Why Post-Association Conditioning Needs a Causal Check
Genetic mediation analyses often restrict attention to variants that pass an association threshold, have a particular functional annotation, or satisfy both criteria. Such restrictions can be useful for defining a candidate set, but they also change which variants enter the analysis. If selection depends on association evidence, annotation, or other causes of the estimated effects, the selected set may have relationships that were absent in the set considered before filtering.
The central diagnostic question is not simply whether a filter was used. It is what caused inclusion, and what other variables caused the association or mediation quantities being interpreted? Conditioning on a common effect of two variables—a collider—can induce an association between those variables. In genetic analyses, selection can therefore connect a variant's underlying effect with sampling error, or connect a functional feature with other locus properties that relate to the outcome. Neither consequence is automatic: the relevant causal structure must be plausible for the actual selection rule.
Two Selection Structures to Distinguish
For association-based selection, consider a collection of loci and let \(\theta_j\) be the true effect for locus \(j\), \(\hat{\theta}_j\) its estimated effect, and \(\varepsilon_j\) its estimation error. A simple sampling model is \(\hat{\theta}_j=\theta_j+\varepsilon_j\). If a locus is retained when its test statistic passes a threshold, the selection indicator \(S_j\) depends on \(\hat{\theta}_j\). The corresponding diagram is \(\theta_j\rightarrow\hat{\theta}_j\leftarrow\varepsilon_j\), with \(\hat{\theta}_j\rightarrow S_j\). The estimate is a common effect of the true effect and sampling error; selection on its descendant can induce dependence between them across the selected loci.
This is a statement about the distribution of effects and errors among selected loci, not a claim that selection changes a fixed variant's biological effect. It helps explain why selected estimates can be too extreme: among variants with modest true effects, those included are disproportionately likely to have errors that push their estimates toward the threshold. The same concern can affect a mediation analysis if the association estimates used to choose instruments or candidates are also used to estimate or interpret mediation effects.
A different structure can arise with annotation-based selection. Suppose a candidate feature \(A\) and another locus property \(U\) both affect whether a locus enters a functionally filtered set, while \(U\) also affects the outcome \(Y\). The diagram is \(A\rightarrow S\leftarrow U\rightarrow Y\). Before conditioning on \(S\), \(A\) and \(U\) may be independent; among selected loci, they may become associated. Since \(U\) relates to \(Y\), a comparison involving \(A\) in that selected set can then reflect this induced relationship as well as any relationship of interest.
The arrow structure must be justified from the actual filter. An annotation that was fixed independently of the outcome data may not have the second structure. Conversely, an annotation chosen after inspecting association results, or a composite inclusion rule requiring either of two kinds of evidence, may depend on multiple causes. Calling every functional filter a collider would be incorrect; the diagnostic is to trace its causes and consequences.
Worked Example: Tracing Selection on an Association Statistic
Consider a hypothetical set of loci. At each locus, an estimated effect is the sum of its true effect and sampling error, and the analysis retains a locus if its estimated effect exceeds a positive threshold. Identify the variables involved and explain why retained estimates can be optimistic.
Model. For locus \(j\), write \(\hat{\theta}_j=\theta_j+\varepsilon_j\). Selection is \(S_j=1\) when \(\hat{\theta}_j/SE_j>c\), where \(SE_j\) is the standard error and \(c\) is the chosen threshold. Thus, both the true effect and its error contribute to the quantity determining inclusion.
Diagram. The true effect and error point into the estimate: \(\theta_j\rightarrow\hat{\theta}_j\leftarrow\varepsilon_j\). Selection is caused by the estimate, so conditioning on \(S_j=1\) selects on a descendant of this common effect. Among selected loci, a smaller \(\theta_j\) must, on average, be accompanied by a more favorable \(\varepsilon_j\) to cross the threshold.
Interpretation. The retained estimates are not a neutral sample of the estimates that would have been observed across all eligible loci. An estimate that crossed the threshold partly because of positive error can exaggerate the underlying effect. If the same selected estimates are then used to characterize instrument strength or a mediation pathway, the association-based filter needs to be considered when interpreting the result.
