Ascertainment Mechanisms in Family Studies

Why Recruitment Rules Matter

A family study rarely begins by drawing pedigrees at random from every family in a population. Researchers may recruit through a clinic, invite families after an affected child is identified, or seek pedigrees with several affected relatives. These rules are practical, but they determine which families can enter the study. A sample recruited through affected individuals can therefore contain a very different distribution of phenotypes from the distribution among all families.

This selection does not change the underlying Mendelian transmission process or the phenotypes that families would have had in the absence of sampling. It changes which of those families are observed. Keeping that distinction clear is essential when interpreting the counts and proportions in a pedigree sample.

Definition: Ascertainment is the process by which families or pedigrees become identified and recruited into a study. An ascertainment mechanism is the rule, or set of probabilities, governing that process. A proband is the family member whose phenotype or other feature first brings the family to the attention of the study. The source population is the population of families to which the study aims to relate its findings; the ascertained sample is the set of families actually recruited.

For a compact representation, let \(Y\) denote the phenotype pattern in a family and let \(S=1\) mean that the family is selected. The rule can be described by the inclusion probability \(a(y)=P(S=1\mid Y=y)\). A random sample of families has the same inclusion probability for every phenotype pattern. Under affected-proband recruitment, patterns containing an eligible affected proband have higher inclusion probability than patterns without one. Some patterns may have probability zero of selection.

The distribution among recruited families is consequently a selection-weighted version of the source distribution. This statement concerns the distribution of observations, not a change in inheritance or disease risk caused by recruitment.

P(Y=y\mid S=1) = \frac{P(S=1\mid Y=y)P(Y=y)}{P(S=1)} = \frac{a(y)P(Y=y)}{\sum_{y'}a(y')P(Y=y')}

The denominator is the overall probability that a family is selected. The expression requires \(P(S=1)>0\); patterns with \(a(y)=0\) cannot appear in the recruited sample. If families also differ in measured characteristics such as size or clinic access, those characteristics may matter to the selection rule as well. The key question is always which features of a family influence its chance of inclusion.

What an Affected-Proband Rule Does

A study that recruits a family because a specified member is affected guarantees that the proband is affected in every recruited pedigree. That fact alone changes the observed distribution of whole-family phenotypes. What happens to the other relatives depends on the family model: if their phenotypes are independent of the proband's, their marginal probability need not change; if relatives share familial risk, affected-proband recruitment can enrich the sample for high-risk families.

Worked Example: Affected Proband With Independent Sibling Phenotypes

Suppose every source family has two siblings, and each sibling is affected with probability \(q=0.10\), independently of the other. The first sibling is designated as the proband, and a family is recruited exactly when that sibling is affected. Let \(K\) be the number of affected siblings in the family. Find the distribution of \(K\) among recruited families and compare the affected fraction with the source population.

The source probabilities follow from independence. The probability that neither sibling is affected is \(0.9^2=0.81\); exactly one is affected with probability \(2(0.10)(0.90)=0.18\); and both are affected with probability \(0.10^2=0.01\). Selection requires the first sibling to be affected, so \(P(S=1)=0.10\).

For \(K=1\), the proband is affected and the other sibling is unaffected, giving \(P(K=1,S=1)=0.10(0.90)=0.09\). For \(K=2\), both siblings are affected, giving \(P(K=2,S=1)=0.10(0.10)=0.01\). Dividing by \(P(S=1)=0.10\),

P(K=1\mid S=1)=\frac{0.09}{0.10}=0.90, \qquad P(K=2\mid S=1)=\frac{0.01}{0.10}=0.10.

Thus the expected affected fraction among the two siblings in recruited families is \(\{(1)(0.90)+(2)(0.10)\}/2=0.55\), compared with \(0.10\) in the source population. The recruited family's affected fraction is high partly because one sibling must be affected to qualify. However, the nonproband sibling remains affected with probability \(0.10\): under the stated independence assumption, learning that the proband is affected does not change the other sibling's risk.

This example shows why a statement such as “the observed fraction of affected people is 55%” is incomplete unless it specifies who is counted and how families were recruited. The value is not an estimate of the source-population prevalence in this design.

Selection Can Enrich Familial Risk

Phenotypes among relatives are often dependent because relatives share genetic and environmental influences. One simple way to represent such dependence is to imagine that families belong to different risk groups. If affected probands are more likely in one group, recruiting through affected probands changes the mix of groups in the sample. As a result, the relatives of recruited probands may have a different affected proportion from relatives in the source population.

Worked Example: High-Risk Families Are Overrepresented

Consider a hypothetical source population in which 20% of families are in a high-risk group and 80% are in a low-risk group. A specified proband is affected with probability 0.40 in the high-risk group and 0.05 in the low-risk group. A family is recruited if that proband is affected. Among families in the high-risk group, a relative is affected with probability 0.30; in the low-risk group, that probability is 0.05. Assume that, within each risk group, the proband's and relative's phenotypes are independent. Calculate the group composition after recruitment and the affected probability for a relative in the recruited sample.

