spacing=nonfrench

# When Do Natural Mediation Effects Differ from Their Randomized
Interventional Analogues: Test and Theory

Ang Yu, Li Ge, and Felix Elwert Department of Sociology, University of
Wisconsin-Madison. Email<EMAIL_ADDRESS>of Biostatistics and
Medical Informatics, University of Wisconsin-MadisonDepartment of Sociology,
Department of Biostatistics and Medical Informatics, and Department of
Population Health Sciences, University of Wisconsin-Madison

(July 2, 2024)

###### Abstract

In causal mediation analysis, the natural direct and indirect effects (natural
effects) are nonparametrically unidentifiable in the presence of treatment-
induced confounding, which motivated the development of randomized
interventional analogues (RIAs) of the natural effects. The RIAs are easier to
identify and widely used in practice. Applied researchers often interpret RIA
estimates as if they were the natural effects, even though the RIAs could be
poor proxies for the natural effects. This calls for practical and theoretical
guidance on when the RIAs differ from or coincide with the natural effects,
which this paper aims to address. We develop a novel empirical test for the
divergence between the RIAs and the natural effects under the weak assumptions
sufficient for identifying the RIAs and illustrate the test using the Moving
to Opportunity Study. We also provide new theoretical insights on the
relationship between the RIAs and the natural effects from a covariance
perspective and a structural equation perspective. Additionally, we discuss
previously undocumented connections between the natural effects, the RIAs, and
estimands in instrumental variable analysis and Wilcoxon-Mann-Whitney tests.

Keywords— causal mediation analysis, falsification test, nonparametric
structural equation, randomized interventional analogue, Wilcoxon-Mann-Whitney
test

##  1 Introduction

###  1.1 Background

Causal mediation analysis explains the mechanisms of a total causal effect by
decomposing it into direct and indirect effects in terms of some mediators.
The direct effect is the part of the total effect that does not go through the
researcher-specified mediators, and the indirect effect is the part that does.
As a central task in the social and health sciences, causal mediation analysis
is widely used in applied research.

We adopt the conventional notation in causal mediation analysis. $Y$ is the
observed outcome, $A$ is a binary treatment with support $\\{0,1\\}$,111This
is generalizable to any pair of two values for a multivalued treatment. and
$M$ is a vector of mediators. $Y_{a}$ and $M_{a}$ are respectively the
potential values of $Y$ and $M$ under the assignment of treatment value $a$.
We further define two groups of confounders that may be empty, $C$ is a vector
of baseline confounders, and $L$ is a vector of post-treatment confounders.
Figure 1 illustrates the relationship between variables, when any variable may
affect any temporally subsequent variables.

$A$$M$$Y$$C$$L$ Figure 1: Variable Set-up in Causal Mediation Analysis

The most classic approach of causal mediation analysis decomposes the total
effect (TE) into the natural indirect effect (NIE) and the natural direct
effect (NDE) (Robins and Greenland, 1992; Pearl, 2001).

$\underbrace{\operatorname{E}(Y_{1}-Y_{0})}_{\text{TE}}=\underbrace{\operatorname{E}(Y_{1,M_{1}}-Y_{0,M_{0}})}_{\text{TE}}=\underbrace{\operatorname{E}(Y_{1,M_{1}}-Y_{1,M_{0}})}_{\text{NIE}}+\underbrace{\operatorname{E}(Y_{1,M_{0}}-Y_{0,M_{0}})}_{\text{NDE}},$

where $Y_{a,M_{a^{\prime}}}$ denotes the potential outcome of $Y$ under the
assignment of treatment $a$ and the mediator value that would be realized
under the assignment of treatment $a^{\prime}$. The NIE is defined by varying
the mediator assignment from $M_{1}$ to $M_{0}$ but otherwise fixing treatment
assignment at 1, capturing the part of the total effect that only goes through
$M$. The NDE is defined by varying the treatment assignment from 1 to 0 but
holding mediator assignment at the baseline mediator value, capturing the part
of the total effect that does not go through $M$. Since the natural effects
(NIE and NDE) are defined in terms of individual-level potential mediators
($M_{1}$ and $M_{0}$), they capture causal mechanisms at the individual level,
even though they are ultimately summarized as population-average effects
through the expectation operator.

