Identification of causal direct-indirect effects
without untestable assumptions

In recent years, there has been considerable methodological development and applied studies based on potential outcome approaches (Rubin, 1974) to causal mediation analysis to understand causal mechanisms (for example, Pearl 2001; Van der Weele 2009; Imai et al. 2010a; Tchetgen Tchetgen & Shpitser 2012; Ding & Van der Weele 2016; Miles et al. 2020). Let $T$ denote the exposure or treatment of interest, $Y$ the outcome, $M$ the mediator, and the baseline covariates $v$, which are not affected by the exposure and mediator. Following the potential outcome approach, let $Y_{j(m)}$ be the potential outcome when $T=j$ and $M=m$.

Most recent studies in causal mediation analysis consider the natural direct/indirect effects, which are defined using the expectation of the “never-observed” outcome (not “potential outcome”) $Y_{j}(M_{k})\ \ (j\neq k)$, which is the potential outcome under treatment $j$ for the potential mediator for treatment $k$, $M_{k}$. To identify these effects various assumptions are proposed. The following assumptions (Pearl, 2001) are often made:

Assumption 1

$Y_{j}(k)\perp\!\!\!\perp T|v,\ \forall j,k,$

Assumption 2

$M_{j}\perp\!\!\!\perp T|v,\ \forall j,$

Assumption 3

$Y_{j}(k)\perp\!\!\!\perp M|T,\ v,\ \forall j,k,$

Assumption 4

$Y_{j}(k)\perp\!\!\!\perp M_{j^{*}}|v,\ \forall j,j^{*},k,$

Another example for sufficient conditions for identifying the two natural effects includes the following sequential ignorability conditions (SI1 and SI2) by Imai et al. (2010b):

Assumption SI1

$\{Y_{j}(k),M_{j^{*}}\}\perp\!\!\!\perp T|v,\ \forall j,j^{*},k,$

Assumption SI2

$Y_{j}(k)\perp\!\!\!\perp M_{j^{*}}|T=j^{*},v,\ \forall j,j^{*},k,$

For the relationship between these sets of conditions, see Pearl (2014).

As has already been pointed out by various studies, Assumptions 1 and 2 or assumption SI1 are satisfied if $T$ is randomized. Assumption 4 or SI2 is not testable in that this states independence of potential outcomes and potential mediators, some of which we never observe simultaneously. These assumptions do not hold even if both $T$ and $M$ are randomized or ignorable given $v$ (Pearl, 2014), while Assumption 3 holds.

This paper defines new causal mediation effects that are identifiable from observed data without the untastable assumptions when both $T$ and $M$ are randomized or ignorable given $v$. The proposed ones are useful even if the ranzomization for the mediator is not possible in that the assumptions required for identification are weaker than those for the traditional estimands, natural direct/indirect effects.

The proposed direct and indirect effects will have the following properties:

(a)

indirect effect will be zero if $M_{0}=M_{1}$ for all units.

(b)

these effects are identifiable without untestable Assumption 4 or SI2.

(c)

the defined effects are expressed as the potential outcomes $Y_{j}(m)$ and the potential mediators $M_{j}$, not $Y_{j}(M_{k})$. Thus, the causal interpretation is straightforward.

Without loss of generality, we consider binary treatment in this paper (for multi-valued treatment, we can generalise the result using similar arguments to those by Imbens 2000). Using two potential mediators $M_{1}$ (for $T=1$ treatment) and $M_{0}$ (for $T=0$ treatment), $M$ is expressed as

$$M=TM_{1}+(1-T)M_{0}.$$

(1)

We assume that $M$ is a categorical variable (i.e., $M=0,\cdots,M^{*}$). Let $M(m)$ be the binary indicator such that $M(m)=1$ if $M=m$. Similarly, let $M_{j}(m)$ be the binary indicator under $T=j$ treatment, $M_{j}(m)=1$ if the potential mediator under treatment $j$, $M_{j}$, is $m$.

