From a Path Product to a Causal Claim
A genetic variant may be associated with an upstream molecular trait, such as gene expression, and with a downstream outcome, such as a disease phenotype. A mediation model can summarize this pattern with an upstream path from genotype to molecular trait and a downstream path from molecular trait to outcome. Their product is a useful description of a fitted model. It is not, by itself, proof that changing the molecular trait would change the outcome.
The distinction matters because genotype is not the only source of variation in a molecular trait. Environmental exposures, cell composition, technical factors, and other biological processes can influence both the molecular trait and the outcome. Further, the analyzed participants may be a selected subset, and the measured molecular value may differ from the true trait. Each issue can alter the fitted paths or their interpretation.
For a simple additive linear model, write
Here \(C\) is a set of measured pre-exposure covariates, \(a\) is the genotype-to-mediator coefficient, \(b\) is the mediator coefficient conditional on genotype and covariates, and \(c'\) is the direct genotype coefficient conditional on the mediator and covariates. Under the model and causal assumptions discussed below, the indirect effect per unit change in \(G\) is \(ab\), and the total effect is \(c'+ab\). These identities require the stated linear, additive structure; they are not general identities for arbitrary outcomes, interactions, or nonlinear models.
What Must Be Identified
Potential outcomes make the causal target precise. Let \(M(g)\) be the mediator value that would occur under genotype exposure level \(g\), and let \(Y(g,m)\) be the outcome that would occur under genotype level \(g\) and mediator value \(m\). For two genotype levels \(g_0\) and \(g_1\), one natural indirect effect is
It compares outcomes at the same genotype level \(g_1\), changing only the mediator from the value it would take under \(g_0\) to the value it would take under \(g_1\). The corresponding natural direct effect is \(E\{Y(g_1,M(g_0))-Y(g_0,M(g_0))\}\). Their sum is the total contrast \(E\{Y(g_1,M(g_1))-Y(g_0,M(g_0))\}\). These natural effects involve counterfactual quantities under different exposure levels and therefore require strong assumptions to identify from observed data.
A standard identification argument requires consistency (observed outcomes equal the relevant potential outcomes under the exposure and mediator actually received), no interference between participants, and positivity (each relevant exposure and mediator value has support within covariate strata used for comparison). It also requires adequate control of confounding: conditional on \(C\), there must be no unmeasured confounding of genotype and mediator, genotype and outcome, or mediator and outcome. In addition, identifying natural effects requires a cross-world condition linking outcomes under one exposure level to mediator values under another. A common sufficient formulation is sequential exchangeability together with no exposure-induced mediator–outcome confounder. These assumptions are causal conditions, not consequences of a small p-value or a strong genetic association.
For genotype, exchangeability can be threatened by population structure, assortative mating, relatedness, or selection into the analyzed sample. Genetic assignment is not automatically equivalent to randomized assignment in the study population. Earlier material on colocalization helps assess whether molecular and outcome associations are compatible with a shared causal variant, but even convincing shared-variant evidence does not establish that the molecular trait mediates the outcome association. Horizontal pathways from the variant to the outcome that do not pass through \(M\) must also be represented or ruled out for a full mediation interpretation.
Why Mediator–Outcome Confounding Is Central
Even when \(G\) is a useful source of variation in \(M\), it does not randomize all differences in \(M\). A factor \(U\) that affects both \(M\) and \(Y\) can make the conditional association between them differ from the causal effect of changing \(M\). For example, inflammation could alter measured expression and independently affect disease risk. Adjusting for \(G\) does not remove this problem unless the common causes of \(M\) and \(Y\) are otherwise controlled.
The following short derivation shows the issue. First residualize \(M\) and \(Y\) with respect to \(G\) and \(C\); denote the residuals by \(M_r\) and \(Y_r\). If the outcome equation has true mediator coefficient \(b\), then \(Y_r=bM_r+U_r\). The regression coefficient of \(Y_r\) on \(M_r\) is
The second equality follows by substituting \(Y_r=bM_r+U_r\), taking covariance with \(M_r\), and dividing by \(\operatorname{Var}(M_r)\). Thus \(\tilde b=b\) only when the residual mediator is uncorrelated with the residual outcome disturbance, or when a suitable adjustment strategy makes that condition hold. This is why the coefficient must be interpreted conditional on genotype and covariates, and why unmeasured mediator–outcome confounding can bias the path product.
