Selection Into Studies as Collider Bias

Why Study Participation Can Change Associations

Genetic analyses usually estimate associations among people whose data were collected. But participation in a study is not always unrelated to the genetic, mediator, or outcome variables being analyzed. If inclusion depends on two otherwise independent causes, restricting analysis to participants can make those causes associated. That induced association may then alter the genetic–outcome or mediator–outcome associations used in a mediation analysis.

The central issue is not merely that participants differ from the source population. It is that conditioning on study participation can change the relationships among variables. A directed acyclic graph (DAG) makes the mechanism visible: if two arrows point into the selection variable, selection is a collider on the path between their causes.

Definition: Let \(S\) be an indicator of study inclusion, with \(S=1\) for included individuals. A variable \(S\) is a collider on a path when two arrows on that path point into \(S\), as in \(G \rightarrow S \leftarrow U\). If \(G\) and \(U\) are otherwise independent, conditioning on \(S=1\) can make them dependent. This is called collider bias when the induced dependence distorts an association or causal analysis.

Here \(G\) can represent a genetic variant or genetic exposure, and \(U\) a non-genetic cause such as access to care, health status, or a social factor. If both affect participation, the path \(G \rightarrow S \leftarrow U\) is closed when no condition is placed on \(S\). Restricting the analysis to participants conditions on \(S=1\), opening that path. The resulting association is created by the selection process; it need not reflect a relationship between \(G\) and \(U\) in the source population.

A useful probability expression is

P(G=g,U=u\mid S=1) = \frac{P(S=1\mid G=g,U=u)P(G=g)P(U=u)} {P(S=1)}

when \(G\) and \(U\) are independent before selection. The product \(P(G=g)P(U=u)\) describes their source-population distribution. The selection factor \(P(S=1\mid G=g,U=u)\) can disrupt that product, so the conditional distribution among participants need not preserve independence.

Worked Examples of Collider Bias

Worked Example: Two Independent Causes Become Associated

Suppose a hypothetical source population has a binary genetic factor \(G\) and a binary participation-related factor \(U\). Each has probability \(0.5\) of being 1, and they are independent. For this illustration only, a person is included if \(G=1\) or \(U=1\). Thus \(S=1\) for \((G,U)=(1,0),(0,1),(1,1)\), but not for \((0,0)\).

Each of the four source-population combinations has probability \(0.25\). After restricting to \(S=1\), the three remaining combinations each have conditional probability \(0.25/0.75=1/3\). Therefore,

P(G=1\mid S=1)=\frac{2}{3},\qquad P(U=1\mid S=1)=\frac{2}{3},\qquad P(G=1,U=1\mid S=1)=\frac{1}{3}.

The selected-sample covariance is

\operatorname{Cov}(G,U\mid S=1) = E(GU\mid S=1)-E(G\mid S=1)E(U\mid S=1) = \frac{1}{3}-\frac{2}{3}\frac{2}{3} = -\frac{1}{9}.

The variables were independent in the source population, where their covariance was 0, but are negatively associated among selected people. The reason is visible in the selection rule: among included people with \(G=0\), \(U\) must equal 1; people with \(G=1\) can have either value of \(U\). The association is induced by conditioning on the common effect \(S\), not by a causal effect of \(G\) on \(U\) or vice versa.

Worked Example: A Genetic–Outcome Association Changes After Selection

Consider a hypothetical source population with independent binary \(G\) and \(U\), each having probability \(0.5\) of being 1. Let the outcome, measured in arbitrary units, follow \(Y=G+U\). Assume selection follows the same rule as above: \(S=1\) if \(G=1\) or \(U=1\). In the source population, changing \(G\) from 0 to 1 changes \(Y\) by 1 unit when \(U\) is held fixed.

Among selected people, the three equally likely observations are \((G,U,Y)=(1,0,1),(0,1,1),(1,1,2)\). Thus \(E(G\mid S=1)=2/3\), \(E(Y\mid S=1)=4/3\), and \(E(GY\mid S=1)=(1+0+2)/3=1\). The covariance and variance are

\operatorname{Cov}(G,Y\mid S=1) = 1-\frac{2}{3}\frac{4}{3} = \frac{1}{9}, \qquad \operatorname{Var}(G\mid S=1) = \frac{2}{3}-\left(\frac{2}{3}\right)^2 = \frac{2}{9}.

The simple linear regression slope of \(Y\) on \(G\) among selected people is therefore

\hat{\beta}_{G,Y\mid S=1} = \frac{\operatorname{Cov}(G,Y\mid S=1)} {\operatorname{Var}(G\mid S=1)} = \frac{1/9}{2/9} = 0.5\text{ outcome units per unit of }G.

In the source population the slope is 1, because \(Y=G+U\) and \(G\) is independent of \(U\). In the selected population, \(G\) and \(U\) are negatively associated, so the selected-sample association does not equal the structural change of 1 unit. This simplified calculation shows how selection can alter a genetic–outcome association; it does not imply that selection always attenuates an association.

