Separating Effects of Correlated Molecular Mediators

Why Correlated Mediators Need to Be Separated Carefully

Expression, methylation, and protein abundance can vary together. A genetic variant may be associated with all three, while each molecular measurement is also associated with an outcome. The central question is whether an indirect effect can be attributed to one mediator after accounting for the others. A marginal association for each mediator does not answer that question: it can include variation shared with other mediators.

This tutorial considers parallel mediators: multiple molecular traits measured as potential intermediate variables, without specifying a causal arrow from one mediator to another. The aim is to distinguish their model-based, mediator-specific indirect effects and to state what those quantities do—and do not—establish causally.

Definition: Let \(X\) be an exposure, such as genetic-variant dosage; let \(M_1,\ldots,M_K\) be molecular mediators; let \(Y\) be an outcome; and let \(C\) be a set of measured covariates. In a parallel mediation model, each mediator is modelled as a function of \(X\) and \(C\), and the outcome is modelled as a function of \(X\), all mediators jointly, and \(C\). The specific indirect effect through \(M_j\) is the product of the \(X\)-to-\(M_j\) coefficient and the conditional \(M_j\)-to-\(Y\) coefficient.

The Parallel Linear Model and Its Interpretation

For two mediators, a common linear specification is

M_1=\alpha_1+a_1X+\boldsymbol{\gamma}_1^{\mathsf T}C+\varepsilon_1,\qquad M_2=\alpha_2+a_2X+\boldsymbol{\gamma}_2^{\mathsf T}C+\varepsilon_2,

and

Y=\beta_0+c'X+b_1M_1+b_2M_2+\boldsymbol{\delta}^{\mathsf T}C+\varepsilon_Y.

The mediator residuals \(\varepsilon_1\) and \(\varepsilon_2\) may be correlated. That correlation is one way the model accommodates shared unexplained variation between the mediators. In the outcome equation, \(b_1\) is the association between \(M_1\) and \(Y\) conditional on \(X\), \(M_2\), and \(C\); \(b_2\) is conditional on \(X\), \(M_1\), and \(C\). These are not the corresponding marginal associations.

IE_1=a_1b_1,\qquad IE_2=a_2b_2,\qquad IE_{\mathrm{total}}=a_1b_1+a_2b_2.

Here \(IE_j\) is the model-based specific indirect effect through mediator \(M_j\). In a linear model with no exposure-by-mediator interactions, the total effect coefficient \(c\) from a regression of \(Y\) on \(X\) and \(C\) satisfies

$$ c=c'+a_1b_1+a_2b_2. $$

This coefficient identity requires compatible linear models using the same observations and covariates. It is an algebraic decomposition of regression coefficients; by itself, it does not prove that the products are causal effects. A causal interpretation additionally requires, among other conditions, appropriate temporal ordering, no unmeasured confounding of the exposure–outcome, exposure–mediator, or mediator–outcome relationships, and no exposure-induced mediator–outcome confounder. The exposure and mediators must also be measured and modelled appropriately, with sufficient overlap in the covariate patterns being compared. These assumptions are substantive and are not established merely because the regressions fit well.

Worked Example: Calculating Two Specific Indirect Effects

Suppose a hypothetical analysis uses allele dosage as \(X\), expression in standard-deviation units as \(M_1\), methylation in standard-deviation units as \(M_2\), and an outcome measured in clinical score points as \(Y\). After adjustment for the same covariates, the fitted estimates are \(a_1=0.40\) expression SD per allele, \(a_2=0.30\) methylation SD per allele, \(b_1=0.50\) outcome points per expression SD, \(b_2=-0.20\) outcome points per methylation SD, and \(c'=0.36\) outcome points per allele.

The mediator-specific products are

\widehat{IE}_1=0.40\times0.50=0.20\ \text{outcome points per allele}, \qquad \widehat{IE}_2=0.30\times(-0.20)=-0.060\ \text{outcome points per allele}.

Thus the fitted indirect effect through expression is positive, while the fitted indirect effect through methylation is negative. Their sum is \(0.20-0.060=0.14\) outcome points per allele. If the compatible total-effect estimate is \(c=0.50\) outcome points per allele, the coefficient identity is satisfied: \(0.36+0.14=0.50\).