The natural effects are notoriously difficult to identify. Without parametric
assumptions, they are unidentifiable when there exists any treatment-induced
confounder $L$, regardless of whether $L$ is observed (Robins, 2003; Avin et
al., 2005). This severely limits the application of the natural effects in
practice, as ruling out $L$ is impossible in most empirical settings.

Motivated by the difficulty of identifying the natural effects, an alternative
decomposition has been proposed (VanderWeele et al., 2014). Nonparametrically,
its identification does not require the absence of treatment-induced
confounders. This alternative decomposition is based on the randomized
interventional analogues (RIA) of the TE, the NIE, and the NDE, namely the
TER, the NIER, and the NDER:

$\underbrace{\operatorname{E}(Y_{1,G_{1}}-Y_{0,G_{0}})}_{\text{TE}^{R}}=\underbrace{\operatorname{E}(Y_{1,G_{1}}-Y_{1,G_{0}})}_{\text{NIE}^{R}}+\underbrace{\operatorname{E}(Y_{1,G_{0}}-Y_{0,G_{0}})}_{\text{NDE}^{R}},$

where $G_{a^{\prime}}$ is a value randomly drawn from the mediator
distribution that would realize under the assignment of treatment value
$a^{\prime}$ given $C$, and $Y_{a,G_{a^{\prime}}}$ is the potential outcome of
$Y$ under the assignment of the treatment value $a$ and the mediator value
$G_{a^{\prime}}$. Clearly, the RIAs differ from the natural effects in
mediator assignments. Instead of $M_{1}$ and $M_{0}$, the mediator assignments
for the RIAs are $G_{1}$ and $G_{0}$. As $G_{1}$ and $G_{0}$ are random draws
from population distributions, the RIAs are not aggregations of individual-
level causal contrasts like the natural effects.

Seen as much less demanding and more widely applicable than the natural
effects, the RIAs are popular in empirical research.222 Methodological work
that further develops the RIA decomposition includes (VanderWeele and Tchetgen
Tchetgen, 2017; Vansteelandt and Daniel, 2017; Zheng and Van Der Laan, 2017;
Rudolph et al., 2018; Wodtke and Zhou, 2020; Loh et al., 2020; Díaz et al.,
2021; Benkeser and Ran, 2021; Hejazi et al., 2022; Nguyen et al., 2022; Díaz
et al., 2023; Rudolph et al., 2024). In practice, applied researchers
frequently estimate the RIAs as proxies of the natural effects. In fact, the
RIA estimates are often interpreted as if they were estimates of the natural
effects (after all, the RIAs are named as “analogues”!). Sarvet et al. (2023)
reviewed 16 applied studies that estimate RIAs, all of which contain
interpretive statements that elide the difference between the RIAs and the
natural effects. Indeed, the methodological literature has encouraged this
ambiguity. For example, VanderWeele and Tchetgen Tchetgen (2017) write that
“it will only be in extremely unusual settings that the interventional
analogue is non-zero, with there being no natural indirect effects”.

However, there are reasons to suspect that the RIAs can be poor proxies of the
natural effects. Unlike the natural effects, they are not interpretable as
explanatory mechanisms at the individual as opposed to the population level.
Formalizing this intuition, Miles (2023) proposes a set of null criteria that
valid causal indirect effect measures should satisfy and shows that the NIE is
valid by these criteria while the NIER is not. In particular, the NIER can be
nonzero even if the mediator does not “mediate” the treatment effect for any
individual. In addition, it has been frequently noted in the methodological
literature that the NIER and the NDER do not generally sum to the TE, which is
problematic because the canonical task of causal mediation analysis is to
understand the TE (Vansteelandt and Daniel, 2017; Nguyen et al., 2021).

Beyond the violation of null criteria, which focuses on a knife-edge scenario,
we draw attention to possible quantitative differences between the natural
effects and the RIAs in a wide range of data generating processes (DGPs).
These quantitative differences may be large and even involve sign reversal. In
the illustration of Figure 2, data are simulated according to a set of very
simple and seemingly innocuous DGPs. By varying one parameter of the DGP, we
observe areas of significant divergence and sign reversal, where the RIAs can
hardly be used to draw conclusions about the natural effects.

