Integrating Uncertain IBD Into the Linkage Likelihood

Why IBD Uncertainty Belongs in the Likelihood

In a family linkage analysis, marker data may not identify exactly which alleles are identical by descent (IBD) at a quantitative-trait locus. Instead, they provide probabilities over possible locus-specific IBD configurations. A phenotype likelihood should account for that uncertainty. Treating the most likely configuration as known, or replacing the uncertain sharing with its mean, can give a different likelihood and can understate what the data leave unresolved.

The key operation is to average the phenotype likelihood over configurations, weighting each by its probability conditional on the marker data. This is not the same as first averaging IBD and then calculating a Gaussian likelihood. The distinction arises because the likelihood is a nonlinear function of its covariance parameters.

The weights \(q(z\mid M)\) are nonnegative and sum to one. They describe uncertainty about IBD after using the marker data; they are not phenotype likelihood contributions. The phenotype density for each configuration supplies that contribution. Once phenotypes are observed, configurations with larger density can receive greater posterior probability, but the likelihood itself averages the densities using the marker-based weights.

Gaussian Family Model With Latent IBD

For a family with \(n\) observed members, let \(Y\) be the \(n\)-vector of quantitative phenotypes and let \(X\beta\) be its mean. A commonly used variance-components model, conditional on configuration \(z\), is $$ Y\mid Z=z \sim N(X\beta,V_z), \qquad V_z=2\sigma_A^2\Phi+\sigma_Q^2\Pi_z+\sigma_e^2 I. $$ Here \(\Phi\) is the pedigree kinship matrix, \(\Pi_z\) is the matrix of pairwise IBD sharing at the QTL under configuration \(z\), and \(I\) is the identity matrix. The parameters \(\sigma_A^2,\sigma_Q^2,\sigma_e^2\) are respectively the polygenic, QTL, and residual variance components. The diagonal entries of \(\Pi_z\) are one; its off-diagonal entries represent the IBD fractions for member pairs under \(z\).

This model assumes that, conditional on \(z\), the family phenotype vector is multivariate normal with the stated mean and covariance, and that the residual component has covariance \(\sigma_e^2 I\). It also assumes that the marker-based probabilities \(q(z\mid M)\) correctly represent uncertainty about locus-specific IBD. Other covariance terms or mean structures may be appropriate in other applications, but they must be specified rather than silently absorbed into the likelihood.

For a fixed configuration, the density is $$ f(Y\mid z;\theta)= \frac{\exp\!\left[-\frac12(Y-X\beta)^\mathsf{T}V_z^{-1}(Y-X\beta)\right]} {(2\pi)^{n/2}|V_z|^{1/2}}. $$ Thus each configuration can change both the quadratic fit to the observed phenotypes and the determinant that scales the density. The marginal likelihood is a weighted sum of these densities, not a density evaluated at an average configuration.

Phenotypes Can Update Configuration Probabilities

Although the likelihood weights configurations by their probabilities given markers, the phenotypes can also inform which configuration is more plausible. Bayes’ rule gives the configuration probability conditional on both marker and phenotype data: $$ P(Z=z\mid Y,M;\theta)= \frac{q(z\mid M)f(Y\mid z;\theta)} {\sum_{z'}q(z'\mid M)f(Y\mid z';\theta)}. $$ This conditional probability is different from the marker-only weight \(q(z\mid M)\). It is useful for interpreting the latent configurations, but calculating it does not replace the marginal likelihood.

Why Expected IBD Is Not Generally Enough

A tempting shortcut is to average the IBD configuration matrices first, using $$ \overline{\Pi}=\sum_z q(z\mid M)\Pi_z, $$ and then evaluate a single Gaussian likelihood with covariance \(2\sigma_A^2\Phi+\sigma_Q^2\overline{\Pi}+\sigma_e^2I\). This produces a plug-in likelihood. It can be convenient, but it is not generally the likelihood obtained by integrating over the uncertain configuration.

The distinction can be seen directly in the two-configuration example. The average correlation is \(0.5(0)+0.5(0.8)=0.4\). The single Gaussian density at that correlation is $$ f(1,1\mid 0.4)= \frac{\exp(-1/1.4)}{2\pi\sqrt{1-0.4^2}} \approx 0.08501. $$ The integrated likelihood was approximately \(0.105375\), not \(0.08501\). In this example the plug-in value is lower, but there is no universal direction: depending on the observations and covariance values, it may be higher or lower.

This difference is about the likelihood model, not merely numerical rounding. Integration retains the distribution over possible IBD states. The plug-in approach collapses that distribution to its expected sharing and imposes a single Gaussian form. It may be a useful approximation in some settings, but it should be identified as an approximation and not described as exact marginalization.

Practical Checks and Common Pitfalls

A reliable calculation starts by listing the configurations and verifying that their marker-based probabilities sum to one. For each configuration, use the same phenotype model and parameter values, changing only the IBD-dependent covariance terms. Check that each covariance matrix is positive definite so its Gaussian density is defined. Then calculate each density, multiply by its marker weight, and sum.

For a pedigree with several relatives, configurations are joint states of the family’s inheritance pattern. Pairwise IBD probabilities are summaries of those states; they do not necessarily determine the full joint distribution of configurations. If only marginal probabilities are available, do not treat all pairwise sharing indicators as independent unless that independence is justified by the inheritance model. Use the multipoint IBD probabilities or configuration weights that preserve the dependence implied by the pedigree and marker data.

• Mistake: substitute the most likely configuration. The configuration with the largest marker probability need not dominate the phenotype likelihood. The correct likelihood averages all supported configurations with their weights.
• Mistake: evaluate at expected IBD and call it integrated. A density at average covariance is generally not an average of densities. Label this as a plug-in approximation when it is used.
• Mistake: use phenotype-updated weights as the original likelihood weights. The marginal likelihood uses \(q(z\mid M)\); phenotype-updated probabilities arise only after applying Bayes’ rule to the completed likelihood.
• Mistake: interpret a density as a probability. For continuous phenotypes, the density at a point is not the probability of observing exactly that point. It is a likelihood contribution relative to other parameter values or models.