Why Regression Coefficients Need a Causal Translation
Mediation analyses often fit one regression for the mediator and another for the outcome. The coefficients are easy to calculate and familiar to interpret, but they are not automatically natural direct or indirect effects. A coefficient describes a feature of a specified statistical model; a natural effect compares outcomes under particular combinations of treatment and mediator counterfactuals. Moving from one to the other requires compatible scales, model structure, and causal assumptions.
This distinction matters even when both regressions fit the observed data well. Confounding can make a mediator coefficient differ from the causal effect of changing the mediator. Treatment–mediator interaction can make a treatment coefficient depend on the mediator value at which it is evaluated. With a nonlinear outcome model, coefficients may be on a scale such as log odds, while natural effects are often reported as risk differences. The following examples isolate these reasons for mismatch.
The two effects add to the total effect on the mean scale: \(E[Y(1,M(1))-Y(0,M(0))]\). This follows by adding and subtracting \(E[Y(1,M(0))]\). Identification from observed data requires the relevant assumptions for natural effects, including appropriate control of treatment and mediator–outcome confounding, positivity, consistency, and the cross-world condition described in the earlier material on natural effects. A regression alone does not establish these conditions.
Confounding Changes the Meaning of a Mediator Slope
Consider a linear outcome model with no treatment–mediator interaction. If all relevant confounders are measured and correctly adjusted for, its mediator coefficient may represent a causal effect per unit of mediator under additional assumptions. But if a cause of both the mediator and outcome is omitted, the fitted mediator slope generally combines the mediator’s causal effect with the association induced by that common cause. The next example calculates this distinction directly.
Worked Example: A Confounded Mediator–Outcome Regression
Suppose a hypothetical population has binary randomized treatment \(A\), and a binary baseline factor \(C\) independent of \(A\). Let \(C\) have mean 0 and variance 1 for this illustrative linear model, and let an independent mediator disturbance \(\varepsilon\) have mean 0 and variance 1. The data-generating equations are \(M=A+C+\varepsilon\) and \(Y=2A+3M+4C+\eta\), where \(\eta\) has mean 0 and is independent of \(A,C,\varepsilon\). The units of \(Y\) are outcome points. Thus \(C\) causes both \(M\) and \(Y\), but the regressions below omit \(C\).
Under an intervention on \(A\), the mediator changes by one unit: \(M(1)-M(0)=1\). Holding the mediator fixed, the outcome changes by two points, so the natural direct effect is 2 points. The mediator’s structural effect on the outcome is 3 points per unit; because treatment shifts the mediator by one unit, the natural indirect effect is \(3\times1=3\) points. These values follow from the stated structural equations, not from the omitted-confounder regressions.
The mediator regression \(M\) on \(A\) has slope 1, because \(A\) is independent of \(C+\varepsilon\). Now fit the outcome regression on \(A\) and \(M\), omitting \(C\). Write \(R=C+\varepsilon\), so \(M=A+R\), and rewrite the outcome as \(Y=5A+3R+4C+\eta\). Since \(\operatorname{Var}(R)=2\) and \(\operatorname{Cov}(R,C)=1\), the population regression slope of \(Y\) on \(R\), after accounting for \(A\), is
The regressors \(A,M\) span the same linear space as \(A,R\), since \(R=M-A\). In the regression on \(A,R\), the coefficient of \(A\) is 5 and the coefficient of \(R\) is 5. Substituting \(R=M-A\) gives the equivalent fitted expression \(5A+5(M-A)=5M\). Therefore the regression of \(Y\) on \(A,M\) has treatment coefficient 0 and mediator coefficient 5. The mediator coefficient is not the causal effect of 3 points per unit, and its product with the mediator-regression slope, \(5\times1=5\) points, is not the natural indirect effect of 3 points. In this construction the treatment coefficient also differs from the natural direct effect of 2 points.
The mismatch arises because conditioning on \(A\) does not remove the unmeasured common cause \(C\) from the mediator–outcome relationship. Here the regression slopes are calculable, but they do not recover the causal contrasts. In an actual analysis, if \(C\) were measured and the causal assumptions held, it would need to be handled appropriately rather than simply omitted.
Interaction Makes a Treatment Coefficient Depend on the Mediator
When the outcome model includes an \(A\times M\) term, there is no single treatment effect coefficient that applies at every mediator value. The coefficient of \(A\) in that model is the treatment contrast at \(M=0\); at another mediator value, the contrast includes the interaction term. Natural effects average contrasts over mediator counterfactuals, so the coefficient at zero is generally not the natural direct effect.
Worked Example: Treatment–Mediator Interaction
In a hypothetical setting with no confounding, suppose \(M(a)=U+2a\), where \(U\) is independent of treatment and has mean 1. Suppose the outcome mean under interventions is \(E[Y(a,m)]=10+2a+3m+4am\), measured in outcome points. These equations specify a causal model with treatment–mediator interaction; assume the required interventions and counterfactuals are well-defined.
The population mean natural direct contrast, under the stated causal assumptions, is \(E[Y(1,U)-Y(0,U)]=2+4E[U]\). Averaging over \(U\), the natural direct effect is \(2+4E[U]=2+4(1)=6\) points. For the natural indirect effect, treatment is held at 1 while the mediator changes from \(M(0)=U\) to \(M(1)=U+2\). The outcome contrast is \((3+4)(2)=14\) points. The total effect is therefore \(6+14=20\) points.
