From a Molecular Association to a Mediation Estimate
A genetic variant associated with both gene expression and a disease-related outcome can suggest a molecular pathway, but two associations alone do not establish that expression mediates the genetic effect. A worked analysis therefore needs to keep several questions distinct: are the genetic effects aligned to the same allele, do the molecular and outcome signals plausibly share a causal variant, how large is the estimated indirect effect, and which assumptions are still needed for a causal interpretation?
The analysis below uses a hypothetical variant \(G\), expression \(M\) of one gene, and a continuous disease-related outcome \(Y\). The outcome is a standardized disease-relevant score, not a binary diagnosis. Effects are expressed per additional copy of the designated effect allele. The numerical results illustrate the calculations; they are not from a real study.
The identity \(\gamma=\alpha\beta+\delta\) is a model relationship, not an automatic property of any three association estimates. It requires a compatible linear model with no exposure–mediator interaction, and estimates that target the same population and scale. In a genetic analysis, causal interpretation also requires valid genetic instruments for the effects being estimated. In particular, an instrument for \(M\) must be relevant to \(M\), independent of confounders of \(M\) and \(Y\), and affect \(Y\) only through \(M\) under the instrumental-variable model. The focal variant may itself have a direct effect \(\delta\); mediation does not require that effect to be zero.
First Check: Are the Signals Compatible?
Before combining component estimates, harmonize their allele definitions. Each effect must refer to the same effect allele, with the same direction of effect, and the same variant representation. Otherwise a sign reversal can make an apparent positive indirect effect negative, or vice versa. Also record the phenotype scales: an expression effect per allele cannot be multiplied meaningfully by an outcome effect unless the mediator–outcome estimate is per corresponding expression unit.
A second check is whether the expression and outcome associations are consistent with a shared causal variant in the region. Colocalization compares hypotheses about the underlying signals, including \(H_3\), in which distinct causal variants affect the two traits, and \(H_4\), in which one shared causal variant affects both. Posterior support for \(H_4\) makes a shared signal more plausible under the colocalization model. It does not show that the gene is the causal mediator, establish the direction of mediation, or rule out a direct variant effect on the outcome.
Worked Example: Reading Shared-Variant Evidence
Suppose the effect allele of a hypothetical variant increases gene expression by an estimated \(0.20\) expression standard deviations per allele, with \(SE=0.05\). In a separate outcome dataset, the same allele increases the standardized disease-related score by \(0.10\) outcome standard deviations per allele, with \(SE=0.04\). Assume the alleles and variant representation have been harmonized.
The association statistics are \(0.20/0.05=4.00\) for expression and \(0.10/0.04=2.50\) for the outcome. These indicate evidence of association for both traits, but they do not by themselves tell us whether the associations arise from the same causal variant.
Suppose a regional colocalization analysis gives posterior probabilities \(0.01\) for \(H_0\) (neither trait associated), \(0.01\) for \(H_1\) (expression only), \(0.02\) for \(H_2\) (outcome only), \(0.04\) for \(H_3\) (distinct causal variants), and \(0.92\) for \(H_4\) (a shared causal variant). They sum to \(1.00\). The posterior probability assigned to the shared-variant hypothesis is \(0.92\), compared with \(0.04\) for distinct variants. This is strong shared-signal evidence under the specified model and priors.
The result supports taking the expression signal seriously as a candidate part of the pathway. It does not turn the \(0.10\) outcome association into a mediated effect: the variant could affect the outcome through another route, and the gene could be correlated with the actual mediator. Colocalization is evidence about shared genetic signal, not a mediation test.
Estimating the Indirect Effect
For a product estimate, \(\alpha\) is the effect of the focal variant on the mediator, and \(\beta\) is an estimate of the causal effect of the mediator on the outcome. A two-step genetic analysis may estimate the latter using genetic instruments for the mediator. For this calculation to target mediation by the specified expression phenotype, the effect scales, allele directions, and target population must be compatible. The analysis also relies on the instrument assumptions stated above and on the absence of exposure–mediator interaction in the model.
Worked Example: Calculating the Product Estimate
Continue with the hypothetical focal-variant estimate \(\hat{\alpha}=0.20\) expression standard deviations per allele. Suppose two other, independent genetic instruments for expression give mediator–outcome ratio estimates of \(0.28\) and \(0.32\) outcome standard deviations per expression standard deviation, each with standard error \(0.12\). Assume for this illustration that both instruments satisfy the required relevance, independence, and exclusion assumptions.
With equal standard errors, the inverse-variance weighted estimate is the mean of the two ratio estimates:
Its standard error is \(0.12/\sqrt{2}=0.0849\), approximately \(0.085\), under the assumed independence and inverse-variance weighting. The estimated indirect effect per allele of the focal variant is therefore
Thus, under the linear mediation model and its causal assumptions, the estimated indirect effect is \(0.060\) outcome standard deviations per additional copy of the focal effect allele. It is not a \(0.060\)-unit change in gene expression: the expression unit has been multiplied by the mediator–outcome effect, leaving outcome units per allele.