Figure 2: Illustration of Possible Divergence Between the Natural Mediation
Estimands and Their RIAs. Panel (a) depicts NIE and NIER, and Panel (b)
depicts NDE and NDER. The DGP is as follows. $L\sim\mathcal{N}(A,1)$;
$M\sim\text{expit}(A+L+bAL)$; $Y\sim\mathcal{N}(A+L+M+LM,1)$. Thus, the $x$
axis is the $b$ value that we vary, which is the coefficient of the $A$-$L$
interaction in the generative model for $M$. In Panel (a), the shaded area
indicates sign reversal between NIE and NIER.

Therefore, it is natural to ask when the natural effects differ from their
RIAs. If they are identical or at least close to each other, then it might be
warranted to interpret estimates of the RIAs as the natural effects.
Conversely, if they substantially differ, then more caution and precision in
interpretation is called for. In this paper, we answer this question with one
practical test and two theoretical perspectives.

###  1.2 Contributions

We make a practical contribution by proposing a novel test for the differences
between the NIE, the NDE, and their respective RIAs. The empirical testability
of these differences may be surprising, because under the standard assumptions
for identifying the NIE and the NDE, the natural effects necessarily coincide
with their RIAs (VanderWeele and Tchetgen Tchetgen, 2017, p.921). And under
the standard assumptions identifying the NIER and the NDER, the NIE and the
NDE are unidentified. Thus, it may appear that under no set of common
assumptions can one test the differences. However, our test is made possible
by leveraging two simple facts. First, the TE and the TER are identified under
the standard assumptions for the NIER and the NDER. Second, when
$\text{TE}-\text{TE}^{R}\neq 0$, it is necessarily the case that either
$\text{NIE}\neq\text{NIE}^{R}$ or $\text{NDE}\neq\text{NDE}^{R}$. Hence,
instead of hoping that “the natural and interventional effects may coincide
empirically” (Loh et al., 2020), we can actually test their divergence by
testing $\text{TE}-\text{TE}^{R}=0$ under weak identifying assumptions that
are sufficient for the RIAs but not the natural effects.

We make a theoretical contribution by clarifying and illustrating the
substantive conditions under which the natural effects differ from or coincide
with their RIAs. We do so from a nonparametric covariance perspective and a
structural equation perspective. First, we derive a covariance-based
representation of the differences between the natural effects and their RIAs.
Second, we derive parametric constraints on the structural equations
generating the data under which the the natural effects will coincide with the
RIAs. These two novel perspectives provide intuitive insights on the
substantive mechanisms underpinning the relationship between the natural
effects and the RIAs. In Miles’ (2023) discussion of the relationship between
NIE and NIER, he proves the null criteria violation using one specific
numerical counterexample. With two new analytic perspectives that are general
and intuitive, we thus demystify and expand on Miles’ (2023) results.

The remaining of this paper is organized as follows. In Section 2, we review
some standard assumptions in causal mediation analysis that were referred to
above. In Section 3, we present our empirical test for the differences between
the natural effects and the RIAs and apply it to the Moving to Opportunity
(MTO) study. Section 4 and 5, respectively, introduce the covariance
perspective and the structural equation perspective. Section 6 discusses
related estimands, including those in the instrumental variable (IV) settings
and those underlying the Wilcoxon-Mann-Whitney tests. We present novel results
that unify causal mediation analysis with these other fields of causal
inference. Technical proofs are collected in the appendix. R code for
simulating Figure 2 and empirical data analysis in Section 3 can be found at
https://github.com/ang-yu/diff_naturals_rias.

##  2 Review of Conventional Mediation Assumptions

We review conventional assumptions included in the literature of causal
mediation analysis.

######  Assumption 1 (Consistency).

$f(M_{a}\mid a,C)=f(M\mid a,C)$ and $\operatorname{E}(Y_{a,m}\mid
a,m,C)=\operatorname{E}(Y\mid a,m,C)$, for all $a$ and $m$.

######  Assumption 2 (Ignorability of $A$ conditional on $C$).

$Y_{a,m}\rotatebox[origin={c}]{90.0}{$\models$}A\mid C$ for all $a$ and $m$;
$M_{a}\rotatebox[origin={c}]{90.0}{$\models$}A\mid C$ for all $a$.

