Cohort Participation as a Collider

Why Participation Can Change a Genetic Association

A polygenic score is often evaluated in people who enrolled in a cohort, not in everyone from the population the score is intended to describe. Enrollment may depend both on genetic liability and on health characteristics related to the outcome. Restricting analysis to enrolled people can then create or change an association between the score and outcome, even when the score is not associated with the outcome in the source population.

This is a form of collider bias. It is a consequence of conditioning on a common effect of two variables. The score–outcome association observed among participants can therefore differ from the population association; the difference is not, by itself, evidence that the score has a different biological effect in the cohort.

Participation as a Collider

Let \(G\) represent genetic liability summarized by a score, \(H\) a health characteristic related to the outcome, \(Y\) the outcome, and \(S\) an indicator of cohort participation (\(S=1\) for enrolled people). The diagram below is a simplified model: genetic liability and health can each influence enrollment, while health and possibly genetic liability are related to the outcome.

G \longrightarrow S \longleftarrow H \longrightarrow Y, \qquad\text{and possibly}\qquad G \longrightarrow Y.

The arrows involving \(G\) summarize dependencies associated with underlying genetic liability; they do not mean that changing a numerical score itself would change a person's outcome. The node \(S\) is a collider because arrows from both \(G\) and \(H\) point into it. In the source population, the path from \(G\) through \(S\) to \(H\) is closed. Restricting analysis to \(S=1\) can open that path, making \(G\) and \(H\) associated among participants. If \(H\) predicts \(Y\), this induced association can be transmitted to the score–outcome association.

Definition: Participation-as-collider bias occurs when analysis conditions on enrollment \(S=1\), a common effect of genetic liability \(G\) and another characteristic \(H\), thereby inducing or changing an association between \(G\) and \(H\). If \(H\) is related to outcome \(Y\), the score–outcome association among participants can consequently differ from that in the source population.

This mechanism does not require the score to influence participation directly in a literal or manipulable sense. A genetic liability that the score measures may influence health, study eligibility, or willingness to enroll. Likewise, \(H\) may represent a current health state, a symptom, or a health-related factor that affects participation and predicts the study outcome. A diagram is a simplified account of the dependencies being considered; other causes of participation or outcome may also matter.

A Short Derivation: How Conditioning Creates Dependence

Suppose \(G\) and \(H\) are binary and independent in the source population. Write \(\pi_{gh}=P(S=1\mid G=g,H=h)\), the probability of participation for people with values \(g\) and \(h\). Among participants, the odds ratio comparing \(G\) and \(H\) is

\frac{P(G=1,H=1\mid S=1)\,P(G=0,H=0\mid S=1)} {P(G=1,H=0\mid S=1)\,P(G=0,H=1\mid S=1)} = \frac{\pi_{11}\pi_{00}}{\pi_{10}\pi_{01}}.

To see why, each joint probability in the numerator or denominator is proportional to the source-population joint probability times the relevant participation probability. The common normalizing constant for conditioning on \(S=1\) cancels. Source independence means \(P(G=1,H=1)P(G=0,H=0)=P(G=1,H=0)P(G=0,H=1)\), so the remaining ratio is the one shown. If this ratio is not 1, \(G\) and \(H\) are associated among participants even though they were independent in the source population. Its direction depends on the pattern of participation probabilities; conditioning does not always induce the same sign of association.

This calculation explains the open path in the diagram, but it does not determine the size or direction of the resulting score–outcome association on its own. That also depends on how \(H\) relates to \(Y\), whether \(G\) already predicts \(Y\), and the particular way participation selects people.

Worked Examples

Worked Example: An Association Appears From Selection

Hypothetical data-generating model. In a source population, binary \(G\) and \(H\) are independent, each with prevalence 0.5. The outcome is \(Y=H\), so \(H=1\) always means the outcome is present and \(G\) has no source-population association with \(Y\). Participation probabilities are as follows:

\(G\)	\(H\)	\(P(S=1\mid G,H)\)
0	0	0.20
1	0	0.60
0	1	0.60
1	1	0.80

Each source-population cell has probability \(0.25\). Among participants with \(G=1\), the probabilities contributed by \((G,H)=(1,0)\) and \((1,1)\) are \(0.25(0.60)=0.15\) and \(0.25(0.80)=0.20\). Therefore,

P(Y=1\mid G=1,S=1)=P(H=1\mid G=1,S=1) =\frac{0.20}{0.15+0.20}=0.5714.

For \(G=0\), the corresponding participant probabilities are \(0.25(0.20)=0.05\) for \(H=0\) and \(0.25(0.60)=0.15\) for \(H=1\). Thus,

P(Y=1\mid G=0,S=1)=\frac{0.15}{0.05+0.15}=0.7500, \qquad \text{risk difference}=0.5714-0.7500=-0.1786.

In the source population, \(P(Y=1\mid G=1)=P(Y=1\mid G=0)=0.5\). The source association is zero, but the participant-only risk difference is \(-0.1786\), or 17.86 percentage points lower outcome risk for \(G=1\) than for \(G=0\). The participation probabilities favor enrollment for people with higher values of either variable, but enrollment has made \(G\) and \(H\) negatively associated among participants. Since \(Y=H\), that induced dependence appears as a negative score–outcome association.

