Path Decomposition in Genetic Mediation Models

Why Decompose a Genetic Effect?

A genetic variant may be associated with a downstream clinical trait through several routes. One proposed route is that genotype changes a molecular phenotype, which in turn changes the outcome. Other routes may connect genotype to the outcome without passing through that measured mediator. A path decomposition quantifies these routes within a specified model: it separates the indirect effect through the mediator from the direct effect not passing through it, and relates both to the total effect.

This decomposition is useful for making a model’s assumptions and scale explicit. It does not, by itself, establish that the mediator causes the outcome. In particular, a product of fitted regression coefficients can be a valid algebraic summary of regressions while still lacking a causal interpretation.

Definition: Let \(G\) denote a genetic variant, usually coded as allele dosage 0, 1, or 2; let \(M\) denote a molecular mediator; let \(Y\) denote a downstream molecular or clinical outcome; and let \(C\) denote measured covariates included in the model. In a linear mediation model, \(a\) is the path coefficient from \(G\) to \(M\), \(b\) is the path coefficient from \(M\) to \(Y\) conditional on \(G\) and \(C\), \(c'\) is the direct path coefficient from \(G\) to \(Y\) conditional on \(M\) and \(C\), and \(c\) is the total path coefficient from \(G\) to \(Y\) conditional on \(C\).

The Linear Path Model

A common model represents the mediator and outcome with two structural equations. Intercepts are included; covariates may have different coefficients in the two equations.

M=\alpha_M+aG+\boldsymbol{\gamma}_M^{T}C+\varepsilon_M

Y=\alpha_Y+c'G+bM+\boldsymbol{\gamma}_Y^{T}C+\varepsilon_Y

The coefficient \(a\) is the expected change in \(M\) per one-unit increase in \(G\), holding \(C\) fixed. The coefficient \(b\) is the expected change in \(Y\) per one-unit increase in \(M\), holding \(G\) and \(C\) fixed. The coefficient \(c'\) is the expected change in \(Y\) per one-unit increase in \(G\), holding \(M\) and \(C\) fixed. With dosage coding, one unit of \(G\) is one additional copy of the coded allele. These are slopes on the measurement scales used for \(M\) and \(Y\).

Substituting the mediator equation into the outcome equation gives the path decomposition. The resulting \(G\) coefficient consists of a direct route and a route through \(M\).

\text{Indirect effect}=ab,\qquad \text{Direct effect}=c',\qquad \text{Total effect}=c=c'+ab

The product \(ab\) is the indirect effect per one-unit increase in \(G\), on the scale of \(Y\). It multiplies the change in the mediator per dosage unit by the change in the outcome per mediator unit. The direct effect \(c'\) and indirect effect \(ab\) therefore have the same units: units of \(Y\) per dosage unit. For a contrast of \(\Delta G\) dosage units, each of these effects is multiplied by \(\Delta G\).

The notation describes a particular model, not a universal biological partition. If the mediator is transformed, measured with error, or replaced by another molecular trait, the coefficients and their interpretation can change. Likewise, a direct path means “not through this mediator in this model”; it does not identify a single biological mechanism.

Why the Decomposition Holds

The algebra is straightforward, but its conditions matter. In the structural model, substitute the equation for \(M\) into the equation for \(Y\):

\begin{aligned} Y &=\alpha_Y+c'G+b\left(\alpha_M+aG+\boldsymbol{\gamma}_M^{T}C+\varepsilon_M\right) +\boldsymbol{\gamma}_Y^{T}C+\varepsilon_Y\\ &=\left(\alpha_Y+b\alpha_M\right) +\left(c'+ab\right)G +\left(b\boldsymbol{\gamma}_M+\boldsymbol{\gamma}_Y\right)^{T}C +b\varepsilon_M+\varepsilon_Y. \end{aligned}

The coefficient multiplying \(G\) after substitution is \(c'+ab\). Thus the model’s total effect is the sum of the direct path and the product along the path \(G\rightarrow M\rightarrow Y\). This algebraic result requires the stated linear equations and a common additive scale, with no \(G\)-by-\(M\) interaction. For the total coefficient to have the intended conditional-mean interpretation, the combined error term must not have a systematic association with \(G\) after adjustment for \(C\).

There is also a regression version of the identity: when the mediator and outcome regressions use the same observations and the same covariate adjustment, the coefficient of \(G\) from the regression of \(Y\) on \(G\) and \(C\) equals the coefficient of \(G\) from the regression of \(Y\) on \(G\), \(M\), and \(C\), plus the product of the coefficient of \(G\) in the regression of \(M\) on \(G\) and \(C\) and the coefficient of \(M\) in the outcome regression. This is a linear-projection identity; it does not prove that any of these coefficients is causal.

Assumptions for a causal path interpretation: The equations must represent the relevant intervention contrasts: the genotype precedes the mediator, and the mediator precedes the outcome; the measured variables and their scales must be well defined; the relationships must be adequately represented by linear additive terms over the contrasts of interest; and there must be no relevant unmeasured confounding of the genotype–mediator, genotype–outcome, or mediator–outcome relationships after conditioning on \(C\). The interpretation also requires that no exposure-induced variable confounds the mediator–outcome relationship. The mediator and outcome models must use compatible covariate adjustment. These conditions are substantive assumptions, not consequences of observing a product \(ab\).

Genotype is determined before most measured molecular phenotypes, but that ordering alone does not guarantee the assumptions. For example, ancestry can be associated with both allele frequencies and outcomes, and a molecular mediator can share causes with the outcome. The decomposition gives a causal interpretation only when the model and the relevant confounding assumptions support it. Otherwise, describe the quantities as regression paths or associations.