######  Assumption 3 (Ignorability of $M$ conditional on $C,A,L$).

$Y_{a,m}\rotatebox[origin={c}]{90.0}{$\models$}M\mid C,A=a,L$ for all $a$ and
$m$.

######  Assumption 4 (Ignorability of $M$ conditional on $C,A$).

$Y_{a,m}\rotatebox[origin={c}]{90.0}{$\models$}M\mid C,A=a$ for all $a$ and
$m$.

######  Assumption 5 (Cross-world Independence).

$Y_{a,m}\rotatebox[origin={c}]{90.0}{$\models$}M_{a^{\prime}}\mid C$ for all
$a$, $a^{\prime}$, and $m$.

Assumption 1 is a standard statement linking the potential values to the
observed values. Assumption 2 requires the treatment $A$ be ignorable
conditional on baseline confounders $C$. Assumption 3 states that the mediator
$M$ is conditionally ignorable given both baseline confounders $C$ and post-
treatment confounders $L$, as well as the treatment. Assumption 4 imposes
conditional ignorability of the mediator given only baseline confounders and
the treatment, which is stronger than Assumption 3. Finally, Assumption 5
requires the conditional independence between the potential outcomes $Y_{a,m}$
and potential mediators $M_{a^{\prime}}$ under two possibly different
treatment assignments $a$ and $a^{\prime}$, hence its name (cross-world
independence).

In the literature, Assumptions 1, 2, and 3, are the standard identifying
assumptions for the RIAs (VanderWeele et al., 2014), while Assumptions 1, 2,
4, and 5 are the standard assumptions for identifying the NIE and the NDE
(Pearl, 2001; VanderWeele, 2015, p.463-4; Imai, Keele, and Yamamoto, 2010 for
a slightly stronger version). Notably, the cross-world independence assumption
requires the absence of any post-treatment confounding of the mediator-outcome
relationship ($L=\varnothing$) (Robins, 2003; Avin et al., 2005; Andrews and
Didelez, 2021). Hence, it is clear that the standard assumptions for the RIAs
are weaker, as they allow for the existence of post-treatment confounders.
Furthermore, when the cross-world independence assumption holds, the natural
effects are necessarily equivalent to their RIAs.

##  3 Empirical Test

We propose to use the empirical estimate of $\text{TE}-\text{TE}^{R}$ as a
test statistic for the divergence between the NIE and the NIER and the
divergence between the NDE and the NDER. This test relies on the fact that if
$\text{TE}-\text{TE}^{R}\neq 0$, it is necessarily the case that either
$\text{NIE}\neq\text{NIE}^{R}$ or $\text{NDE}\neq\text{NDE}^{R}$, or both.
Thus, if we reject the null hypothesis that $\text{TE}-\text{TE}^{R}=0$, we
also reject the null hypothesis that $\text{NIE}=\text{NIE}^{R}$ and
$\text{NDE}=\text{NDE}^{R}$.333In a recent work, Vo et al. (2024) constructs a
test using a similar premise. In the context of path-specific effects, they
propose using the difference between the TE and the sum of a set of RIA-type
path-specific estimands to test the absence of intermediate confounding. In
addition, since
$|\text{TE}-\text{TE}^{R}|\leq|\text{NIE}-\text{NIE}^{R}|+|\text{NDE}-\text{NDE}^{R}|$
by the triangle inequality, $|\text{TE}-\text{TE}^{R}|$ also provides a lower
bound for the sum of the absolute differences between the NIE and the NIER and
between the NDE and the NDER.

Under assumptions 1, 2 and 3,
$\text{TE}-\text{TE}^{R}=\operatorname{E}(Y_{1})-\operatorname{E}(Y_{0})-\operatorname{E}(Y_{1,G_{1}})+\operatorname{E}(Y_{0,G_{0}})$
is identified by the functionals below (VanderWeele et al., 2014).