Worked Example: Selection Distorts a Mediator–Outcome Association

Now use a hypothetical simplified mediation model. Let \(G\) be binary with probability \(0.5\), let the mediator be \(M=G\), and let \(U\) be an independent binary cause with probability \(0.5\). Suppose \(Y=M+U\), in outcome units, and inclusion occurs when \(M=1\) or \(U=1\). In the source population, the structural effect of increasing \(M\) from 0 to 1 while holding \(U\) fixed is 1 outcome unit. The DAG contains \(G\rightarrow M\rightarrow Y\), \(U\rightarrow Y\), and the selection collider \(M\rightarrow S\leftarrow U\).

The selected \((M,U,Y)\) combinations are \((1,0,1),(0,1,1),(1,1,2)\), each with probability \(1/3\). Therefore \(E(M\mid S=1)=2/3\), \(E(Y\mid S=1)=4/3\), and \(E(MY\mid S=1)=1\). It follows that

\operatorname{Cov}(M,Y\mid S=1) = 1-\frac{2}{3}\frac{4}{3} = \frac{1}{9}, \qquad \operatorname{Var}(M\mid S=1) = \frac{2}{3}-\left(\frac{2}{3}\right)^2 = \frac{2}{9}.

The selected-sample regression slope of \(Y\) on \(M\) is \((1/9)/(2/9)=0.5\) outcome units per mediator unit, rather than the structural effect of 1. Conditioning on \(S=1\) has associated \(M\) with \(U\), which also affects \(Y\). In this example the induced association is negative and reduces the observed mediator–outcome slope. The example isolates one way selection can distort an association used in mediation analysis; it does not establish that a particular mediation estimator has this exact bias in other settings.

What the Examples Establish

The arithmetic in the examples follows a general argument. In the source population, independent causes have a joint distribution that factors into their marginal distributions. Among selected people, that joint distribution is reweighted by the probability of inclusion. If inclusion depends jointly on both causes in a way that does not preserve the factorization, the causes can become associated.

P(G,U\mid S=1) \propto P(S=1\mid G,U)P(G)P(U).

For the induced association to distort a genetic–outcome association, the non-genetic cause must also be related to the outcome, or selection must otherwise alter the association being estimated. For the mediator–outcome association, the analogous concern is that selection associates the mediator with a cause of the outcome. In either case, an association calculated among participants can combine the relationship of interest with a relationship created by selection.

Collider bias is not inevitable whenever a study has an inclusion criterion. For example, suppose \(G\) and \(U\) are independent in the source population and inclusion depends only on \(G\), not on \(U\). Then conditioning on inclusion can change the distribution of \(G\), but it does not by itself induce an association between \(G\) and \(U\): the selection factor depends only on \(G\), so the conditional joint distribution still factors. More generally, a selection rule can preserve independence if its probability factorizes appropriately. The relevant question is whether the selection mechanism opens a path between causes that matter for the association or causal effect under study.

Implications for Genetic Mediation

Genetic mediation analyses often combine evidence about genetic associations with an exposure or mediator, the mediator with the outcome, and the genetic factor with the outcome. Selection into a genetic study can affect any of these associations if inclusion depends on variables along the relevant paths. For example, if both a genetically influenced trait and a non-genetic cause of the outcome influence participation, conditioning on participation may associate them. The resulting genetic–outcome association may then reflect both the genetic pathway and the induced relationship with the non-genetic cause.

Likewise, if a mediator and a cause of the outcome both influence inclusion, participants can show a mediator–outcome association that differs from the association in the source population. Because mediation methods use associations to learn about pathways, distortion of one input can affect the resulting interpretation of direct or indirect effects. This is a threat to the validity of those associations, not evidence that a mediated effect exists.

There is no universal direction of collider bias. In the worked examples, selection produced negative dependence and attenuated slopes, but other selection probabilities, outcome relationships, or variable scales can produce a different direction or magnitude. Nor does a strong association among participants establish that the variables caused one another: it may partly reflect the selection process. Correcting the problem requires information and assumptions about how selection occurred; merely analyzing the selected sample does not remove the conditioning.

Common Mistake

Wrong reasoning: “The genetic factor and \(U\) were independent before the study, so they must remain independent in the participants.”
Correction: Independence in the source population does not guarantee independence after conditioning. If both factors affect study inclusion, the selected distribution is reweighted by the inclusion probability, which can associate them.
Another wrong reasoning: “Selection always makes the genetic association smaller.”
Correction: Bias direction depends on the selection mechanism and the relationships among the causes and outcome. The examples show attenuation under one specific rule, not a general direction.

Key takeaway: When two otherwise independent causes both influence study inclusion, conditioning on participation can open a collider path between them. If one cause also predicts the mediator or outcome, the induced association can distort genetic and mediation associations among participants.

Check Your Understanding

Use the DAG and conditioning arguments from this tutorial to answer the following.

In \(G\rightarrow S\leftarrow U\), why can \(G\) and \(U\) be associated among people with \(S=1\) even if they are independent in the source population?
In the first worked example, what is \(\operatorname{Cov}(G,U\mid S=1)\), and what feature of the selection rule produces its sign?
Why can selection distort a mediator–outcome association when the mediator and a cause of the outcome both affect inclusion?
If inclusion depends only on \(G\) and \(G\) is independent of \(U\) in the source population, does conditioning on inclusion necessarily associate \(G\) and \(U\)? Explain.
Why does the negative association in the worked examples not establish that collider bias always attenuates genetic or mediation associations?