How Thresholding Shifts an Estimate
The direction and size of the selection effect can be made explicit under a simple model. Suppose a single estimate has \(\hat{\theta}=\theta+\varepsilon\), with \(\varepsilon\sim N(0,\sigma^2)\), and it is selected when \(\hat{\theta}>c_0\). Here \(\theta\) and \(\sigma\) are treated as fixed for this calculation. Define \(a=(c_0-\theta)/\sigma\), and let \(\phi\) and \(\Phi\) denote the standard normal density and cumulative distribution functions.
To justify the second line, standardize the error as \(Z=\varepsilon/\sigma\sim N(0,1)\). The conditional mean of \(Z\) above \(a\) is \(\int_a^\infty z\phi(z)\,dz/[1-\Phi(a)]\). Since \(\phi'(z)=-z\phi(z)\), the numerator is \([-\phi(z)]_a^\infty=\phi(a)\). The ratio is positive for every finite \(a\), so the expected selected estimate exceeds \(\theta\) under this one-sided rule.
This formula is not a universal correction for genome-wide significance. Real selection may be two-sided, involve many correlated variants, or depend on several association results. It does establish a clear diagnostic: when estimates are selected for being large in a specified direction, their conditional mean is shifted in that direction even if the estimator was unbiased before selection.
Worked Example: Quantifying a One-Sided Selection Shift
Suppose, hypothetically, that a locus has true effect \(\theta=0.15\) outcome units per allele and sampling standard deviation \(\sigma=0.10\) outcome units per allele. Retain it when \(\hat{\theta}/0.10>2\), so the estimate must exceed \(c_0=0.20\). Calculate the expected estimate among loci with this true effect that pass the rule.
The standardized cutoff relative to the true effect is \(a=(0.20-0.15)/0.10=0.50\). For a standard normal distribution, \(\phi(0.50)\approx0.3521\) and \(1-\Phi(0.50)\approx0.3085\). Therefore,
Among these hypothetical selected estimates, the expected estimate is about \(0.2641\) outcome units per allele, roughly \(0.1141\) units per allele above the true effect. The calculation does not say that every selected estimate is inflated by exactly that amount, nor does it give the probability that this particular locus is selected. It gives the conditional mean under the stated normal model and one-sided threshold.
Diagnosing Functional and Combined Filters
For a functional filter, a useful first step is to write down the inclusion rule in words and then identify its plausible causes. A rule that selects loci with either strong mediator association or a particular annotation can have a different structure from a rule requiring both. In either case, ask whether some other determinant of inclusion also predicts mediator or outcome effects, or is related to the variants' biological mechanisms. If so, conditioning on inclusion may connect that determinant to the feature being compared.
A compact example shows how an association can be created by selection alone. Let \(A\) and \(U\) be independent binary properties, each with probability \(0.5\). Let selection occur if either property is present, \(S=1\) when \(A=1\) or \(U=1\). Suppose, for illustration, that the outcome is \(Y=U\). In the full set, \(A\) is independent of \(Y\). In the selected set, however, the distribution of \(U\) differs by \(A\): if \(A=0\), selection requires \(U=1\), whereas if \(A=1\), either value of \(U\) is allowed.
Worked Example: Selection Creates an Association With an Outcome
Use the hypothetical binary setup just described. The four equally likely combinations of \(A\) and \(U\) are \((0,0),(0,1),(1,0),(1,1)\); the first is excluded and the other three are selected. Calculate the outcome mean in each selected \(A\) group.
For selected loci with \(A=0\), the only possible combination is \((0,1)\). Thus \(Y=U=1\) for every such locus and \(E(Y\mid A=0,S=1)=1\).
For selected loci with \(A=1\), both \((1,0)\) and \((1,1)\) occur with equal probability. Their outcomes are 0 and 1, so \(E(Y\mid A=1,S=1)=(0+1)/2=0.5\). The selected-set difference is \(0.5-1=-0.5\).