$\displaystyle\operatorname{E}(Y_{a})$ $\displaystyle=\iint yf(y\mid
c,a)f(c)\differential y\differential c$
$\displaystyle\operatorname{E}(Y_{a,G_{a}})$ $\displaystyle=\iiiint yf(y\mid
c,a,l,m)f(m\mid c,a)f(l\mid c,a)f(c)\differential y\differential
m\differential l\differential c.$

Hence, importantly, our test is nonparametrically identifiable when there are
treatment-induced confounders and Assumption 5 is invalid. This is because
although the NIE and the NDE are not nonparametrically identifiable under
treatment-induced confounding, their sum is.

The task now is to estimate $\text{TE}-\text{TE}^{R}$. This can be done using
various estimators of TE and TER. Below, we discuss a nonparametric approach
based on the efficient influence functions (EIF) (Bickel et al., 1998; Hines
et al., 2022).

The EIF for $\operatorname{E}(Y_{a})$ is well-known (Hahn, 1998):

$\phi_{a}\vcentcolon=\frac{\mathbbm{1}(A=a)}{\operatorname{Pr}(A=a\mid
C)}[Y-\operatorname{E}(Y\mid C,a)]+\operatorname{E}(Y\mid
C,a)-\operatorname{E}[\operatorname{E}(Y\mid C,a)\mid a].$

The EIF for $\operatorname{E}(Y_{a,G_{a}})$ is derived by Díaz et al. (2021).
Let $\operatorname{E}(Y\mid C,A,L,M)=\mu(C,A,L,M)$ and
$\xi(C,A)=\iint\mu(C,A,l,m)f(m\mid C,A)f(l\mid C,A)\differential
m\differential l$, this EIF is

$\displaystyle\psi_{a}\vcentcolon=\frac{\mathbbm{1}(A=a)}{\operatorname{Pr}(A=a\mid
C)}$ $\displaystyle\left\\{\frac{\operatorname{Pr}(M\mid
C,a)}{\operatorname{Pr}(M\mid C,a,L)}\left[Y-\mu(C,a,L,M)\right]\right.$
$\displaystyle\left.\phantom{\left\\{\right.}+\int\mu(C,a,L,m)f(m\mid
C,a)\,\differential m-\xi(C,a)\right.$
$\displaystyle\left.\phantom{\left\\{\right.}+\int\mu(C,a,l,M)f(l\mid
C,a)\,\differential l-\xi(C,a)\right\\}-\operatorname{E}[\xi(C,a)].$

Then the EIF for $\text{TE}-\text{TE}^{R}$ is
$\phi_{1}-\phi_{0}+\psi_{1}-\psi_{0}$. Using these EIFs, we can construct
either double machine learning (Chernozhukov et al., 2018) or targeted maximum
likelihood (Van der Laan and Rose, 2011) estimators that allow nonparametric
estimation with desirable theoretical properties (Díaz et al., 2021; Rudolph
et al., 2024). In particular, these estimators are data-adaptive and can
handle high-dimensional covariates. They are also multiply robust to
misspecification of components of the EIF. Furthermore, using cross-fitting,
they attain semiparametric efficiency and asymptotic normality under
relatively weak conditions. In practice, researchers may take advantage of the
output of _medoutcon_ package (Hejazi et al., 2022) in R to estimate
$\text{TE}-\text{TE}^{R}$ and the associated confidence interval.444Another R
package, _HDmediation_ (Williams et al., 2024), may also be useful.
_medoutcon_ only accommodates a single binary $L$ variable but directly
outputs individual-level EIF estimates, which allows the calculation of $p$
value and confidence interval. On the other hand, _HDmediation_ accommodates
vector-valued and non-binary $L$ but does not directly outputs EIF estimates.

###  3.1 Empirical Illustration

We apply our test to mediation analysis of the Moving to Opportunity (MTO)
study, a large-scale longitudinal randomized control trial conducted by the
Department of Housing and Urban Development of the United States (Ludwig et
al., 2013; Kling et al., 2005). We follow the conceptual set-up of Rudolph et
al. (2021) and Rudolph et al. (2024), who estimated the RIAs.555Due to lack of
access to the restricted-use dataset, we follow their variable and sample
choices only conceptually, not precisely. Hence, our estimates should be
regarded as purely illustrative. The treatment is a binary indicator of
whether or not a family living in a high-poverty neighborhood was randomized
to receive a Section 8 housing voucher that allowed them to move to a less
poor neighborhood. We consider two mediators measured between 10-15 years of
follow up, neighborhood poverty and the number of residential moves. The
outcome is a composite score for mental health. For causal identification, we
account for a post-treatment confounder which is whether the family used the
voucher to move within the 90 days allotted. We also account for 12 baseline
confounders, which capture baseline household socioeconomic and demographic
characteristics, as well as neighborhood-related perceptions and aspirations.