The mediator coefficients in the outcome model must be joint, conditional estimates for these products. If the mediators are correlated, substituting a marginal association of \(Y\) with expression for \(b_1\) would generally calculate a different quantity. The units also follow from multiplication: allele-to-mediator units times outcome-per-mediator units give outcome units per allele.

For a causal reading, the analysis would need to justify the stated confounding and ordering assumptions, use compatible covariate adjustment in each equation, and assess whether a linear, no-interaction specification is reasonable. A fitted model and a coefficient identity do not verify those causal conditions.

Why Joint Mediator Coefficients Differ From Marginal Ones

A simple standardized calculation shows how correlation changes the interpretation of the outcome coefficients. Suppose the variables have first been adjusted for \(X\) and \(C\), and are standardized. Write \(r_{12}\) for the correlation between the residualized mediators, and \(r_{Y1}\) and \(r_{Y2}\) for their correlations with the residualized outcome. The two-predictor least-squares equations are

\begin{pmatrix}1&r_{12}\\r_{12}&1\end{pmatrix} \begin{pmatrix}b_1\\b_2\end{pmatrix} = \begin{pmatrix}r_{Y1}\\r_{Y2}\end{pmatrix}.

Solving these equations, provided the mediators are not perfectly correlated, gives

b_1=\frac{r_{Y1}-r_{12}r_{Y2}}{1-r_{12}^2}, \qquad b_2=\frac{r_{Y2}-r_{12}r_{Y1}}{1-r_{12}^2}.

The numerator subtracts the part of the outcome association that is shared with the other mediator; the denominator accounts for the mediators’ shared variation. When \(|r_{12}|\) is close to one, the denominator is small, so the separate coefficient estimates can be unstable. Correlation does not make a joint regression impossible, but it can make the apparent separation imprecise and sensitive to modelling choices.

Worked Example: Obtaining Conditional Effects From Correlations

Consider hypothetical standardized residual correlations after accounting for \(X\) and \(C\): \(r_{12}=0.80\), \(r_{Y1}=0.70\), and \(r_{Y2}=0.65\). The joint coefficients are

b_1=\frac{0.70-(0.80)(0.65)}{1-(0.80)^2} =\frac{0.18}{0.36}=0.50, \qquad b_2=\frac{0.65-(0.80)(0.70)}{0.36} =\frac{0.09}{0.36}=0.25.

The marginal residual correlations with the outcome are 0.70 and 0.65, but the conditional standardized regression coefficients are 0.50 and 0.25. They answer different questions: the correlations describe separate pairwise relationships, while the joint coefficients describe each mediator’s association with the outcome after accounting for the other mediator.

If the fitted exposure-to-mediator coefficients were \(a_1=0.30\) and \(a_2=0.20\), in standardized mediator units per allele, the corresponding products would be \(0.30\times0.50=0.15\) and \(0.20\times0.25=0.050\) standardized outcome units per allele. These products are conditional-model estimates, not proof that intervening on either mediator would change the outcome by that amount.

The calculation assumes that the correlations refer to the same adjusted, standardized variables and that the mediator correlation is not exactly one. The high correlation of 0.80 also warrants attention to coefficient uncertainty: the point estimates separate the mediators algebraically, but do not make their separate contributions precisely known.

Shared Causes Can Produce Apparent Mediator Effects

A shared cause of a mediator and the outcome can make their conditional association look like a mediator effect even when changing the mediator would not change the outcome. Correlated mediators do not, by themselves, reveal whether their shared variation comes from a common cause, a causal relationship between them, or other sources. Including both mediators in the outcome regression is not a substitute for measuring and appropriately handling mediator–outcome confounders.

Worked Example: A Shared Cause Creates Misleading Products

Suppose hypothetical data follow the equations \(M_1=0.40X+U+e_1\), \(M_2=0.30X+U+e_2\), and \(Y=0.50X+0.60U+e_Y\). The unmeasured shared cause \(U\) has variance 1; the independent errors \(e_1\), \(e_2\), and \(e_Y\) each have variance 1; and \(X\) is independent of \(U\) and the errors. Crucially, neither mediator appears in the equation for \(Y\), so there is no causal effect of either mediator on \(Y\) in this data-generating model.

