Index Event Bias in Genetic Mediation

Why Conditioning on a Diagnosis Can Change Associations

A genetic mediation analysis may include only people who experienced a diagnosis, such as a disease onset. That restriction can seem natural when the scientific question concerns progression or complications among affected people. But a diagnosis can be a common effect of several causes. Restricting the analysis to those with the diagnosis can then associate causes that were independent in the population. This selection-induced association is called index event bias.

The consequences matter for mediation. A selected sample can show an association between a mediator and a later outcome even when the mediator has no causal effect on that outcome. It can also show a genotype–mediator association that was absent before selection. If those associations are read as evidence for a genetic pathway, the resulting pathway may be misleading.

Definition: An index event is an event that defines entry into the group being analyzed, often a diagnosis. Index event bias is bias induced when analysis is restricted to people with that event and, as a result, conditioning on the event opens a noncausal association between its causes or their descendants. The event induces this bias when it is a common effect (a collider) of relevant variables in the causal structure.

For example, suppose genetic liability \(G\) and an environmental or biological factor \(U\) both affect diagnosis \(D\). In the source population, \(G\) and \(U\) may be independent. Among people with \(D=1\), however, a person with less genetic liability may need more of \(U\), on average, to have reached the diagnosis. Conditioning on \(D\) can therefore make \(G\) and \(U\) negatively associated. If \(U\) also affects a mediator \(M\) or later outcome \(Y\), this induced association can be passed on to analyses of those variables.

G \longrightarrow D \longleftarrow U \longrightarrow M \longrightarrow Y

In this diagram, \(D\) is a collider on the path from \(G\) to \(U\). The path is closed in the source population, but conditioning on \(D\) can open it. The diagram describes a possible mechanism, not a guarantee of bias in every case-only analysis. The size and direction of any bias depend on the causes of the index event, their effects on the variables being analyzed, and the selection process.

A Simple Selection Calculation

The basic calculation is Bayes' rule. For causes \(A\) and \(B\) of a diagnosis \(D\), the distribution of those causes among diagnosed people is

P(A,B\mid D=1) = \frac{P(D=1\mid A,B)P(A,B)}{P(D=1)}.

Even if \(P(A,B)=P(A)P(B)\) in the source population, the diagnosis-specific probability \(P(D=1\mid A,B)\) can change their joint distribution among cases. The following examples make that change explicit. All data and probabilities are hypothetical.

Worked Example: A Spurious Mediator–Outcome Association in Cases

Let \(M\) and \(U\) be independent binary variables, each with probability \(1/2\) of being 1. Suppose diagnosis occurs exactly when at least one is 1: \(D=1\) if \(M=1\) or \(U=1\). Let a later outcome be \(Y=U\). In this model, changing \(M\) does not change \(Y\); \(U\), not \(M\), determines the outcome.

In the source population, the four \((M,U)\) combinations each have probability \(1/4\). Among diagnosed people, the \((0,0)\) combination is absent. The remaining three combinations are equally likely:

\(M\)	\(U\)	\(Y=U\)	Probability among \(D=1\)
0	1	1	\(1/3\)
1	0	0	\(1/3\)
1	1	1	\(1/3\)

Thus, among cases, \(E(M\mid D=1)=2/3\), \(E(Y\mid D=1)=2/3\), and \(E(MY\mid D=1)=1/3\). The covariance is

\operatorname{Cov}(M,Y\mid D=1) = \frac{1}{3}-\left(\frac{2}{3}\right)\left(\frac{2}{3}\right) = -\frac{1}{9}.

Since \(\operatorname{Var}(M\mid D=1)= (2/3)(1/3)=2/9\), the slope from regressing \(Y\) on \(M\) among cases is

\frac{\operatorname{Cov}(M,Y\mid D=1)} {\operatorname{Var}(M\mid D=1)} = \frac{-1/9}{2/9} = -\frac{1}{2}.

The negative case-only association is not a causal effect of \(M\) on \(Y\): by construction, \(Y=U\) and \(M\) does not cause \(Y\). Conditioning on the diagnosis, which is caused by both \(M\) and \(U\), makes \(M\) and \(U\) negatively associated. Because \(U\) determines \(Y\), that induced association appears as a negative \(M\)–\(Y\) association in cases.

How a Genetic Pathway Can Appear After Selection

The same logic can affect the first link in a proposed genetic mediation pathway. If genotype and another cause of diagnosis are independent in the population, they can become associated among diagnosed people. If that other cause influences the mediator and outcome, the case-only data can then resemble a genetic pathway even when the genotype has no causal effect on either one.

Worked Example: An Apparent Genetic Association With a Mediator

Suppose \(G\) and \(U\) are independent binary variables, each with probability \(1/2\) of being 1. Diagnosis occurs if either is 1. Let \(M=U\) and \(Y=U\). There is no causal arrow from \(G\) to \(M\) or \(Y\); \(U\) determines both. Consider the association between \(G\) and \(M\) among people with the diagnosis.

The three case combinations \((G,M)=(0,1),(1,0),(1,1)\) each have conditional probability \(1/3\). Therefore, \(E(G\mid D=1)=E(M\mid D=1)=2/3\), while \(E(GM\mid D=1)=1/3\). The covariance is

\operatorname{Cov}(G,M\mid D=1) = \frac{1}{3}-\frac{4}{9} = -\frac{1}{9}.

