Estimating Segregation Model Parameters by Likelihood

Why Parameter Estimation Must Account for Selection

A segregation model can describe how genotype relates to disease, but its parameters still need to be estimated from observed families. Family studies often recruit a pedigree because a member has the disease. The resulting sample is not a random collection of families: affected probands are more likely to appear by design. Estimating parameters as if the families had been sampled at random can therefore distort the evidence about disease probability and allele frequency.

The likelihood provides a direct way to combine the model with the observed family data. Its construction must account for Mendelian transmission, uncertain genotypes, and the rule that led to a family being included. This tutorial develops that construction for a simple dominant model, then shows how to maximize the likelihood and describe uncertainty in the resulting estimates.

Parameters and the Family Likelihood

For a concrete model, suppose a biallelic locus has a disease-associated allele \(A\), with population frequency \(q\). Assume random mating and Hardy–Weinberg equilibrium among founders. The genotype \(AA\) or \(Aa\) is called a carrier genotype; \(aa\) is noncarrier. Let \(f\) be the probability of disease for a carrier and \(c\) the probability of disease for a noncarrier. Thus \(f\) is the penetrance in carriers and \(c\) is the phenocopy probability. The parameter vector is \(\theta=(q,f,c)\), with \(0\leq q,f,c\leq1\); a model in which the allele increases risk additionally assumes \(f\geq c\).

Under Hardy–Weinberg equilibrium, founder genotype probabilities are \(P(AA)=q^2\), \(P(Aa)=2q(1-q)\), and \(P(aa)=(1-q)^2\). Given parental genotypes, offspring genotype probabilities follow Mendelian transmission. For each pedigree, the probability of its observed phenotypes is obtained by summing over genotype configurations that were not observed. Conditional on the genotypes, the model treats relatives’ disease statuses as independent Bernoulli outcomes with probability \(f\) for carriers and \(c\) for noncarriers. This is a modeling assumption, not a claim that relatives’ marginal disease statuses are independent.

Write \(D_i\) for the observed data in family \(i\), and \(S_i\) for the event that the family meets the ascertainment rule. A data record may include phenotypes, genotypes when measured, and which family members were observed. The ascertainment-aware likelihood contribution is the conditional probability of the data given selection:

If the observed data \(D_i\) already imply selection, then \(P(D_i,S_i;\theta)=P(D_i;\theta)\). The denominator is still necessary: it is the probability that a family would have met the ascertainment rule under the model. It must be calculated using the actual selection procedure, not chosen merely to simplify the algebra.

For independent families sampled under the stated scheme, the likelihood is the product of their conditional contributions. A log-likelihood is easier to optimize numerically because it converts products into sums:

When genotypes are uncertain, each probability in the numerator or denominator is evaluated by summing over possible genotype configurations. Each configuration is weighted by its founder genotype probabilities and Mendelian transmission probabilities, then by the appropriate phenotype probabilities. In practice, the same pedigree probability calculations used for likelihood contributions can be evaluated repeatedly at candidate parameter values during optimization.

Maximization and a Useful Factorization

The maximization is usually numerical when pedigrees have unknown genotypes or complex ascertainment rules. An optimizer searches over allowed parameter values, evaluating the log-likelihood at each candidate. Parameter constraints matter: estimates must remain within their probability ranges, and a model that requires \(f\geq c\) must enforce that restriction. The reported estimate should be a maximizer, not merely the last value returned by an algorithm; checking alternative starting values and the log-likelihood near the solution can help detect a local or boundary maximum.

A simpler conditional calculation illustrates how penetrance and phenocopy estimates arise. Suppose a set of independent ascertained families has a selected, affected proband and one additional relative per family. Assume both relatives’ genotypes are observed, and condition on the observed genotypes and on proband selection. The extra relatives’ disease statuses are then Bernoulli observations: carrier relatives have disease probability \(f\), and noncarrier relatives have disease probability \(c\). The proband’s affected status is fixed by conditioning on selection, so it does not supply an ordinary Bernoulli observation in this conditional likelihood.