The outcome regression has treatment coefficient 2, mediator coefficient 3, and interaction coefficient 4. The treatment coefficient 2 is the treatment contrast specifically at \(M=0\), not the average natural direct effect of 6. The familiar product of the mediator-regression slope and the mediator coefficient is \(2\times3=6\), not the natural indirect effect of 14. In fact, the outcome contrast for a one-unit mediator change under treatment 1 is \(3+4=7\) points, and the mediator changes by two units; hence the indirect contrast is \(7\times2=14\) points. Ignoring interaction discards the part of the mediator effect that depends on treatment.
A useful algebraic check makes the issue explicit. Under the outcome model \(E[Y(a,m)]=\beta_0+\beta_Aa+\beta_Mm+\beta_{AM}am\), the treatment contrast at mediator value \(m\) is \(\beta_A+\beta_{AM}m\), not simply \(\beta_A\) unless \(\beta_{AM}=0\). Likewise, the mediator contrast between \(m_1\) and \(m_0\) at treatment \(a\) is \((\beta_M+\beta_{AM}a)(m_1-m_0)\). Natural effects average these contrasts over the relevant counterfactual mediator values. The regression coefficients alone do not perform that averaging.
Nonlinear Models Put Coefficients on Different Scales
In a logistic outcome regression, coefficients describe changes in conditional log odds, not changes in risk. Natural effects may instead be requested as differences in mean outcomes, such as risk differences. Even with no confounding and a correctly specified logistic model, the coefficient values cannot generally be read off as natural effects on the risk-difference scale. The following small hypothetical calculation illustrates the scale mismatch; its purpose is to compare interpretations, not to prescribe a general estimation procedure.
Worked Example: Logistic Coefficients and Risk-Difference Effects
Suppose a hypothetical randomized study has a binary mediator with \(P(M(0)=1)=0.20\) and \(P(M(1)=1)=0.60\). Assume no confounding of the mediator–outcome relation, no exposure-induced mediator–outcome confounder, and the cross-world and other identification conditions needed for natural effects. Let the conditional outcome risk follow the correctly specified model \(P(Y=1\mid A=a,M=m)=\operatorname{expit}(-2+0.5a+1.2m)\), where \(\operatorname{expit}(x)=1/(1+e^{-x})\).
The treatment coefficient is 0.5 log-odds units, and the mediator coefficient is 1.2 log-odds units. The four model-implied risks are \(P(Y=1\mid 0,0)=0.1192\), \(P(Y=1\mid 1,0)=0.1824\), \(P(Y=1\mid 0,1)=0.3100\), and \(P(Y=1\mid 1,1)=0.4256\). For the natural direct effect on the risk-difference scale, average the treatment contrast over the mediator distribution under \(A=0\):
Thus the natural direct effect is approximately 0.0737, or 7.37 percentage points. For the natural indirect effect, use the mediator distributions under treatment 1 and treatment 0, while holding treatment at 1. The risk difference between mediator values 1 and 0 at \(A=1\) is \(0.4255575-0.1824255=0.2431320\), and the probability of mediator value 1 increases by \(0.60-0.20=0.40\). Therefore the natural indirect effect is \(0.40(0.2431320)=0.0972528\), or about 9.73 percentage points.
Neither logistic coefficient equals either risk-difference natural effect: the coefficients are on the log-odds scale, while the effects just calculated are in probability points. Even comparing on the log-odds scale would not make a coefficient a natural effect automatically, because the latter averages outcome contrasts over a counterfactual mediator distribution. Here the total risk difference is \(0.3283047-0.1573674=0.1709373\), which agrees with \(0.0736845+0.0972528=0.1709373\) up to rounding. That decomposition does not turn the regression coefficients into the component effects.
When Can a Coefficient Be Useful?
Regression remains useful for describing conditional outcome patterns, modeling mediator distributions, and estimating causal quantities when the model and identification assumptions are appropriate. Under restrictive conditions, coefficients can coincide with particular causal effects. For example, in a correctly specified linear outcome model with no treatment–mediator interaction, the treatment coefficient may represent a controlled direct effect, and under suitable identification assumptions it can also correspond to the natural direct effect on the additive scale. Such coincidences depend on the model, scale, and assumptions; they are not a general property of mediation regressions.
Before assigning a causal interpretation to a coefficient, specify the target contrast and its scale, write the outcome model including any interaction, and ask whether common causes of the mediator and outcome have been controlled. Then check whether the coefficient is conditional at a fixed mediator value or averages over a counterfactual mediator distribution. If the requested effect is a natural effect, the cross-world identification assumptions also matter. A model can fit well and still target the wrong quantity.
Common Mistakes
- Using the product-of-coefficients rule without checking assumptions. Multiplying the treatment-to-mediator slope by the mediator coefficient can fail with mediator–outcome confounding, treatment–mediator interaction, or a nonlinear outcome model.
- Calling the treatment coefficient the natural direct effect. With interaction it is the conditional treatment contrast at the model’s reference mediator value; without interaction it still requires causal identification assumptions.
- Comparing coefficients and effects with incompatible units. A logistic coefficient is in log-odds units; a risk-difference effect is in probability points. Numerical comparison does not make them the same estimand.
- Treating adjustment as proof of no confounding. Including measured covariates does not control unmeasured common causes or establish the cross-world assumptions required for natural effects.
Check Your Understanding
Use the model structure and target effect, not the coefficient labels alone, to answer these questions.
- In the confounding example, what causes the outcome-regression mediator coefficient to differ from the structural mediator effect?
- For a model containing \(A\times M\), what treatment contrast does the coefficient of \(A\) represent?
- In the interaction example, why does multiplying the mediator-regression slope by the mediator coefficient omit part of the natural indirect effect?
- Why can a logistic regression coefficient not be interpreted directly as a natural effect on the risk-difference scale?
- Name two causal assumptions or modeling features that must be checked before interpreting regression coefficients as natural effects.