The observed outcome $Y$ is expressed by the potential outcomes, potential mediators, and treatment indicator as follows:

$$Y=\sum_{m=0}^{M^{*}}\Bigl{[}TM(m)Y_{1}(m)+(1-T)M(m)Y_{0}(m)\Bigr{]}=\sum_{m=0}^{M^{*}}\Bigl{[}TM_{1}(m)Y_{1}(m)+(1-T)M_{0}(m)Y_{0}(m)\Bigr{]}$$

(2)

The observed outcome can be expressed by two potential outcomes $Y_{1}=Y_{1}(M_{1})$ (for $T=1$ treatment) and $Y_{0}(M_{0})$ (for $T=0$ treatment) as follows:

$$Y=TY_{1}+(1-T)Y_{0}=TY_{1}(M_{1})+(1-T)Y_{0}(M_{0})$$

(3)

under the composition assumption (Pearl, 2009).

The average treatment effect ($ATE$) is defined as the expectation of the difference between two potential outcomes:

$$ATE\equiv E[Y_{1}-Y_{0}],$$

(4)

A straightforward way of defining the direct effect is to set the mediator to a pre-specified level $M=m$. Pearl (2001) defined the controlled direct effect with mediator fixed at $M=m$, $CDE(m)$, in which the mediator is set to $m$ uniformly over the entire population:

$$ICDE(m)\equiv Y_{1}(m)-Y_{0}(m),\ \ \ CDE(m)\equiv E[ICDE(m)].$$

(5)

where $ICDE(m)$ is an unit-level version of the $CDE(m)$ that we define here to use later in this paper. As pointed out in existing studies, the quantity defined as $ATE-CDE$ is not a proper measure of indirect effect in that this quantity may not be zero even when $M_{0}=M_{1}$ for all units (Van der Weele, 2009)

In the literature on causal mediation analysis (Robins & Greenland, 1992; Pearl, 2001, 2009), ATE is expressed as the sum of the natural direct effect ($NDE$) and the natural indirect effect ($NIE$), instead of using $CDE$. Following Imai et al. (2010b),

	$\displaystyle NDE(t)$	$\displaystyle\equiv E[Y_{1}(M_{t})-Y_{0}(M_{t})],\ \ \ NIE(t)\equiv E[Y_{t}(M_{1})-Y_{t}(M_{0})],\ (t=0,1)$
	$\displaystyle NDE$	$\displaystyle\equiv\frac{1}{2}[NDE(1)+NDE(0)]\ \ \ NIE\equiv\frac{1}{2}[NIE(1)+NIE(0)]$
	$\displaystyle ATE$	$\displaystyle=E[Y_{1}-Y_{0}]=E[Y_{1}(M_{1})-Y_{0}(M_{0})]=NDE+NIE.$		(6)

Note that the NDE and NIE are not identified without further assumptions because quantity $Y_{1}(M_{0})$ is not observable. For identification, the existing studies assume independence between potential outcomes and mediator (given some observable covariates), or related conditions such as sequential ignorability. In Section 5, we show that NIE may be biased in that under no mediation, NIE is not zero when the assumption is violated while the proposed one is not.

Identification of NDE and IDE requires assumption 4 or SI2 because the causal mediation effect is defined by using the “never-observable”outcome (not “potentially observable” outcome) $Y_{j}(M_{k})\ \ (j\neq k)$. However, $Y_{j}(k)$ and $M_{j}$ are observed for some portion of the units.

We redefine the potential outcomes $Y_{j}(M_{k})$ by using the functions of potential outcomes and mediators as follows:

$\displaystyle Y_{j}(M_{k})\equiv\sum_{m=0}^{M^{*}}M_{k}(m)Y_{j}(m).$

(7)

Under this definition, $Y_{j}(M_{k})=Y_{j}(m)$ when $M_{k}(m)=1$.