The probability of recruiting a source family is \(P(S=1)=(0.20)(0.40)+(0.80)(0.05)=0.08+0.04=0.12\). Of this total, 0.08 comes from high-risk families. Therefore,

P(\text{high risk}\mid S=1)=\frac{0.08}{0.12}=0.6667, \qquad P(\text{low risk}\mid S=1)=0.3333.

The source population was 20% high-risk, but two-thirds of recruited families are high-risk under this rule. The relative's affected probability in the recruited sample is therefore \((0.6667)(0.30)+(0.3333)(0.05)=0.2167\). In the source population it is \((0.20)(0.30)+(0.80)(0.05)=0.1000\).

The difference arises from selection of families, not from a change in the relative's phenotype model. Conditional independence within each risk group was used to calculate the group composition from the proband's status; the relative's probability was then averaged over that selected composition. Without the stated risk groups and independence assumption, these calculations would not follow from the given numbers alone.

Different Recruitment Rules Select Different Pedigrees

“Recruit families through affected cases” is not a single precise design. A rule may require an affected member in a specified position, at least one affected person anywhere in a pedigree, or multiple affected relatives. Rules can also involve a clinic diagnosis, age at onset, family size, or willingness to participate. These designs select different phenotype patterns, even when the source population is the same.

Worked Example: Multiplex Recruitment Among Three Siblings

Suppose a source family has three siblings, each independently affected with probability \(q=0.20\). A study recruits only families with at least two affected siblings. Let \(K\) be the number affected. Find the distribution of \(K\) among recruited families and their mean affected fraction.

Before selection, \(K\) has a binomial distribution. The probabilities for \(K=0,1,2,3\) are respectively \(0.8^3=0.512\), \(3(0.2)(0.8^2)=0.384\), \(3(0.2^2)(0.8)=0.096\), and \(0.2^3=0.008\). Recruitment occurs for \(K=2\) or \(K=3\), so its probability is \(0.096+0.008=0.104\). Hence,

P(K=2\mid K\geq 2)=\frac{0.096}{0.104}=0.9231, \qquad P(K=3\mid K\geq 2)=\frac{0.008}{0.104}=0.07692.

The mean affected fraction among recruited families is \(\{(2)(0.9231)+(3)(0.07692)\}/3=0.6923\), compared with the source affected probability \(0.20\) for each sibling. This large difference is expected: families with zero or one affected sibling are ineligible by design. A sample recruited for multiple affected relatives therefore cannot be treated as if it were a random sample of all families.

A Short Justification of Selection Weighting

The selection-weighted distribution follows directly from the definition of conditional probability. For any phenotype pattern \(y\), the event that a family has pattern \(y\) and is selected has probability \(P(Y=y,S=1)=P(S=1\mid Y=y)P(Y=y)\). Dividing this joint probability by the probability of selection gives the distribution among recruited families. Summing the numerator over all possible patterns gives the normalizing denominator.

P(Y=y\mid S=1) = \frac{P(Y=y,S=1)}{P(S=1)} = \frac{P(S=1\mid Y=y)P(Y=y)} {\sum_{y'}P(S=1\mid Y=y')P(Y=y')}

This argument requires no assumption that phenotype patterns are independent within a family. Independence may be used to calculate source probabilities in a particular model, but the selection-weighting identity itself applies whether or not family members' phenotypes are dependent.

Describe the Mechanism Before Interpreting the Sample

A useful description of a family study identifies the sampling unit, the qualifying rule, and the feature that triggers recruitment. For example: “Families were eligible if a specified child had the phenotype,” or “Families were recruited only when at least two siblings were affected.” It should also be clear whether the study records the proband, other relatives, or the entire pedigree. These details determine what observed proportions mean.

Ascertainment matters because an apparent concentration of affected relatives may partly reflect the route by which families entered the study. It does not, by itself, show that the underlying population has the same concentration, or that a particular genetic model is correct. The segregation and penetrance model describes phenotype probabilities; the recruitment rule describes how those families are filtered into the data.

Common pitfall: “Most people in the recruited pedigrees are affected, so most people in the source population must be affected.” This reasoning ignores selection. If a family is included because a proband is affected, or because several relatives are affected, the sample is deliberately enriched for those patterns. First state the ascertainment rule; then interpret observed phenotype proportions as properties of the recruited sample unless the sampling design supports a source-population interpretation.

Check Your Understanding

Use the recruitment rule as part of every interpretation.

A study enrolls a family whenever a specified proband is affected. Which phenotype is guaranteed among recruited families? Does this alone establish that every relative has elevated risk?
In the independent-sibling example, why does the nonproband sibling's affected probability remain 0.10 even though the affected fraction among all recruited siblings is 0.55?
In the high- and low-risk family example, what quantity changes from 0.20 to 0.6667 after recruitment? Explain why this changes the relative's observed affected probability.
For three independent siblings with affected probability 0.20, why are families with zero or one affected sibling absent from a sample recruited only when at least two are affected?
State the selection-weighting formula in words: what does the inclusion probability do to each phenotype pattern's source-population probability?