from the father and the homolog inherited by each sibling from the mother. For either parent, the siblings share that parent's transmitted allele IBD when they inherited the same homolog, and do not share it when they inherited different homologs. Thus \(K_{ij}\) is the sum of the paternal and maternal sharing indicators.

Across an interval, a transmitted homolog can switch when a recombination occurs between the marker positions. Under the Haldane map, crossovers in a meiosis follow a Poisson process along map distance, with no crossover interference. If the interval has length \(d\) Morgans, the probability of an odd number of crossovers—and therefore a switch in inherited homolog—is:

For distinct meioses, switches are independent under the standard transmission model. Therefore, for a pair of siblings, each of the four transmission bits has its own interval transition probability \(\theta(d)\). This is a model assumption, not a consequence of observed marker data. A different genetic map or crossover model can imply different transition probabilities.

Marker genotypes are evidence about inheritance states, rather than direct observations of them in every case. When parental phase and marker alleles are informative, a marker may identify a transmitted homolog. With uninformative genotypes, unknown phase, or possible genotyping error, the marker likelihood should reflect that uncertainty. In a multipoint calculation, the inheritance states form a hidden Markov model: transitions are determined by the map, and each marker contributes an emission likelihood for the observed genotypes given the state.

Worked Example: Converting Map Distance to Recombination Probability

Using Flanking Markers to Infer Sharing at a QTL

A simple bridge calculation illustrates why two flanking markers can be informative. For a specified parent and a pair of siblings, define \(D=0\) if the siblings inherited the same parental homolog and \(D=1\) if they inherited different homologs. Across an interval, \(D\) switches if exactly one of the two sibling transmissions switches. If each transmission switches with probability \(\theta\), independently, then:

If the marker states at both flanks establish \(D=0\), the state at a QTL between them is still uncertain. It can switch from 0 to 1 in the first interval and return to 0 in the second. The posterior probability at the QTL is obtained by considering the possible paths and conditioning on the observed right-flank state.

From Expected IBD Sharing to a QTL Covariance Matrix

To translate sharing into covariance, use the additive QTL model. Suppose each allele has an additive effect with mean zero and variance \(\tau^2\), and effects of distinct ancestral alleles are independent. An individual's QTL value is the sum of the effects of their two alleles, so its QTL variance is \(\sigma_Q^2=2\tau^2\). Two relatives have covariance \(\tau^2\) for each allele they share IBD. Averaging over an uncertain IBD state gives the following result.

This result follows directly from the allele-effect model. Conditional on sharing \(K_{ij}\) alleles IBD, exactly \(K_{ij}\) pairs of allele effects are the same random effect. Each contributes variance \(\tau^2\) to the covariance, while non-IBD allele effects have covariance zero. Thus the conditional covariance is \(K_{ij}\tau^2\). Taking its expectation over the marker-informed IBD distribution gives \(E(K_{ij}\mid M)\tau^2=\pi_{ij}\sigma_Q^2\).

Practical Checks and Common Pitfalls

A multipoint calculation is sensitive to the inheritance model, marker informativeness, phase, and map. The Haldane expression is appropriate only when its crossover assumptions are being used. In applications, marker likelihoods should account for the observed genotypes and any relevant uncertainty; treating an ambiguous marker pattern as a known transmission can make posterior sharing too certain.

A common mistake is to replace uncertain sharing by the most likely IBD category. In the worked sibling example, \(K=2\) is the most probable category, but it has probability only 0.6250. Setting \(K=2\) as if it were certain would produce \(\pi_{AB}=1\), whereas the posterior expected sharing is 0.7906. The covariance uses the expectation over all IBD states, not the category with the largest probability.

Another important distinction is between constructing a covariance matrix from posterior expected sharing and integrating a phenotype likelihood over IBD states. The former is the task here: each entry uses \(\pi_{ij}\). Whether a likelihood should instead average over complete latent inheritance configurations is a separate modeling step. Finally, the entries must describe a coherent pedigree and marker model; independently chosen pairwise sharing values need not form a valid family covariance matrix.