Natural Direct / Indirect Effect for continuous treatment - Causal Mediation

Question

I'm reading in the topic of causal mediation and would like to apply it to a case with all continuous variables, treatment mediator interaction, and assuming linearity. I'd like to calculate the Natural Indirect (NIE) and Natural Direct Effect (NDE). Here is my current understanding from what I read here. Having the following models with Treatment T, Mediator M, target Y and controls Cm and Cy as shown in the following DAG:

mediation model: $M = a_0 + a_1 * T + a_2 * C_m + e_m$

outcome model: $Y = x_0 + x_1 * T + x_2 * M + x_3 * T * M + x_4 * C_y + e_y$

we can estimate the 2 effects the following way:

$NDE = (x_1 + x_3 * (a_0 + a_1 * T^* + a_2 * C_m)) * (T-T^*)$

$NIE = (a_1 * (x_2 + x_3 * T)) * (T-T^*)$

What I don't really understand is $T$ and $T^*$ in the continuous case. My approach would be to loop through a reasonable range of values of $T$ and and have $T = T^* + 1$ and than plot the results as a line plot as the range of effects depending of the treatment level. So the lines would basically have the following lines:

$NDE = x_1 + x_3 * (a_0 + a_1 * T + a_2 * C_m)$

$NIE = a_1 * (x_2 + x_3 * (T+1))$

Does that make sense or am I missing a specific way to cope with continuous treatment?

Answer 1

In classical mediation analysis, effects are typically defined with respect to two different exposure levels, where one serves as a reference value. This means that changes in the outcome are measured relative to this reference. For instance, the natural direct effect (NDE) of exposure level $X = x$ compared to a reference level $X = x_0$ on outcome $Y$, through mediator $Z$, is formally given by:

$$ \text{NDE}(x,x_0) := \mathbb{E}\left[Y^{x,Z^{x_0}}\right]-\mathbb{E}\left[Y^{x_0} \right]. $$

This definition remains valid for continuous exposures, where the NDE is a function of two exposure levels. Provided certain assumptions are met —such as sequential ignorability and positivity— the NDE is identifiable. In your graphical scenario, these assumptions hold with the adjustment set $W = {C_M, C_Y}$, so the NDE can be expressed as:

$$ \text{NDE}(x,x_0) = [\chi_1+\chi_3(\alpha_0+\alpha_1x_0+\alpha_2\bar{C}_M)]\cdot (x-x_0), $$

where $\bar{C}_M$ denotes the mean of ${C}_M$.

Alternatively, for continuous exposures, you can parameterize the NDE in terms of a reference level $x_0$ and a small increase $\delta > 0$ from it, yielding:

$$ \text{NDE}(x_0+\delta,x_0) = [\chi_1+\chi_3(\alpha_0+\alpha_1x_0+\alpha_2\bar{C}_M)]\cdot \delta. $$

You can set $\delta=1$ if that makes sense for the application case. Moreover, if zero is within the support of the exposure, and it's a meaningful reference point, you can consider the NDE of a small increase $\delta > 0$ in exposure from zero:

$$ \text{NDE}(\delta,0) = [\chi_1+\chi_3(\alpha_0+\alpha_2\bar{C}_M)]\cdot \delta. $$

Now, to analyze infinitesimal changes in exposure, you can define the continuous analog of the NDE: the natural direct derivative (NDD). This can be set to be a function of the reference level alone, as:

$$ \text{NDD}(x_0) := \frac{\partial}{\partial x}\mathbb{E}\left[Y^{x,Z^{x_0}}\right]\Big\vert_{x=x_0}, $$

which, in your graphical setting, is identified as:

$$ \text{NDD}(x_0) = \chi_1+\chi_3(\alpha_0+\alpha_1x_0+\alpha_2\bar{C}_M). $$

Some references:

• Judea Pearl (2001). Direct and Indirect Effects

• Judea Pearl (2014). Interpretation and Identification of Causal Mediation

• George Knafl et al (2017). Incorporating nonlinearity into mediation analyses