Estimand | Estimate | 95% Confidence Interval
---|---|---
TE | 0.084 | (0.081, 0.087)
TER | 0.002 | (-0.002, 0.005)
$\text{TE}-\text{TE}^{R}$ | 0.083 | (0.069, 0.072)
Table 1: Empirical Estimates from the MTO study. The treatment is the receipt
of a randomized housing voucher. The mediators are neighborhood poverty and
the number of residential moves. The outcome is mental health. Estimation is
implemented using double machine learning estimators with cross-fitting.
Confidence intervals are Wald-type and calculated using the estimated EIF of
the estimands.

We implement our test using double machine learning estimators with two-fold
cross-fitting. The nuisance functions are estimated using random forests
(Wright and Ziegler, 2017). For confidence intervals, we leverage the
asymptotic normality of the estimators and calculate the variance estimates
using the mean squared estimated EIFs. We present our estimates in Table 1.
Our estimate of $\text{TE}-\text{TE}^{R}$ is significantly different from 0
(95% Confidence Interval$=(0.081,0.087)$). Therefore, we reject the null
hypothesis that $\text{NIE}=\text{NIE}^{R}$ and $\text{NDE}=\text{NDE}^{R}$.
In this empirical example, one should not interpret the RIA estimates as the
natural effects. Furthermore, the sum of the absolute differences between the
NIE and the NIER and between the NDE and the NDER is greater than
$|\text{TE}-\text{TE}^{R}|$, which is estimated to be 0.083.

##  4 Covariance Perspective

We present a covariance-based representation of the differences between the
natural effects and their RIAs. We first focus on the simple case with a
scalar binary mediator and a randomized treatment such that $C$ is empty. This
simple case most easily captures the core intuition. Then we generalize the
covariance representation to vector mediators with arbitrary distributions and
a non-randomized treatment. The expressions are derived just using the
definitions of the estimands, without any identifying assumptions or
functional form restrictions.

###  4.1 Single Binary Mediator, Randomized Treatment

######  Proposition 1.

When the treatment is randomized and the support of $M$ is $\\{0,1\\}$,

$\displaystyle\text{TE}-\text{TE}^{R}$
$\displaystyle=\operatorname{Cov}(M_{1},Y_{1,1}-Y_{1,0})-\operatorname{Cov}(M_{0},Y_{0,1}-Y_{0,0})$
$\displaystyle\text{NIE}-\text{NIE}^{R}$
$\displaystyle=\operatorname{Cov}(M_{1}-M_{0},Y_{1,1}-Y_{1,0})$
$\displaystyle\text{NDE}-\text{NDE}^{R}$
$\displaystyle=\operatorname{Cov}(M_{0},Y_{1,1}-Y_{1,0}-Y_{0,1}+Y_{0,0}).$

First, the difference between the TE and the TER reflects how the treatment
changes the covariance between an individual’s mediator value ($M_{a}$) and
their mediator effect on the outcome ($Y_{a,1}-Y_{a,0}$).666In the causal
decomposition of group disparities proposed by Yu and Elwert (2024), the
“selection” component captures the contribution of group-differential
selection into treatment to an outcome disparity. Relabelling the group and
the treatment in the framework of Yu and Elwert (2024) as the treatment and
mediator, the selection component can be written as
$\operatorname{Cov}(M,Y_{M=1}-Y_{M=0}\mid
A=1)-\operatorname{Cov}(M,Y_{M=1}-Y_{M=0}\mid A=0)$, which coincides with
$\text{TE}-\text{TE}^{R}$ when treatment is randomized. The TE will be greater
than the TER if the treatment (rather than the control) induces more accurate
ex-ante expectations of the individual-level mediator effect such that
individuals with a higher mediator effect are more likely to select into the
mediator value 1.