Quantifying Uncertainty in the Product
Uncertainty in both component estimates contributes to uncertainty in their product. A first-order delta-method approximation follows by expanding \(g(\alpha,\beta)=\alpha\beta\) around the estimated values. The gradient is \((\beta,\alpha)\), so the approximate variance is the gradient multiplied by the covariance matrix of the component estimates and by the gradient transpose. If the estimates are independent, their covariance is zero.
The independence condition matters: if component estimates share participants or data, the covariance term may not be zero and should not be silently discarded. Even with independent estimates, a normal confidence interval for the product is an approximation; the product of uncertain estimates need not have a normal sampling distribution, especially when a component estimate is imprecise.
Worked Example: Delta-Method Interval and Sensitivity
Use \(\hat{\alpha}=0.20\), \(SE(\hat{\alpha})=0.05\), \(\hat{\beta}=0.30\), and \(SE(\hat{\beta})=0.085\). Assume the estimates are independent because they come from non-overlapping samples. Then \(\operatorname{Var}(\hat{\alpha})=0.05^2=0.0025\) and \(\operatorname{Var}(\hat{\beta})=0.085^2=0.007225\). The approximate variance of the product is
Therefore, \(SE(\widehat{\alpha\beta})\approx\sqrt{0.000514}=0.0227\). A normal-approximation 95% confidence interval is
Under the assumptions of this analysis, the interval estimates the indirect effect at about \(0.0155\) to \(0.1045\) outcome standard deviations per allele. Its repeated-sampling interpretation is that intervals constructed this way would cover the true indirect effect about 95% of the time under the model and approximation; it is not a 95% probability statement about this particular interval containing the parameter.
As a simple sensitivity check, if the mediator–outcome effect were \(0.20\) rather than \(0.30\), the product would be \((0.20)(0.20)=0.040\). If it were \(0.40\), the product would be \((0.20)(0.40)=0.080\). This range shows how much the indirect-effect estimate depends on the mediator–outcome component; it does not replace an uncertainty analysis or assess whether the instruments are valid.
Relating the Product to the Total Effect
The total effect of the focal variant on the outcome provides a useful coherence check when it is estimated on the same outcome scale and in a compatible population. For example, suppose the total-effect estimate is \(\hat{\gamma}=0.10\) outcome standard deviations per allele. The model-based direct-effect estimate obtained by subtraction is \(\hat{\delta}=\hat{\gamma}-\hat{\alpha}\hat{\beta}=0.10-0.06=0.04\) outcome standard deviations per allele. This subtraction describes a direct effect only if the linear decomposition and compatible-estimand assumptions hold; it is not an independent confirmation of the mediation model.
A confidence interval for this difference cannot be obtained by subtracting the component standard errors or by treating the total and indirect estimates as independent without justification. Their covariance may matter, particularly when estimates share data. A formal uncertainty estimate for the direct effect must account for that covariance and for uncertainty in all of its components. Likewise, a small or imprecise total effect does not by itself rule out an indirect effect: direct and indirect effects can differ in sign and partially cancel.
Confirm that effect alleles and variant definitions match, and that the mediator and outcome effects use compatible units and target populations.
Interpret colocalization as evidence about whether traits share a causal variant, not as proof of a causal mediation pathway.
Obtain the variant–mediator effect and a valid mediator–outcome causal estimate, then multiply them on the specified scale.
Report component uncertainty and the assumptions needed for causal interpretation, including any limits of the instrument and mediation model.
What the Analysis Does Not Establish
Even strong shared-variant evidence and a nonzero product estimate leave important causal uncertainties. The variant might affect the outcome through a pathway other than the measured expression phenotype. The expression measurement may not represent the relevant tissue, cell type, or biological time. Genetic instruments for expression can have horizontal pleiotropic effects, and mediator–outcome confounding or selection can distort causal interpretation. Earlier material on horizontal pleiotropy, sample selection, missing mediator or outcome data, and mediator–outcome confounding gives specific reasons to examine these possibilities.
The mediation interpretation also depends on the specified model. With exposure–mediator interaction, the indirect effect need not be represented by one constant product. With a binary outcome, effect scales and noncollapsibility complicate simple decomposition. The present calculations deliberately use a continuous outcome and a linear, additive model so that the product and sum relationships are clear; they should not be carried over to those settings without a suitable model.
Check Your Understanding
Use the hypothetical analysis above to check both the calculations and their interpretation.
- What are the units of \(\alpha\), \(\beta\), and \(\alpha\beta\) in this example?
- Why does a posterior probability of \(0.92\) for \(H_4\) not by itself establish that expression mediates the variant’s effect on the outcome?
- If \(\hat{\alpha}=0.20\) and a new mediator–outcome estimate is \(\hat{\beta}=0.25\), what is the estimated indirect effect, in context?
- What additional term belongs in the delta-method variance if the component estimates are correlated?
- Under what model relationship can the direct effect be estimated by subtracting the indirect effect from the total effect, and why does that subtraction not automatically provide its standard error?