Worked Example: A Path Product Under Explicit Assumptions
Consider hypothetical data analyzed with the additive linear model above. Suppose the estimated \(G\)-to-\(M\) coefficient is \(a=0.30\) standard deviations of expression per additional effect allele. Suppose the conditional \(M\)-to-\(Y\) coefficient is \(b=0.40\) outcome units per standard deviation of expression, and the conditional direct coefficient is \(c'=0.08\) outcome units per allele.
Under linearity, no genotype-by-mediator interaction, adequate covariate adjustment, and the identification assumptions just described, the indirect-effect estimate per allele is
The corresponding total-effect estimate is
This arithmetic establishes the path-product value under the model; it does not verify the assumptions. In particular, the interpretation as mediation would require that the conditional \(M\)-to-\(Y\) coefficient is not confounded, that the molecular trait is measured adequately, and that selection has not distorted the relevant associations. Without those conditions, \(0.120\) is a model-based decomposition, not an identified causal indirect effect.
Worked Example: Residual Mediator–Outcome Confounding
Suppose that, after controlling for genotype and measured covariates, a hypothetical unmeasured factor \(U\) has variance 1. Let the residual mediator be \(M_r=0.70U+\epsilon_M\), where \(\operatorname{Var}(\epsilon_M)=0.51\), and let the residual outcome satisfy \(Y_r=0.40M_r+0.50U+\epsilon_Y\). Assume \(U\), \(\epsilon_M\), and \(\epsilon_Y\) are mutually uncorrelated, and \(\epsilon_Y\) is uncorrelated with \(M_r\).
The residual mediator variance and its covariance with \(U\) are
The regression of \(Y_r\) on \(M_r\) therefore has coefficient
The causal mediator coefficient in the stated data-generating model is \(0.40\), but the fitted conditional coefficient is \(0.75\) because the unmeasured factor raises both the mediator and outcome. If \(a=0.30\), the fitted path product would be \(0.30(0.75)=0.225\), whereas the product using the true mediator coefficient is \(0.30(0.40)=0.120\). The difference is not a sampling fluctuation in this calculation; it is bias from the specified confounding structure.
Measurement Error in the Molecular Trait
An assay often records a noisy value \(W=M^*+e\), where \(M^*\) is the true molecular trait and \(e\) is measurement error. Under classical measurement error, \(e\) has mean zero and is uncorrelated with \(M^*\), \(G\), the covariates, and the outcome disturbance. If the outcome regression adjusts for \(G\) and \(C\), the relevant attenuation factor is based on the variance of \(M^*\) remaining after those variables are controlled. The unconditional variance of \(M^*\) is generally not the correct quantity because some of it may be explained by genotype or covariates.
To see this, residualize the true mediator and measured mediator with respect to \(G,C\). Then \(W_r=M_r^*+e\). If \(Y_r=bM_r^*+U_r\) and \(M_r^*\) is unconfounded in this conditional regression, the coefficient using \(W\) is
This derivation uses the assumptions that \(e\) is independent of the residual true mediator and outcome disturbance and that the model includes the same adjustment variables. If assay error depends on genotype, outcome, or sample inclusion, this simple attenuation formula need not apply. Measurement error in \(M\) can also distort the estimated \(G\)-to-\(M\) path, especially when assay properties vary across genotype or covariate strata.
Worked Example: Conditional Attenuation From Assay Error
In hypothetical data, let \(M^*=0.40G+U\), with \(\operatorname{Var}(G)=1\), \(U\) independent of \(G\), and \(\operatorname{Var}(U)=0.84\). Thus the variance of the true mediator remaining after adjusting for \(G\) is \(0.84\). Suppose \(W=M^*+e\), where the independent assay error has variance \(0.16\), and the true conditional outcome coefficient is \(b=0.50\) outcome units per molecular-trait unit.