Second, the difference between the NIE and the NIER equals the covariance
between the treatment effect on the mediator ($M_{1}-M_{0}$) and a net
mediator effect on the outcome ($Y_{1,1}-Y_{1,0}$). Thus, if there are common
determinants of these two effects, the NIE will differ from the NIER. These
determinants could be either pre-treatment modifiers of both effects or post-
treatment mediators of the first effect which also modify the second effect.
In the MTO example, those who are better able to take advantage of the housing
voucher (the treatment) to move to a lower-poverty neighborhood (the mediator)
may, in turn, be better able to leverage the resources in their new lower-
poverty neighborhood to improve mental health outcomes. In that case, the
covariance between the treatment effect on the mediator and the net mediator
effect on the outcome will be positive.

Third, the difference between the NDE and the NDER is the covariance between
the mediator value under control ($M_{0}$) and the interaction effect between
treatment and mediator on outcome ($Y_{1,1}-Y_{1,0}-Y_{0,1}+Y_{0,0}$).

Generally, the natural effects and the RIAs differ to the extent that the
potential mediators ($M_{a}$) and the potential outcomes ($Y_{a^{\prime},m}$)
are correlated with one another. This makes sense as the RIAs are defined
using random draws of potential mediators, $G_{a}$, that are independent of
everything else, whereas the natural effects do not remove the naturally
occurring dependency between the potential mediators and the potential
outcomes.

Miles (2023) proposes a set of mediation null criteria. In particular, the
definition of the “sharper mediation null” is: For each individual in the
population of interest, either $M_{1}=M_{0}$ or $Y_{a,m}=Y_{a,m^{\prime}}$ for
$a$, $m$, and $m^{\prime}$. And a valid measure of indirect effect should be
zero when the sharper mediation null is true. We note that
$\text{NIE}=\operatorname{E}[(M_{1}-M_{0})(Y_{1,1}-Y_{1,0})]$. Thus, the NIE
clearly satisfies this criterion, while, by Proposition 1, NIER does not. For
example, if half of the population has $M_{1}-M_{0}=1$ and $Y_{1,1}-Y_{1,0}=0$
while the other half has $M_{1}-M_{0}=0$ and $Y_{1,1}-Y_{1,0}=1$, the NIE will
be zero, but the NIER will be $1/4$. This is consistent with Miles (2023)’s
results. However, Miles (2023) proves that the NIER does not satisfy the null
criterion using a specific counterexample, which might be viewed as a
contrived example. In contrast, Proposition 1 analytically reveals why and
when the NIER deviates from the null criterion: it is because the NIER omits
the natural dependency between the treatment effect on the mediator and the
mediator effect on the outcome, which is a part of the NIE. To the extent that
the correlation between these effects is pervasive in practice, there is
nothing “contrived” in the deviation of the NIER from the null criterion.

###  4.2 General Case

In last subsection, we focused on the case of a binary $M$ and a randomized
treatment. Now we generalize our results to a continuous or multivalued
discrete vector of mediators and a non-randomized treatment. Again, we do not
make any identifying assumptions or parametric restrictions.

######  Proposition 2.

$\displaystyle\text{TE}-\text{TE}^{R}$
$\displaystyle=\sum_{m\in\mathcal{M}}\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m),Y_{1,m}\mid
C]\\}-\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{0}=m),Y_{0,m}\mid
C]\\}$ $\displaystyle\text{NIE}-\text{NIE}^{R}$
$\displaystyle=\sum_{m\in\mathcal{M}}\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}$ $\displaystyle\text{NDE}-\text{NDE}^{R}$
$\displaystyle=\sum_{m\in\mathcal{M}}\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{0}=m),Y_{1,m}-Y_{0,m}\mid
C]\\},$

where $\mathbbm{1}(\cdot)$ is the indicator function, and $\mathcal{M}$ is the
support of $M$. The relationships above directly hold for discrete mediators,
but they also hold for continuous mediators if summations are replaced with
integrals and the indicator function is replaced with the Dirac delta
function.