Before selection, \(A\) and \(U\) are independent, so \(E(Y\mid A=1)=E(Y\mid A=0)=0.5\), and the difference is zero. The selected-set association is produced by the rule \(S=1\) when \(A=1\) or \(U=1\), not by an effect of \(A\) on \(Y\). In a genetic analysis, this example is a schematic warning: the actual causes of the annotation or inclusion rule must support the analogous arrows before the same diagnosis applies.
Comparisons That Help Diagnose Selection
A causal diagram identifies possible bias; comparisons across analyses can show whether the result is sensitive to the selection mechanism. These comparisons do not prove that a particular path is present or absent. They are most informative when each analysis has a clearly stated eligibility rule and when the quantities being compared target the same scientific question.
- Compare with a broader eligible set. Where scientifically justified, examine estimates using variants defined by prespecified criteria rather than by the association result under study. A change in effect estimates after the significance filter is a warning that selection may matter, though differences can also reflect instrument relevance or other changes in the analysis set.
- Use independent evidence for selection or estimation. Selecting candidates in one data source and estimating effects in independent data can reduce dependence between the selection noise and estimation noise. Independence must be credible; shared participants, overlapping measurements, or reuse of the same summary statistics can retain dependence.
- Separate annotation from outcome evidence. Where possible, define functional categories without using the outcome associations being interpreted, then compare them with categories chosen after examining those associations. This can help reveal whether the filtering step itself is entangled with the result.
- Inspect both retained and excluded candidates. Record why candidates were excluded and compare the relevant estimates and characteristics. Excluded variants are not automatically valid controls, but their patterns can help identify which variables drive inclusion.
- Report the selection rule and its timing. State whether the threshold was prespecified, which association supplied the evidence, whether the same data were reused, and how multiple or correlated filters were combined. These details determine which causal paths are plausible.
In genetic mediation, an observed association among selected variants is not, by itself, evidence that a mediator carries a causal effect. Selection may have opened a path between a genetic feature and other determinants of the outcome; alternatively, the filter may be independent of those determinants and create no such path. The conclusion depends on the structure, not on the word “selection” alone.
Common Mistakes and a Practical Diagnostic
- Wrong: “The variants passed a stringent significance threshold, so their estimates are unbiased.” Correction: A stringent threshold changes the distribution of included estimates; under directional selection, sampling error among selected estimates tends to favor threshold crossing.
- Wrong: “Every functional annotation filter is a collider.” Correction: A collider requires at least two relevant causes of the conditioned variable. Trace whether the annotation or selection indicator has multiple causes that connect to the effects of interest.
- Wrong: “A difference between selected and unselected analyses proves collider bias.” Correction: The analyses may differ in relevance, precision, or target set as well as selection. Use the diagram and design details to distinguish these explanations.
- Wrong: “Selection bias means the biology is false.” Correction: Selection affects what the analysis can support; it does not establish that a biological pathway is absent or present. Interpret causal claims in light of the selection mechanism and other assumptions.
Write the threshold, annotation, or combined rule that determines which loci enter the analysis.
Identify the estimated association, sampling error, annotations, locus properties, and outcome-related variables that could affect inclusion.
Ask whether conditioning on inclusion connects causes that otherwise would not be associated and whether that connection can affect the mediation interpretation.
Where possible, contrast the selected analysis with one using independent or prespecified information, and describe remaining differences and limitations.
Check Your Understanding
Use a causal diagram and the selection rule to justify each answer.
- In the model \(\hat{\theta}=\theta+\varepsilon\), which variable is the common effect, and how can selecting on a threshold for \(\hat{\theta}\) connect its causes?
- Under one-sided selection \(\hat{\theta}>c_0\), why is the conditional expected estimate above the true effect in the normal model?
- Give one reason why selecting variants by functional annotation might create collider bias, and one reason why it might not.
- In the binary example, why does the selected-set outcome difference not establish an effect of \(A\) on \(Y\)?
- Name two design or reporting choices that can help assess whether post-association conditioning affects a mediation analysis.