Proband Ascertainment in Family Sequencing

Why the Route Into a Family Study Matters

A family sequencing study often begins when a person with a relevant phenotype is referred for testing. That person is the proband, the family member through whom the family is identified or recruited. Relatives may then be invited for sequencing and phenotyping. The resulting families are not a random sample of families: entry has depended on an affected person being present, recognized, and eligible for referral.

This selection can change both the apparent family structure and the evidence used to evaluate a rare variant. Families with more affected members, higher underlying disease risk, or more opportunity for diagnosis may be more likely to enter. In addition, the proband's affected status is part of the reason the family was sampled. Treating the proband and relatives as though they were an unselected, independent set of cases and controls can therefore misrepresent the sampling process.

Definition: Proband ascertainment is recruitment of a family through a designated index person, usually because that person meets a specified phenotype or eligibility rule. The ascertainment event, denoted \(A\), is the precise event that makes the family eligible or selected—for example, “the designated proband is affected” or “at least one family member is affected.” The rule must specify who could be the proband and which phenotypes count.

Ascertainment is not itself evidence that a genetic association is false. It determines the probability that a family with particular genotypes and phenotypes appears in the study. An analysis that represents this sampling rule can use family data appropriately; an analysis that ignores it may give the proband's outcome too much weight, misstate the distribution of family outcomes, or use an inappropriate reference population.

How Ascertainment Changes the Observed Families

Consider a population in which families differ in an unobserved familial liability, such as a combination of inherited risk and shared background. Even if family members' outcomes are independent once that liability is known, selecting a family because one member is affected makes high-liability families more common in the recruited sample. The relatives of recruited probands will then have a higher affected proportion than relatives in the source population.

Worked Example: Enrichment of High-Risk Families

Suppose, hypothetically, that 10% of families have high familial liability, denoted \(H\), and 90% have low liability. Given \(H\), each family member has probability \(0.50\) of being affected; given low liability, that probability is \(0.02\). Assume members' outcomes are conditionally independent given the family's liability. A family is recruited when its designated proband is affected.

Before recruitment, the probability that a proband is affected is \(P(A)=P(A\mid H)P(H)+P(A\mid H^c)P(H^c)\). Substituting the hypothetical values gives \(P(A)=(0.50)(0.10)+(0.02)(0.90)=0.050+0.018=0.068\).

Among recruited families, the proportion with high liability is, by Bayes' rule, \(P(H\mid A)=P(A\mid H)P(H)/P(A)\). Thus \(P(H\mid A)=(0.50)(0.10)/0.068=0.7353\). High-liability families are 10% of the source population but about 73.5% of families recruited through an affected proband.

For a relative, the probability of being affected in the recruited sample is therefore \(P(Y_{\mathrm{rel}}=1\mid A)=P(Y_{\mathrm{rel}}=1\mid H)P(H\mid A)+P(Y_{\mathrm{rel}}=1\mid H^c)P(H^c\mid A)\). The numerical result is \((0.50)(0.7353)+(0.02)(1-0.7353)=0.3677+0.005294=0.3730\). It is about 37.3%, rather than the source-population value \((0.50)(0.10)+(0.02)(0.90)=6.8\%\). Recruitment has changed the mix of families, even though no relative's risk changed because of being recruited.

This example uses a simple latent family category, not a claim that all familial clustering has one cause. In a sequencing study, shared rare variants, other inherited factors, shared environment, and family-specific diagnosis or referral processes may all contribute to differences between families. The practical point is that observed family structure is conditional on the recruitment rule.

Representing the Ascertainment Event in a Likelihood

Let \(D\) denote the observed family data, including the relevant genotypes and phenotypes, and let \(\theta\) denote the parameters of a specified genetic and phenotype model. If families are included only when event \(A\) occurs, the likelihood for a sampled family is based on the distribution of \(D\) conditional on \(A\), rather than the unselected distribution of \(D\).

