Single- and Multiple-Proband Ascertainment Corrections

Why the Selection Rule Belongs in the Likelihood

Families in genetic studies are often not sampled at random from all families. Recruitment may begin when one relative is diagnosed, or when a family has several affected members. Consequently, the observed sample over-represents families that satisfy the recruitment rule. A likelihood that treats these families as if they were randomly sampled can mistake this selection effect for evidence about disease risk or inheritance.

The correction is determined by the event that caused a family to enter the study. Conditioning on one specified relative being affected is not the same as conditioning on at least one affected member of a family. Likewise, requiring two specified relatives to be affected differs from recruiting any family with two or more affected members. The denominator of the conditional likelihood must describe the actual event.

Definition: Let \(Y\) denote the observed phenotype data for a family, and let \(A\) denote the event that makes the family eligible for recruitment. Under model parameter \(\theta\), the ascertainment-corrected likelihood for an observed family with \(Y=y\) is proportional to \[ L_c(\theta;y)=P_\theta(Y=y\mid A) =\frac{P_\theta(Y=y,\ A)}{P_\theta(A)}, \] provided \(P_\theta(A)>0\). If the observed data \(y\) necessarily satisfy \(A\), then the numerator is \(P_\theta(Y=y)\). The probability \(P_\theta(A)\) is the ascertainment correction.

In a pedigree model, probabilities of observed phenotypes generally account for unobserved genotypes by summing over their possible configurations. The same rule applies to the ascertainment probability: calculate it under the genetic and penetrance model, summing over latent genotypes where needed. The correction is not automatically a product of affected probabilities. Relatives’ phenotypes can be dependent because they share inherited genotypes.

One Specified Affected Proband

Suppose a family is recruited because a designated relative, the proband, is affected. The selection event is \(A=\{Y_p=1\}\), where \(Y_p=1\) records that the proband is affected. If the full observed family phenotype vector is \(y\), its conditional likelihood is \(P_\theta(Y=y)/P_\theta(Y_p=1)\), assuming the observed vector includes the proband’s affected status. If the proband’s phenotype is not included in the recorded data, the numerator instead describes the recorded relatives jointly with the selection event.

The denominator is the probability that the designated proband would be affected under the model, in the population and sampling frame relevant to recruitment. It is not generally the probability that anyone in the pedigree is affected. This distinction matters even if the proband is chosen from among family members after an initial clinical contact: the model must reflect the protocol that actually determines inclusion and the identity of the proband.

Worked Example: Conditioning on One Affected Proband

Consider a hypothetical family of four whose disease indicators are independent Bernoulli variables, each with affected probability \(p\). This is a deliberately simple model for illustrating the correction; real pedigree phenotypes may be dependent. The family is recruited because person 1, the designated proband, is affected. The observed statuses are \((1,1,0,0)\), where 1 means affected and 0 means unaffected. Calculate the conditional likelihood at \(p=0.10\).

Under the independence assumption, the probability of the complete observed vector is \(P_p(Y=(1,1,0,0))=p^2(1-p)^2\). The selection event is only that person 1 is affected, so \(P_p(A)=p\). Therefore,

L_c(p;y)=\frac{p^2(1-p)^2}{p}=p(1-p)^2.

Substituting \(p=0.10\),

$$ L_c(0.10;y)=0.10(0.90)^2 =0.10(0.81)=0.0810. $$

Thus, conditional on person 1 being affected, the probability of the particular remaining pattern—person 2 affected and persons 3 and 4 unaffected—is 0.0810 under this model. The proband’s affected status is part of the observed vector, but it is also guaranteed by selection; dividing by its selection probability removes that contribution from the conditional likelihood.

Several Specified Probands and Shared Genetic Risk

When recruitment requires several specified relatives to be affected, define \(A\) as the joint event that those relatives are affected. Its probability is their joint affected probability under the pedigree model. Even if phenotypes are conditionally independent given genotypes, they may not be independent after unobserved genotypes are marginalized out. Using the product of marginal affected probabilities can therefore give the wrong denominator.

Worked Example: Two Specified Affected Probands

Consider a hypothetical pedigree with two designated probands and one additional relative. For a compact illustration of genotype marginalization, suppose the family has one of two latent inherited genotype configurations, \(G_1\) or \(G_2\), with probabilities 0.60 and 0.40. Conditional on the configuration, the three disease indicators are independent. In configuration \(G_1\), each person’s affected probability is 0.80; in \(G_2\), it is 0.20. The study requires both designated probands to be affected. The observed statuses are \((1,1,0)\).

The ascertainment event \(A\) is that both specified probands are affected. Summing over configurations gives

$$ P(A)=0.60(0.80^2)+0.40(0.20^2) =0.60(0.64)+0.40(0.04) =0.384+0.016=0.400. $$

The probability of the complete observed vector, including the third relative’s unaffected status, is

$$ P(Y=(1,1,0)) =0.60(0.80^2)(0.20)+0.40(0.20^2)(0.80) =0.0768+0.0128=0.0896. $$

Because that vector satisfies the selection event, the conditional likelihood contribution is

L_c=\frac{0.0896}{0.400}=0.224.