By using this expression, ATE is determined as follows:

$\displaystyle ATE=E[Y_{1}(M_{1})-Y_{0}(M_{0})]=E[\sum_{m=0}^{M^{*}}\{M_{1}(m)Y_{1}(m)-M_{0}(m)Y_{0}(m)\}]$

(8)

Then, we defined the weighted controlled direct effect ($WCDE$)

	$\displaystyle WCDE$	$\displaystyle\equiv E[\sum_{m=0}^{M^{*}}\{M(m)Y_{1}(m)-M(m)Y_{0}(m)\}]$
		$\displaystyle=E[\sum_{m=0}^{M^{*}}M(m)(Y_{1}(m)-Y_{0}(m))]=E[\sum_{m=0}^{M^{*}}M(m)ICDE(m)].$		(9)

Note that in $CDE(m)$, the mediator is set to be the specific value, $M=m$, while $WCDE$ is the weighted average of $ICDE(m)$ over the observed distribution of $M$.

The implied indirect effect ($IIE$) is expressed as:

	$\displaystyle IIE$	$\displaystyle\equiv ATE-WCDE$
		$\displaystyle=E[\sum_{m=0}^{M^{*}}[(M_{1}(m)-M(m))Y_{1}(m)+(M(m)-M_{0}(m))Y_{0}(m)]]$		(10)

While the quantity defined as $ATE-CDE$ may not be zero even when $M_{0}=M_{1}$ for all units, $IIE$ is always zero if $M_{1}=M_{0}$ for all units.

Instead of Assumptions 1-4 (or SI1 and SI2), we introduce the following mean independence versions of Assumptions 1-4, SI1 and SI2:

Assumption $1^{{}^{\prime}}$

$E[Y_{j}(k)|T,v]=E[Y_{j}(k)|v]\ \forall j,k,$

Assumption $2^{{}^{\prime}}$

$E[M_{j}|T,v]=E[M_{j}|v]\ \forall j,$

Assumption $3^{{}^{\prime}}$

$E[Y_{j}(k)|M,T=j,v]=E[Y_{j}(k)|T=j,v]\ \forall j,k,$

Assumption $4^{{}^{\prime}}$

$E[Y_{j}(k)|M_{j^{*}},v]=E[Y_{j}(k)|v]\ \forall j,j^{*},k,$

Assumption $SI1^{{}^{\prime}}$

$E[M_{j^{*}}Y_{j}(k)|T,v]=E[M_{j^{*}}Y_{j}(k)|v]\ \forall j,j^{*},k,$

Assumption $SI2^{{}^{\prime}}$

$E[Y_{j}(k)|M_{j^{*}},T=j,v]=E[Y_{j}(k)|T=j,v]\ \forall j,j^{*},k,$

Assumption $SI1^{{}^{\prime}}$ implies the mean independence version of the ignorability assumption (Rosenbaum & Rubin, 1983), $E[Y_{j}|T,v]=E[Y_{j}|v]\ \forall j$, which is sufficient for identifying ATE.

For identification of $WCDE$, we consider the following two cases: Case 1, when both $M$ and $T$ are randomized or ignorable given $v$, and Case 2, when $M$ is not directly maipulable.

For illustrative purposes, we present a simulation study that compares the defined effects with the previously proposed ones. We numerically evaluated bias from the true values (population WCDE for the proposed method in Case 1 and population NDE for the existing method with assumptions SI1 and SI2). We consider the data-generating model in which assumption SI2 in Section 1 can be violated.

For simplicity, we consider binary mediator $M$, and define two latent continuous potential mediators $M_{0}^{L}$ and $M_{1}^{L}$ so that $M_{j}=1$ if $M_{j}^{L}>0$ otherwise $M_{j}=0\ \ (j=0,1)$. We generated 10,000 samples of size $n=4,000$ from the joint vector of potential mediators and potential , $W=(M_{0}^{L},M_{1}^{L},Y_{0}(0),Y_{0}(1),Y_{1}(0),Y_{1}(1))^{t}$ which follows a finite scale mixture of multivariate-normal distributions;