Definition: For a fixed family design, if selection is deterministic and occurs exactly when \(A\) holds, the ascertainment-conditioned likelihood contribution is \(L(\theta;D\mid A)=P_\theta(D\mid A)=P_\theta(D)/P_\theta(A)\) for observed data \(D\) that satisfy \(A\). The denominator is calculated under the same model and for the specified ascertainment rule. If the rule depends on known design information \(C\), use the corresponding conditional probability \(P_\theta(A\mid C)\).

More generally, if a family with data \(D\) is selected with probability \(s(D)\), its sampled-data distribution is proportional to \(s(D)P_\theta(D)\); the normalizing denominator is the average selection probability under the model. This makes explicit why the exact recruitment procedure matters. “Recruited through an affected family” is not sufficiently precise if one design requires an affected designated proband and another requires any affected member.

The model for \(P_\theta(D)\) must also reflect dependence within pedigrees. For example, genotypes are related through inheritance, and phenotypes may be modeled as conditionally independent only after specifying the genotypes and any other modeled family-level factors. Ascertainment correction does not replace a correct genetic or phenotype model, nor does it repair an unrecorded difference in how families were found.

Conditioning on a Designated Affected Proband

A useful special case makes the correction transparent. Suppose the proband is fixed in advance as the index person, the family genotypes are treated as fixed, and phenotypes are conditionally independent given those genotypes. Let \(f_i\) be the probability that family member \(i\) is affected under the phenotype model. If selection requires only that the designated proband \(p\) be affected, then \(P_\theta(A\mid G)=f_p\), where \(G\) is the fixed family genotype information.

P_\theta(Y\mid G,A) = \frac{\prod_i P_\theta(Y_i\mid G_i)} {P_\theta(Y_p=1\mid G_p)} \quad\text{for data with }Y_p=1.

Because the proband's observed phenotype contributes the factor \(f_p\) to the numerator, that factor cancels the denominator. The conditional likelihood for the other members' phenotypes is then their ordinary phenotype likelihood given their genotypes. This cancellation is limited to the stated assumptions: a designated proband, a rule based only on that proband's affected status, and conditioning on the relevant genotypes. It is not a general license to discard the proband or to ignore how family genotypes were sampled.

Worked Example: One Designated Proband Must Be Affected

Suppose a hypothetical family has a designated proband and two sequenced relatives. Treat the genotypes as fixed and assume phenotype outcomes are conditionally independent given genotype. Under a specified model, their affected probabilities are \(f_p=0.80\), \(f_1=0.40\), and \(f_2=0.10\). Recruitment requires the proband to be affected. The observed outcomes are: proband affected, relative 1 affected, relative 2 unaffected.

The unselected probability of this particular outcome pattern is \(P(D)=f_p f_1(1-f_2)=(0.80)(0.40)(0.90)=0.2880\). The probability of meeting the recruitment rule, given these fixed genotypes, is \(P(A)=f_p=0.80\). Hence the conditional likelihood contribution is \(P(D\mid A)=0.2880/0.80=0.3600\).

Equivalently, after conditioning on the proband being affected, the likelihood for the two relatives' outcomes is \(f_1(1-f_2)=(0.40)(0.90)=0.3600\). The proband's phenotype does not provide an additional independent likelihood factor in this conditional calculation because it was required for entry. The relatives' outcomes still contribute evidence under the model. If inference also models how family genotypes were generated, or the proband's genotype affects inclusion through additional rules, those parts of the sampling process must be represented rather than removed by this simple cancellation.

When the Rule Is “At Least One Affected”

Some studies recruit a family if any eligible member is affected, without selecting a single designated proband in advance. This event differs from the rule in the previous example. If \(m\) members have independent affected probabilities \(f_1,\ldots,f_m\), then the probability of the ascertainment event is one minus the probability that all are unaffected:

P_\theta(A)=1-\prod_{i=1}^{m}(1-f_i), \qquad A=\{\text{at least one of the }m\text{ members is affected}\}.

The denominator depends on the probabilities for all eligible members. If several family members could trigger recruitment, dividing only by the affected probability of the person ultimately labeled “proband” generally does not implement this rule. Likewise, if the study preferentially identifies families with multiple affected members, the event may need to encode that count or the actual referral process.

