Factorizing Likelihoods Across Pedigree Structures

Why Factor a Pedigree Likelihood?

A pedigree likelihood can involve several relatives whose genotypes and phenotypes are connected by inheritance. Writing one large joint probability obscures where each part of the model enters. Factoring the likelihood makes the assumptions visible: founders have genotype probabilities, descendants inherit genotypes from their parents, and phenotypes are generated according to penetrance probabilities.

This factorization is useful whether every genotype is observed or some relatives have missing genotype data. In the latter case, the likelihood sums over possible genotypes for the unobserved relatives; it does not simply discard them. The central task is to preserve the pedigree’s dependencies while organizing the probability into interpretable pieces.

The Model Behind the Product

Let \(G_i\) be person \(i\)’s genotype and \(Y_i\) their phenotype. For a nonfounder \(i\), let \(m_i\) and \(f_i\) denote their mother and father. Write \(q_i(g)\) for the genotype probability of founder \(i\), and \(T_i(g\mid g_{m_i},g_{f_i})\) for the probability that a child has genotype \(g\) given the parental genotypes. Write \(h_i(y\mid g)\) for the phenotype probability, often called a penetrance probability when \(y\) denotes disease status.

The simple product below uses a single-locus Mendelian model with no mutation. It assumes that founder genotypes are independent under the specified founder-genotype model and that the meioses producing different children are conditionally independent given their parents’ genotypes. Most importantly, it assumes phenotypes are conditionally independent across relatives given their genotypes (and any other variables included in the model). Specifying each person’s penetrance alone does not imply this last assumption. If relatives’ phenotypes remain dependent after conditioning on genotypes, their product must be replaced by a joint phenotype model.

Here \(\mathcal{F}\) is the set of founders, and the products run over all people in the pedigree as appropriate. The first product covers founders only; the second covers nonfounders only; the third covers the modeled phenotypes. If founders share ancestry or otherwise have dependent genotypes, their factors must instead be represented by an appropriate joint founder-genotype distribution.

Why the Factors Multiply

Order people so that parents appear before their children. By the chain rule, the joint genotype probability is a product of each person’s genotype probability conditional on the genotypes already listed. Under the stated inheritance model, a founder’s factor is \(q_i(G_i)\), while a nonfounder’s factor depends on the genotypes of their parents through \(T_i\). This gives the first two products. Conditional independence of phenotypes given the genotypes then gives the third product.

More explicitly, under these assumptions the conditional probability of all genotypes given the pedigree structure is

Multiplying \(P(\mathbf{G})\) by \(P(\mathbf{Y}\mid\mathbf{G})\) yields the joint factorization. This argument also identifies exactly where a dependence assumption is needed: without conditional independence of phenotypes, the displayed product for \(P(\mathbf{Y}\mid\mathbf{G})\) is not justified.

Observed Data and Missing Genotypes

Let \(O_i\) denote the genotype information recorded for person \(i\). Define \(e_i(O_i\mid G_i)\) as the probability of that observation given the person’s true genotype. For an exact genotype call, this is an indicator: it equals 1 when the genotype agrees with the call and 0 otherwise. If the genotype is missing, set \(e_i=1\), since no genotype observation is being imposed. A model for uncertain or error-prone calls can use a non-indicator observation probability.

For observed phenotypes, the likelihood sums the joint model over genotype configurations compatible with the observations. If \(I_Y\) is the set of people whose phenotypes were observed, the likelihood is

The sum is over all genotype configurations not fixed by exact genotype observations. For an exactly observed genotype, its indicator factor restricts the sum to configurations consistent with that call. If a person’s phenotype is missing and the missingness mechanism is ignorable for this likelihood, no phenotype factor is included for that person: summing their phenotype probability over all possible outcomes would give 1. If missingness itself carries information, it needs its own model rather than being silently ignored.

Factorization in a Multigenerational Pedigree

The same structure applies when the pedigree has several generations. Each nonfounder contributes one transmission factor conditional on their own parents, regardless of how many generations separate those parents from the founders. A person who married into the pedigree and has no parents represented is a founder for this pedigree model, even if that person has biological relatives outside the diagram.

Common Mistakes and Practical Checks

• Multiplying penetrances without a conditional-independence assumption. Knowing each person’s \(h_i(Y_i\mid G_i)\) does not, by itself, specify the joint phenotype probability. Use a product only when phenotypes are conditionally independent given the modeled variables; otherwise specify a joint distribution.
• Giving a founder a transmission factor. In the pedigree as drawn, a founder has no represented parents. Their genotype enters through the founder-genotype distribution.
• Dropping a person with missing genotype. Missing genotype does not mean zero contribution or an arbitrary genotype assignment. Sum over the possible genotypes with their model probabilities.
• Multiplying instead of summing over alternatives. When one person’s genotype is unknown, the mutually exclusive genotype possibilities are added. Within each possibility, the relevant founder, transmission, and phenotype factors are multiplied.
• Treating observed calls as extra inheritance probabilities. Exact genotype calls constrain the genotype configurations in the likelihood; they do not replace the founder or transmission probabilities that generated those genotypes.

A useful check is to label each person as founder or nonfounder, identify which genotypes are observed, and verify that every nonfounder’s transmission factor uses the genotypes of the correct parents. Then check that every product of phenotype terms is supported by conditional independence and that each missing genotype is summed over rather than fixed.