$\displaystyle W\sim 0.6\times N(\mu,\Sigma_{1})+0.4\times N(\mu,\Sigma_{2})$

(20)

where $\mu=(-1,1,0,0.2,0.6,1)$, $\Sigma_{1}=\Sigma$, $\Sigma_{2}=2\times\Sigma$ and the diagonal elements of $\Sigma$ is set to be $1$. Off-diagonal elements are set such that $Cov(Y_{j}(k)),Y_{l}(m))=0.5$ and $Cov(M_{0},M_{1})=0.6$. For simplicity, the covariances between potential mediators and potential outcomes are set as follows: $Cov(M_{0},Y_{0}(0))=Cov(M_{0},Y_{1}(0)=Cov(M_{1},Y_{0}(1))=Cov(M_{1},Y_{1}(1))=\phi$, and $Cov(M_{0},Y_{0}(1))=Cov(M_{0},Y_{1}(1))=Cov(M_{1},Y_{0}(0))=Cov(M_{1}Y_{1}(0))=-\phi$. In this setup, $p(M_{1}=1)\approx 0.809$ and $p(M_{0}=1)\approx 0.191$.

The treatment indicator $T$ is generated from the Bernoulli distribution with $Pr(T=1)=p$, independently of $W$. We consider four cases with $p=0.01$, $0.1$, $0.3$ and $0.5$ and in each case $\phi$ varies from $-0.15$ to $0.15$ in $0.05$ increments. Note that for all the models assumption 4 or SI2 is violated, except $\phi=0$.

The true values of $WCDE$ and $NDE$ of the model is difficult to calculate analytically. We then evaluate these values using the simulated 10,000 datasets.

We compare the proposed estimatior for $WCDE$ with the existing estimator of $NDE$ implied by Equation 18 in Imai et al. (2010b), which is frequently used in applied studies. The results are shown in Table 1 and those with $p=0.5$ are illustrated in Figure 1, in which the horizontal axis is $\phi$ and the vertical axis is the bias from the true values.

As shown in this figure and Table1, the bias of the previously proposed estimator can be large for large deviations from Assumption SI2 (i.e., large $\phi$) as mentioned in the sensitivity analysis in Imai et al. (2010b), while the proposed method can find true values on average. The tendency of the size of bias does not change according to $p$, the proportion of $T=1$, but in setups with small $p$ the variance of the two estimators is large because the sample size with $T=1$ is very small.

Table 1 shows the numerically evaluated population $ATE$, $WCDE$ and $NDE$ in each setup. The difference between $WCDE$ and $NDE$ for small $p$ exists but not large, while as mentioned in Section 3 the difference is negligible if $p(T=1)=0.5$.

It is also shown that the bias of the existing estimator can be very large compared with the size of true value of the causal mediation effect. In particular, as seen in the setup with $\phi=-0,15$ where $ATE$ is almost the same as $NDE$ (i.e., $NIE$ is almost zero), the existing method wrongly finds a “(pseudo-)mediation” effect under the setup when the true model does not contain a mediation effect, but the mediator and potential outcomes are correlated.

Tab.1.   Simulation results

In this paper, we proposed a new definition of causal direct and indirect effects in causal mediation analysis.

Identification of the previously proposed quantities, natural direct effect, and natural indirect effect is not possible even when both treatment and mediator are randomized. Therefore, it is unavoidable to employ untestable assumption of the independence of potential outcomes and potential mediators, some of which we never observe simultaneously.

The proposed quantities are identifiable without any assumption when both treatment and mediator are randomized. Even when randomization is not possible, the proposed quantities require weaker assumptions than those for the identification of traditional quantities.

When it is difficult to directly manipulate $M$, Assumption $3^{{}^{\prime}}$ is required for identification of the proposed effects. Another approach for identification such as principal stratification approach (Frangakis & Rubin, 2002; Forastiere et al., 2018) is an promising strategy that we will investigate in a future study.

In this paper, we focused on the case with binary treatment, but the definition of WCDE is useful for the case with multi-valued treatment. In multi-valued treatment, the measure of indirect effect should be defined in terms of the sum of squares or variance of the expectations, instead of the traditional “difference of the expectations” formulation in the case of binary treatment, which will be considered elsewhere.