The variance of \(G\) among cases is \(2/9\), so the case-only regression slope of \(M\) on \(G\) is \((-1/9)/(2/9)=-1/2\). In the source population the slope is zero, since \(G\) and \(U\), and hence \(G\) and \(M\), are independent. In the selected cases, the genotype appears associated with the mediator. Since \(U\) also determines \(Y\), the genotype appears associated with the outcome as well. Those associations do not establish a causal pathway from \(G\) through \(M\) to \(Y\).

This example uses perfect relationships to keep the arithmetic transparent. Real data will generally be noisier, but the selection mechanism does not require perfect measurement or a deterministic diagnosis rule. The next example shows the same issue with probabilistic diagnosis.

Worked Example: Probabilistic Diagnosis and the Induced Slope

Let \(G\) and \(U\) again be independent binary variables with probability \(1/2\) of being 1. Suppose the diagnosis probabilities are 0.10 when \(G=0,U=0\), 0.40 when exactly one is 1, and 0.70 when both are 1. Set \(M=U\). The probabilities of the four combinations before selection are each \(1/4\), so their joint probabilities with \(D=1\) are 0.025, 0.100, 0.100, and 0.175. They sum to \(P(D=1)=0.400\).

Dividing each joint probability by 0.400 gives the case distribution:

\(G\)	\(U=M\)	\(P(G,U\mid D=1)\)
0	0	\(1/16\)
0	1	\(4/16\)
1	0	\(4/16\)
1	1	\(7/16\)

Among cases, \(E(G)=E(M)=11/16\), and \(E(GM)=7/16\). Thus

\operatorname{Cov}(G,M\mid D=1) = \frac{7}{16}-\left(\frac{11}{16}\right)^2 = -\frac{9}{256}.

Also, \(\operatorname{Var}(G\mid D=1)=(11/16)(5/16)=55/256\). The selected-sample slope of \(M\) on \(G\) is therefore

\frac{-9/256}{55/256} = -\frac{9}{55} \approx -0.164.

The corresponding source-population slope is zero because \(G\) and \(U\) are independent there. Thus, a non-deterministic index event still induces a genotype–mediator association among cases. The numerical size here is a feature of this hypothetical model, not a general correction factor.

Implications for Genetic Mediation

In a mediation analysis, a genetic association with a mediator and a mediator–outcome association may be used to assess an indirect pathway. Index event bias can compromise either association when estimation is restricted to diagnosed people. For example, if \(G\) and \(U\) both cause diagnosis and \(U\) affects \(M\) or \(Y\), conditioning on diagnosis can associate \(G\) with \(U\). If \(M\) also contributes to the diagnosis and \(U\) predicts a later outcome, conditioning on diagnosis can associate \(M\) with \(U\), creating or distorting the observed \(M\)–\(Y\) relationship.

These are not necessarily the same causal structure, and an analysis should not assume that one diagram describes every disease. The key question is whether the diagnosis is a common effect of variables that also relate to the genetic exposure, mediator, or outcome. A genotype's independence from many confounders in the source population does not ensure independence from them after conditioning on a downstream event. Consequently, random allocation of alleles at conception does not by itself protect a case-only analysis from this selection mechanism.

The association among cases also answers a different descriptive question from the association in the source population: it concerns people who reached the index event. It should not automatically be interpreted as the population-level mediator effect or as a direct or indirect genetic effect. A case-only estimate can be relevant to a carefully defined question about affected people, but its causal interpretation requires assumptions that address the selection process as well as the usual mediation assumptions.

The direction of index event bias is not universal. In the examples, alternative causes competed to produce the diagnosis, leading to negative associations among cases. Different causal effects, diagnosis rules, interactions, or links from the alternative causes to the mediator and outcome can produce weaker, stronger, or differently signed distortions. Nor does every restriction to diagnosed people necessarily induce bias: the collider mechanism requires the relevant common-effect structure and connections to the analyzed variables.

Common Mistake and Practical Interpretation

Wrong reasoning: “The genotype is randomly allocated, so the genotype–mediator association among cases must be causal.”
Correction: Independence in the source population need not persist after restricting to a diagnosis caused by the genotype and other factors.
Wrong reasoning: “The mediator predicts the later outcome among cases, so the mediator must cause progression.”
Correction: The index event may associate the mediator with other causes of the later outcome; the selected-sample association alone does not establish a causal effect.
Wrong reasoning: “Index event bias always makes the estimated association negative.”
Correction: The sign depends on the causal structure and selection process. The negative signs above follow from the specific hypothetical models.

Key takeaway: When a genetic mediation analysis is restricted to people with a diagnosis, ask whether that diagnosis is a common effect of the genetic exposure, mediator, or other causes of the later outcome. Conditioning on such an event can create associations that resemble links in a genetic pathway, even when those links are not causal.

Check Your Understanding

Use the causal structure and calculations in this tutorial to answer the following.

In the first worked example, why are \(M\) and \(U\) independent in the source population but negatively associated among diagnosed people?
If a genotype and an alternative cause of diagnosis are independent before selection, what change in their association might occur after restricting to diagnosed people?
In the probabilistic diagnosis example, verify that the four conditional case probabilities sum to 1.
Why does an observed mediator–outcome association among cases not, by itself, establish that the mediator causes the later outcome?
Is the direction of index event bias necessarily negative? Explain what determines it.