Estimating Allele Frequency and Quantifying Uncertainty

When unrelated founders are sampled without disease-based selection and their genotypes are observed, the allele-frequency component can be especially simple. If \(m\) diploid founders contribute a total of \(2m\) allele copies and \(a\) copies are \(A\), the allele-count likelihood is proportional to \(q^a(1-q)^{2m-a}\). Under independent sampling from a population with allele frequency \(q\), the MLE is the observed allele fraction. If founder sampling is affected by ascertainment, this simple calculation is not automatically valid; founder genotypes must be handled under the actual sampling design.

Likelihood-Based Uncertainty

An estimate alone does not show how strongly the data determine a parameter. In a regular model with an interior maximum, the observed information matrix is the negative Hessian of the log-likelihood evaluated at the MLE. Its inverse approximates the covariance matrix of the estimates:

The square root of a diagonal entry is an approximate standard error for that parameter. Off-diagonal entries describe estimated dependence between parameter estimates. This dependence matters: for example, a small allele frequency paired with high penetrance may produce patterns similar to a larger frequency paired with lower penetrance. A profile likelihood can reveal such trade-offs more clearly than separate standard errors.

For a parameter of interest, the profile log-likelihood is obtained by fixing that parameter and maximizing over the other parameters. Under regular large-sample conditions, an approximate 95% likelihood interval contains values whose profile log-likelihood is no more than \(1.92\) below the maximum, equivalently where twice the log-likelihood drop is at most \(3.84\). This is an approximation, not a guarantee of accurate coverage in small or sparse pedigrees. If an estimate lies at a boundary, or the likelihood is nearly flat, standard-error and likelihood-ratio approximations can be unreliable; the boundary and the weakness of the information should be reported.

In the simplified penetrance example, the binomial standard-error estimates are \(\sqrt{0.75(0.25)/8}=0.1531\) for \(f\) and \(\sqrt{0.10(0.90)/10}=0.09487\) for \(c\). A naive normal interval for \(f\) would extend above 1, which is impossible for a probability. This signals that a symmetric normal approximation is a poor summary here. Profile-likelihood intervals, or a method appropriate to the small sample and probability boundaries, are preferable.

Derivation: Why the Ascertainment Denominator Is Required

The correction follows directly from conditional probability. Among all possible families generated under \(\theta\), only those satisfying \(S\) are eligible for observation. Thus, for any observed record \(D\), the probability assigned to it in the selected sample is its joint probability with selection divided by the probability of selection:

The denominator depends on \(\theta\), so it can change the maximizing parameter value. Omitting it is justified only when the sampling design makes it constant with respect to the parameters being estimated, or when the likelihood has been constructed in a different, explicitly valid way. Otherwise, maximizing the uncorrected numerator treats the ascertained families as though they were randomly sampled.

Interpretation, Identifiability, and a Common Pitfall

A parameter is identifiable when distinct allowed parameter values imply distinct distributions for the data used in the analysis. Family information can help distinguish \(q\), \(f\), and \(c\), but it does not guarantee that the data do so. For example, an affected-only sample may reveal little about the frequency of unaffected genotypes. Sparse pedigrees, few noncarrier relatives, or few informative genotypes can also leave \(f\) and \(c\) weakly distinguished. A flat or strongly elongated likelihood surface is evidence of limited information, not a reason to report extra decimal places.

• Wrong reasoning: “The proband was affected, so I can estimate penetrance by counting affected probands in the recruited families.”
• Correction: If families were selected because the proband was affected, that status is part of the selection rule. Use the likelihood conditional on that rule; it must include the selection probability. Other relatives’ statuses can still inform penetrance and phenocopy probability under the model.
• Wrong reasoning: “The separate estimates of \(f\), \(c\), and \(q\) are automatically independent.”
• Correction: In a joint pedigree likelihood, parameter estimates can be correlated. Use the joint information matrix or profile likelihood to represent uncertainty and parameter trade-offs.