After accounting for \(X\), the residual variances of \(M_1\) and \(M_2\) are each \(1+1=2\), and their residual covariance is \(\operatorname{Var}(U)=1\). Each mediator’s residual covariance with \(Y\) is \(0.60\). The joint regression coefficients therefore solve

\begin{pmatrix}2&1\\1&2\end{pmatrix} \begin{pmatrix}b_1\\b_2\end{pmatrix} = \begin{pmatrix}0.60\\0.60\end{pmatrix}, \qquad \begin{pmatrix}b_1\\b_2\end{pmatrix} = \frac{1}{3} \begin{pmatrix}2&-1\\-1&2\end{pmatrix} \begin{pmatrix}0.60\\0.60\end{pmatrix} = \begin{pmatrix}0.20\\0.20\end{pmatrix}.

The fitted mediator model has \(a_1=0.40\) and \(a_2=0.30\), so the products are \(0.40\times0.20=0.080\) and \(0.30\times0.20=0.060\) outcome units per allele. Both products appear positive, despite the absence of causal mediator effects in the specified data-generating equations. The source is the unadjusted common cause \(U\), which induces mediator–outcome associations.

In an actual analysis, a measured common cause may be included in the covariate set if the causal structure and timing justify that adjustment. An unmeasured cause cannot be removed by adding both mediators to the same regression. The example also shows why the coefficient identity is not a causal test: its regression decomposition can hold even when the product terms have a confounded interpretation.

Mediator Order Changes the Question

Calling mediators “parallel” is a modelling choice, not a finding that they are biologically independent or unordered. If expression changes protein abundance, for example, a model that treats expression and protein as parallel mediators does not represent that causal ordering. Its coefficient for expression, conditional on protein, excludes the association carried through variation shared with protein; it is not automatically the full effect of expression along every downstream route.

For that reason, mediator-specific products from a parallel model should be reported as conditional on the included mediators and the assumed structure. If a causal arrow between mediators is plausible, the analyst must use a model that represents that ordering and define the target effects accordingly. Reordering variables in a regression, or describing one mediator as “upstream” based only on its coefficient, does not establish a causal sequence.

Specify the target structure.
State whether the molecular traits are being treated as parallel mediators or whether a causal order between them is assumed.

Fit compatible equations.
Use the same observations and covariate set for the mediator and outcome models, and include all parallel mediators jointly in the outcome equation.

Calculate and qualify the products.
Multiply each exposure-to-mediator estimate by that mediator’s conditional outcome coefficient. Report the units, the conditioning set, and the causal assumptions needed for a causal interpretation.

Common Mistake and Practical Interpretation

Wrong reasoning: “Expression and protein are correlated, so the larger of their two marginal associations with the outcome identifies the mediator responsible for the genetic effect.” This compares associations that each contain shared variation and does not separate mediator-specific effects.

Correction: In a parallel linear model, estimate each mediator’s exposure coefficient and its outcome coefficient conditional on the other mediators, then form the corresponding product. Even then, the products are causal only under the required identification assumptions and the assumed mediator structure. Strong correlation can make the separate estimates unstable; a measured shared cause, an unmeasured shared cause, or a genuine mediator-to-mediator relationship can also change what the estimates mean. Report these limitations rather than treating a fitted decomposition as proof of mechanism.

Key takeaway: Correlated molecular mediators can be separated statistically by using their joint conditional outcome coefficients, but the resulting products are specific to the chosen model and covariates. Correlation alone neither establishes a causal ordering nor resolves shared-cause confounding.

Check Your Understanding

Use the parallel-mediator definitions and assumptions above to answer these questions.

In a joint outcome model, what does the coefficient \(b_1\) describe that a marginal association between \(M_1\) and \(Y\) does not?
If \(a_1=0.25\) mediator units per allele and \(b_1=0.40\) outcome units per mediator unit, calculate and state the units of the specific indirect effect through \(M_1\).
Why can a high correlation between two mediators make their conditional outcome coefficients unstable?
How can a shared unmeasured cause of a mediator and the outcome produce a nonzero product even when the mediator has no causal effect on the outcome?
What changes in interpretation if one molecular mediator is assumed to cause another?