Worked Example: Selection Attenuates an Existing Association

Hypothetical model. Keep the same independent binary \(G\) and \(H\), each with source prevalence 0.5, and the same participation probabilities as in the first example. Now let the outcome probability be \(P(Y=1\mid G,H)=0.10+0.20G+0.40H\). These are valid probabilities, ranging from 0.10 to 0.70. In the source population, \(P(H=1\mid G)=0.5\), so the outcome risks are \(0.10+0.40(0.5)=0.30\) for \(G=0\) and \(0.10+0.20+0.40(0.5)=0.50\) for \(G=1\). The source-population risk difference is \(0.20\).

The first example established that among participants \(P(H=1\mid G=1,S=1)=0.5714\) and \(P(H=1\mid G=0,S=1)=0.7500\). Averaging the specified outcome probabilities over those participant distributions gives

\begin{aligned} P(Y=1\mid G=1,S=1)&=0.10+0.20+0.40(0.5714)=0.5286,\\ P(Y=1\mid G=0,S=1)&=0.10+0.40(0.7500)=0.4000,\\ \text{participant risk difference}&=0.5286-0.4000=0.1286. \end{aligned}

The association remains positive, but the participant-only risk difference is \(0.1286\), smaller than the source value \(0.20\). Here selection has induced a negative association between \(G\) and the outcome-related \(H\), partially offsetting the positive association between \(G\) and \(Y\). It has not erased the existing association, and it has not established that the difference is a change in the score's biological effect.

Worked Example: Selection Reverses the Association

Hypothetical model. Again, \(G\) and \(H\) are independent binary variables with source prevalence 0.5. Suppose \(P(Y=1\mid G,H)=0.05+0.20G+0.60H\); the possible outcome probabilities range from 0.05 to 0.85. The source-population risk for \(G=0\) is \(0.05+0.60(0.5)=0.35\), and for \(G=1\) it is \(0.05+0.20+0.60(0.5)=0.55\). The source risk difference is \(0.20\).

Now suppose the participation probabilities for cells \((G,H)=(0,0),(1,0),(0,1),(1,1)\) are, respectively, \(0.05,0.90,0.90,0.95\). Among participants with \(G=1\), the conditional prevalence of \(H=1\) is \(0.95/(0.90+0.95)=0.5135\). Among participants with \(G=0\), it is \(0.90/(0.05+0.90)=0.9474\). The selected outcome risks are therefore

\begin{aligned} P(Y=1\mid G=1,S=1)&=0.05+0.20+0.60(0.5135)=0.5581,\\ P(Y=1\mid G=0,S=1)&=0.05+0.60(0.9474)=0.6184,\\ \text{participant risk difference}&=0.5581-0.6184=-0.0603. \end{aligned}

The source association is positive, but the selected association is negative: it is about 6.03 percentage points lower risk for \(G=1\). In this model, the strong selection-induced difference in \(H\), which is strongly related to \(Y\), outweighs the positive source association. A reversed association in a selected cohort is therefore possible even when the source association is positive.

What to Conclude From a Participant-Only Estimate

The examples use known probabilities to isolate the selection mechanism. In an actual cohort, the source-population participation probabilities and the relevant distributions may not be known. A participant-only estimate consequently describes the association in the enrolled group; it does not automatically estimate the corresponding association in the source population. The direction of bias cannot be inferred from the word “collider” alone. It depends on how the causes of participation relate to one another and to the outcome.

A participation diagram is useful because it makes the assumptions behind an interpretation visible. Ask which genetic liability or score-related features may affect enrollment, which health characteristics may affect both enrollment and outcome, and whether analysis is restricted to participants. Then assess whether the estimated target is the enrolled cohort or a broader population. This is especially important when comparing score performance across cohorts: differences may reflect selection as well as differences in the underlying population or measurement.

Common pitfall: “The score predicts the outcome among enrolled participants, so it must predict the outcome by the same amount in the source population.” This reasoning ignores that enrollment may have opened the path \(G\rightarrow S\leftarrow H\rightarrow Y\). The participant association can be induced, attenuated, or reversed. State the population to which the estimate applies, and do not treat a selected-cohort association as a population result without additional justification.

Key takeaway: When genetic liability and outcome-related health both influence cohort participation, conditioning on participation can alter score–outcome associations. The effect may create an association where none existed, change its magnitude, or reverse its direction.

Check Your Understanding

For each question, distinguish the source population from the enrolled participants.

In the diagram \(G\rightarrow S\leftarrow H\rightarrow Y\), which node is the collider, and what does conditioning on \(S=1\) do to the path through it?
If \(G\) and \(H\) are independent before selection, what does a participant-only association between them indicate about conditioning on enrollment?
In the first worked example, why does an association between \(G\) and \(Y\) appear among participants even though the source-population risk is 0.5 for both values of \(G\)?
Can selection bias only create an association when the source association is zero? Use the second and third examples to explain.
Why is it incorrect to interpret a participant-only score–outcome association automatically as the association in the population the cohort is intended to represent?