The value is the probability of the third relative being unaffected, conditional on both probands being affected, under this illustrative model. For comparison, each proband’s marginal affected probability is \(0.60(0.80)+0.40(0.20)=0.56\). Multiplying the two marginal probabilities gives \(0.3136\), not the correct joint selection probability \(0.400\). The shared latent configuration induces dependence between their phenotypes.

In an actual pedigree likelihood, the latent configurations represent possible genotypes and their probabilities follow the pedigree transmission model. If \(\mathbf{G}\) denotes a family’s genotype configuration, a general way to calculate a selection probability is

P_\theta(A)=\sum_{\mathbf{g}} P_\theta(\mathbf{G}=\mathbf{g})P_\theta(A\mid\mathbf{G}=\mathbf{g}).

For conditionally independent disease indicators, the second factor is formed from the relevant penetrance probabilities for the selected relatives. The sum must include all genotype configurations consistent with the model, not just the most likely configuration.

Selection by a Count of Affected Relatives

Sometimes the study recruits a family if it has a minimum number of affected people among a set of eligible relatives. In that case, the selection event is a union of phenotype patterns. For example, “at least two of these three relatives are affected” includes the patterns 110, 101, 011, and 111. Its probability is the sum of the probabilities of those disjoint patterns.

Worked Example: Recruitment When at Least Two of Three Are Affected

Suppose three relatives have independent affected probabilities \(p\), and a family is recruited if at least two are affected. The observed statuses are \((1,1,0)\). At \(p=0.20\), calculate the likelihood conditional on recruitment.

The probability of the observed pattern is

$$ P_p(Y=(1,1,0))=p^2(1-p) =(0.20)^2(0.80)=0.0320. $$

The selection event comprises exactly two affected in any of three ways, or all three affected. Therefore,

P_p(A)=\binom{3}{2}p^2(1-p)+p^3 =3(0.20)^2(0.80)+(0.20)^3 =0.096+0.008=0.104.

The conditional likelihood contribution for the observed, labeled pattern is

L_c(p;y)=\frac{0.0320}{0.104}=0.3077.

The denominator is not \(p^2\), which would apply to the event that two particular designated relatives are both affected. Nor is it \(1-(1-p)^3\), which is the probability that at least one of the three is affected. The recruitment rule here is specifically at least two affected among these three.

Why the Correction Changes Inference

For independent recruited families, a conditional likelihood is formed by multiplying the family-specific conditional contributions, using each family’s actual selection event. If all families share the same ascertainment rule, this can be written as the product of their unconditional likelihood contributions divided by the corresponding selection probabilities. Since those probabilities usually depend on \(\theta\), they affect estimation and cannot be discarded as constants.

A short derivation makes this dependence explicit. For a differentiable model with positive \(P_\theta(A)\), take the logarithm of the conditional likelihood for an observed \(y\) satisfying \(A\):

\ell_c(\theta;y) =\log P_\theta(Y=y)-\log P_\theta(A).

Differentiating with respect to \(\theta\) gives

\frac{\partial \ell_c(\theta;y)}{\partial\theta} = \frac{\partial}{\partial\theta}\log P_\theta(Y=y) - \frac{\partial}{\partial\theta}\log P_\theta(A).

Thus the conditional score is the ordinary data score minus the score contribution from the probability of being selected. If the selection probability changes with the disease or transmission parameters, omitting that term changes the likelihood’s evidence about those parameters.

The conditional formulation relies on a correctly specified model for the family data and a correctly identified recruitment event. The event must be defined at the same level as the sampling unit: for example, whether a particular person had to be affected, whether any person in a specified group could trigger recruitment, or whether a threshold count was required. The selection probability must be positive under parameter values being considered. If families are recruited through different mechanisms, their likelihood contributions should reflect those different mechanisms.

Common Mistakes

Using the wrong event in the denominator. Dividing by the probability that anyone in the pedigree is affected is incorrect when recruitment required one specified proband to be affected. Write the selection event in words, then in probability notation, before calculating it.
Multiplying marginal probabilities for related probands. Shared genetic and familial factors can make their phenotypes dependent. Calculate the joint probability, marginalizing over genotype configurations as required by the pedigree model.
Confusing designated probands with a threshold rule. “These two relatives are affected” is not the same event as “at least two relatives in this group are affected.” They generally have different probabilities and therefore different corrections.
Dropping the denominator because it is called a correction. If it depends on \(\theta\), it is part of the conditional likelihood and can alter parameter estimates. It is only constant with respect to parameters in special cases.
Assuming conditioning repairs every recruitment problem. A correction addresses the stated selection event under the assumed model. It does not correct an unrecorded or misspecified recruitment process.

Key takeaway: An ascertainment correction is the probability of the event that brought a family into the study. Define that event precisely, calculate its probability under the same pedigree model used for the data likelihood, and condition on it.

Check Your Understanding

Answer each question by identifying the selection event before choosing a denominator.

A family is included because a named sibling is affected. What event belongs in the denominator, and why is “at least one family member is affected” not generally equivalent?
Two specified relatives are required to be affected. Explain why their joint affected probability may differ from the product of their marginal affected probabilities.
A study recruits families with at least two affected members among four eligible relatives. What phenotype patterns must be included when calculating the selection probability?
In a conditional likelihood, explain why a selection probability that depends on a model parameter cannot generally be ignored.
State one assumption of the independent Bernoulli example that may fail in a pedigree, and describe how a pedigree model can address that issue.