Worked Example: Ascertainment Through Either of Two Siblings

Suppose, hypothetically, that two named siblings each have affected probability \(0.20\) under a model, with outcomes independent under that model. A family is included if at least one sibling is affected. In one recruited family, sibling 1 is affected and sibling 2 is unaffected.

The unselected probability of this labeled outcome is \(P(D)=(0.20)(1-0.20)=(0.20)(0.80)=0.1600\). The probability that the family qualifies is \(P(A)=1-(1-0.20)(1-0.20)=1-(0.80)(0.80)=0.3600\). Therefore the conditional probability of the observed labeled pattern among eligible families is \(P(D\mid A)=0.1600/0.3600=0.4444\).

The denominator includes both ways the family can qualify: sibling 1 affected, sibling 2 affected, or both affected. If the outcome were recorded only as “exactly one sibling is affected,” without identifying which one, the unselected probability would be \(2(0.1600)=0.3200\), and its conditional probability would be \(0.3200/0.3600=0.8889\). The distinction is whether the outcome is a labeled pattern or an unordered count.

Design and Analysis Choices

Ascertainment should be planned and documented before interpreting segregation or association evidence. In particular, record the eligibility criteria, how a proband was identified, whether only one proband was permitted per family, whether relatives were recruited regardless of phenotype, and whether families with multiple affected members were preferentially referred. Also document the phenotype definition, family members eligible to trigger entry, and whether the analyzed sample includes all eligible relatives or a selected subset.

State the sampling unit and target.
Clarify whether the target is families from a defined source population, disease risk among relatives, or association between variant status and phenotype under a family-based design.

Write the recruitment rule as an event.
Specify the eligible proband or family members, the phenotype threshold, and whether one or multiple affected members can trigger inclusion.

Match the analysis to the rule.
Use a likelihood or sampling design that accounts for the ascertainment event, and represent within-family genotype and phenotype dependence as required by the model.

Interpret the resulting evidence conditionally.
Explain that the findings concern the specified family sample and model; do not treat the recruited families as a random population sample unless the design supports that interpretation.

In some designs, conditioning on the proband's phenotype is a deliberate way to focus on the relatives' outcomes. In others, the proband's genotype and phenotype are part of the information needed to estimate an association or a penetrance parameter. A conditional analysis can remove information as well as correct selection. The conditioning variables and the target parameter therefore need to be stated together.

Common Mistakes and Why They Matter

Counting every affected proband as an unselected case. The probands are affected by design. Their observed proportion affected cannot estimate disease prevalence in the source population.
Assuming relatives are an unselected control group. Their family membership and the affected-proband entry rule can enrich the sample for shared risk. Their phenotype distribution is not automatically the population control distribution.
Using the wrong ascertainment denominator. If any of several members could trigger recruitment, the probability of at least one qualifying phenotype—not merely the designated proband's risk—determines selection under that rule.
Treating the proband and relatives as independent observations. Relatedness creates dependence in genotypes and can also be relevant to phenotype modeling. The ascertainment correction addresses selection; it does not make family members independent.
Assuming ascertainment correction fixes every design problem. Incorrect phenotypes, unmodeled referral differences, population structure, or technical differences in sequencing still require attention. Conditioning on the sampling event does not remove those problems.

Key takeaway: A family recruited through an affected proband is observed conditional on a family-level selection event. Define that event precisely, use a model whose likelihood matches it, and interpret the proband and relatives in light of how the family entered the study.

Check Your Understanding

Use the sampling rule—not just the observed pedigree—to reason through each question.

In a study requiring an affected designated proband, why is the observed fraction of recruited probands who are affected not an estimate of population prevalence?
Two family members could each trigger recruitment. What event should define ascertainment if either one is sufficient?
Under what assumptions does conditioning on an affected designated proband cancel that person's phenotype likelihood factor?
How can recruitment through an affected proband increase the proportion of high-liability families even if relatives' outcomes are independent conditional on family liability?
Why does an ascertainment correction not, by itself, make family members independent or address sequencing batch differences?