The conditional reliability ratio is
The expected coefficient from the regression of \(Y\) on \(G\) and \(W\), under the classical-error and no-confounding conditions, is therefore
The calculation uses the residual variance after conditioning on \(G\), not the unconditional variance of \(M^*\). The genotype-predicted component \(0.40G\) is not part of the mediator variation available to estimate the conditional mediator coefficient once \(G\) is included in the regression. The value \(0.420\) describes attenuation under the assumptions specified; it is not a general correction for arbitrary assay error.
Selection Into the Analyzed Sample
Mediation analyses often require a molecular assay, survival to a particular age, participation in a biobank, or availability of a complete outcome record. Let \(S=1\) indicate inclusion in the analysis. If inclusion depends on genotype, mediator, outcome, or their causes, conditioning on \(S=1\) can change associations. In causal terms, selection may act as a collider: two otherwise independent causes of inclusion can become associated among selected individuals. It can also change the distribution of confounders or remove the covariate overlap needed for positivity.
Worked Example: Selection Creates an Association
Let hypothetical binary variables \(G\) and \(Y\) be independent in a source population, with each taking 0 or 1 with probability \(1/2\). Suppose a study includes a person only when \(S=1\), where \(S=1\) if \(G=1\) or \(Y=1\). Before selection, \(P(Y=1\mid G=1)=P(Y=1\mid G=0)=1/2\), so \(G\) and \(Y\) are independent.
Among selected people, the possible \((G,Y)\) pairs are \((1,0),(0,1),(1,1)\). Each had source-population probability \(1/4\), so each has conditional probability \(1/3\) given selection. Consequently,
The selected sample therefore shows an association even though there was none in the source population. This simplified example is not a model of a particular study; it demonstrates why restricting analysis to participants with available molecular data or a particular clinical status can induce bias. In an actual analysis, investigators should describe the inclusion process, assess whether selection could depend on the relevant variables or their causes, and avoid assuming that a large selected sample removes selection bias.
Practical Checks Before Interpreting Mediation
The assumptions are partly scientific and partly statistical. Some can be investigated with design information and diagnostics, but no single observational diagnostic proves the required causal conditions. A disciplined interpretation asks what would have to be true for the estimated paths to correspond to the desired effects.
Specify the genotype contrast, the molecular trait, the outcome, and whether the target is a direct, indirect, or total effect. State the units of each path coefficient.
Consider population structure, relatedness, assortative mating, and other causes of genotype–outcome or genotype–mediator association. Explain why the genetic contrast is exchangeable after the chosen adjustment.
Identify plausible common causes of the molecular trait and outcome, distinguish pre-exposure confounders from variables affected by genotype, and justify adjustment without conditioning on an inappropriate post-exposure variable.
Describe assay reliability and whether error can depend on genotype or outcome. Describe who was eligible, measured, and analyzed, and consider how conditioning on inclusion could alter associations.
A common mistake is to reason, “The variant is associated with expression and disease, so expression mediates the genetic effect.” The correction is that two associations do not establish the causal mediator–outcome effect. A path product has a causal interpretation only under a defensible model and identification assumptions, including control of mediator–outcome confounding, appropriate measurement, and a selection process that does not induce bias. If these conditions are uncertain, report the result as a mediation-model decomposition or association rather than a demonstrated causal pathway.
Check Your Understanding
Use the assumptions and calculations above to answer the following questions.
- Why does a genotype–mediator association not, by itself, establish that the mediator coefficient in an outcome regression is causal?
- In the residual-confounding derivation, what condition makes the fitted conditional mediator coefficient equal to \(b\)?
- Why is the measurement-error attenuation factor based on mediator variance residualized on genotype and covariates rather than unconditional mediator variance?
- Give one example of how study inclusion could act as a collider and explain what conditioning on inclusion may do.
- Under what model conditions does the path product \(ab\) represent the indirect effect per unit of genotype exposure?