Worked Example: Calculating All Three Paths

Suppose a hypothetical analysis codes \(G\) as dosage of a particular allele. The mediator is protein abundance in nanograms per milliliter, and the outcome is a clinical score in points. After adjustment for the same covariates, the fitted mediator model gives \(a=0.40\) ng/mL per allele, and the fitted outcome model gives \(b=0.30\) points per ng/mL and \(c'=0.05\) points per allele. Calculate the paths per additional allele.

Indirect path. The genotype is associated with a \(0.40\) ng/mL increase in protein abundance per allele; each ng/mL higher protein abundance corresponds to \(0.30\) more outcome points, conditional on genotype and covariates. Therefore,

ab=(0.40\ \text{ng/mL per allele})(0.30\ \text{points per ng/mL}) =0.12\ \text{points per allele}.

The ng/mL units cancel. The direct path is \(c'=0.05\) points per allele, and the total path is

c=c'+ab=0.05+0.12=0.17\ \text{points per allele}.

Under the model and causal assumptions stated above, the decomposition attributes \(0.12\) points per allele to the route through protein abundance and \(0.05\) points per allele to the remaining direct path. Without those assumptions, these are fitted regression-path summaries rather than established causal effects.

Worked Interpretations and Contrasts

A decomposition is easiest to interpret when the units are carried through every multiplication. It is also important to retain the signs of the paths: indirect and direct effects need not point in the same direction.

Worked Example: Opposing Direct and Indirect Paths

Consider a second hypothetical analysis. Let \(M\) be a metabolite concentration in micromoles per liter and \(Y\) be a clinical risk score in points. Suppose \(a=-0.25\) micromoles per liter per allele, \(b=0.80\) points per micromole per liter, and \(c'=0.30\) points per allele. Calculate and interpret the decomposition.

The indirect path is

ab=(-0.25)(0.80)=-0.20\ \text{points per allele}.

The direct path is \(0.30\) points per allele, so the total path is

c=c'+ab=0.30+(-0.20)=0.10\ \text{points per allele}.

The modeled direct path increases the risk score, while the modeled route through the metabolite decreases it. Their sum is a smaller positive total effect of \(0.10\) points per allele. Reporting only the total would hide the opposing paths. A “proportion mediated” would be especially hard to interpret here: the indirect path has the opposite sign from the total, so a simple ratio would not describe a share of the total effect. These numerical paths alone do not show that the metabolite causally offsets the direct path; that conclusion depends on the causal assumptions.

Worked Example: Scaling the Paths to a Two-Allele Contrast

Suppose a hypothetical expression analysis uses \(G=0,1,2\), with \(a=0.50\) expression units per allele, \(b=0.20\) outcome units per expression unit, and \(c'=0.04\) outcome units per allele. What does the decomposition imply for a contrast between dosage 0 and dosage 2?

First calculate the per-allele indirect effect:

ab=(0.50)(0.20)=0.10\ \text{outcome units per allele}.

The per-allele total effect is \(0.04+0.10=0.14\) outcome units per allele. The contrast from \(G=0\) to \(G=2\) is two dosage units, so each path is multiplied by 2:

\begin{aligned} \text{Indirect effect}_{0\to2}&=2(0.10)=0.20\ \text{outcome units},\\ \text{Direct effect}_{0\to2}&=2(0.04)=0.08\ \text{outcome units},\\ \text{Total effect}_{0\to2}&=2(0.14)=0.28\ \text{outcome units}. \end{aligned}

The result is a model-based contrast for two additional copies of the coded allele. It is not a comparison of two randomly assigned groups, and the causal reading still depends on the model and its assumptions. The linear dosage model also assumes that the per-allele slope is appropriate across the 0-to-2 range.

Common Mistakes and Practical Checks

Before interpreting a path decomposition, check the model’s scale, adjustment set, and intended interpretation. A numerically correct product can be misleading if its inputs do not correspond to the same model or if the conclusion exceeds what the design supports.

Calling \(ab\) causal just because it is a product. Multiplying two regression coefficients gives a path product, not automatic evidence that intervening on \(M\) would change \(Y\). State whether the result is an associational decomposition or a causal one supported by additional assumptions.
Mixing units or scales. If \(a\) is expressed per allele and \(b\) per mediator unit, their product is per allele on the outcome scale. Check that the mediator measurement and outcome scale used in both paths are the ones intended.
Ignoring sign cancellation. A small total effect can result from sizable direct and indirect paths with opposite signs. Do not infer that the separate routes are negligible from the total alone.
Using mismatched adjustments. If covariates, observations, or model specifications differ between the component regressions, the simple regression identity may not apply as stated. Report the models used to obtain \(a\), \(b\), and \(c'\).
Assuming additivity without checking the specification. A \(G\)-by-\(M\) interaction or nonlinear relationship generally prevents the single product \(ab\) and constant \(c'\) from representing the full decomposition across all values. The simple formula belongs to the linear additive model.

Key takeaway: In a linear additive mediation model, the indirect path is \(ab\), the direct path is \(c'\), and the total path is \(c=c'+ab\). The multiplication and addition are algebraic; interpreting these paths as causal requires substantive assumptions about the model, confounding, and the meaning of the mediator.

Check Your Understanding

Use the linear additive model and distinguish algebraic conclusions from causal ones.

If \(a=0.6\) mediator units per allele, \(b=-0.5\) outcome units per mediator unit, and \(c'=0.1\) outcome units per allele, calculate the indirect and total effects per allele. What are their signs?
Why does the product \(ab\) have units of outcome units per allele?
In a dosage model, how do the direct, indirect, and total contrasts change when the contrast is from \(G=0\) to \(G=2\), rather than per allele?
Does a valid algebraic identity \(c=c'+ab\) establish that changing the mediator would change the outcome? Explain.
Why can the total effect be small even when both the direct and indirect paths are appreciable?