We thus obtain a covariance-based representation analogous to Proposition 1.
Here, the building blocks are conditional covariances between the potential
mediators ($M_{a}$) and the potential outcomes ($Y_{a^{\prime},m}$) given
baseline confounders $C$. We further summarize the $c$\- and $m$-specific
covariances by taking expectation over the distribution of $C$ and uniformly
taking sum over the support of $M$. The natural effects and the RIAs generally
differ due to the dependency between the mediator and outcome potential values
conditional on baseline confounders. Clearly, the natural effects and the RIAs
coincide when the cross-world independence assumption (Assumption 5) is
satisfied.

In particular, $\text{TE}-\text{TE}^{R}$ still has a highly interpretable
form. It reflects the treatment effect on selection into mediator values based
on the corresponding potential outcomes. Consider a treatment assignment $a$,
a mediator value $m$, and a baseline confounder value $c$. If those who, when
assigned $a$, would take the mediator value $m$ tend to be those whose
corresponding potential outcome is higher (among individuals with $C=c$), then
$\operatorname{Cov}[\mathbbm{1}(M_{a}=m),Y_{1,m}\mid C=c]$ will be positive.
This may happen if treatment $a$ induces somewhat accurate ex-ante
anticipation of what outcome $m$ would bring about, and individuals choose $M$
based on this anticipation. And $\text{TE}-\text{TE}^{R}$ will generally be
non-zero if this induction differs by treatment status for some $m$ and $c$.

An alternative RIA-based decomposition is developed by Lok (2016) and Lok and
Bosch (2021)777With relabelled variables, this decomposition also coincides
with a disparity decomposition in Jackson (2021). In this decomposition, the
TE is decomposed to what are called the organic indirect and direct effects
($\text{NIE}^{\text{organic}}$ and $\text{NDE}^{\text{organic}}$).

$\underbrace{\operatorname{E}(Y_{1}-Y_{0})}_{\text{TE}}=\underbrace{\operatorname{E}(Y_{1}-Y_{1,G_{0}})}_{\text{NIE}^{\text{organic}}}+\underbrace{\operatorname{E}(Y_{1,G_{0}}-Y_{0})}_{\text{NDE}^{\text{organic}}}.$

We again show a corresponding covariance representation in the general case.

######  Proposition 3.

$\displaystyle\text{NIE}-\text{NIE}^{\text{organic}}$
$\displaystyle=-\sum_{m\in\mathcal{M}}\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}$ $\displaystyle\text{NDE}-\text{NDE}^{\text{organic}}$
$\displaystyle=\sum_{m\in\mathcal{M}}\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}.$

Finally, Zheng and Van Der Laan (2017) propose a related decomposition (also
see Nguyen et al., 2022, p.264). The intervention underlying this
decomposition involves assigning to people with $C=c,L_{a}=l$ values of
mediator randomly drawn from the distribution of $M_{a^{\prime}}$ conditional
on $C=c,L_{a^{\prime}}=l$. Denoting these random draws by $G_{a^{\prime}\mid
L_{a}}$, their decomposition is

$\underbrace{\operatorname{E}(Y_{G_{1\mid C,L_{1}}}-Y_{G_{0\mid
L_{0}}})}_{\text{TE}^{RL}}=\underbrace{\operatorname{E}(Y_{1,G_{1\mid
L_{1}}}-Y_{1,G_{0\mid
L_{1}}})}_{\text{NIE}^{RL}}+\underbrace{\operatorname{E}(Y_{1,G_{0\mid
L_{1}}}-Y_{0,G_{0\mid L_{0}}})}_{\text{NDE}^{RL}}.$

The differences between the natural effects and the estimands above do not
have a covariance representation. This is because the way $L$ enters into the
estimands makes the $\text{NIE}^{RL}$ the path-specific effect through $M$ but
not $L$ (see Appendix S2 in Miles, 2023). Thus, these estimands are
conceptually further removed from the natural effects.

##  5 Structural Equation Perspective

To further facilitate substantive reasoning on the differences between the
RIAs and the natural effects, we illustrate some specific data generating
processes (DGPs) that would make the NDE coincide with the NDER or the NIE
with the NIER. We express these DGPs using structural equations (generative
models) with parametric constraints. Throughout this section, we allow for the
existence of $L$, such that the equivalence between the natural estimands and
the RIAs is not guaranteed by cross-world independence. We also do not
restrict the number or the distribution of mediators.

We first present results with assumed linearity and a randomized treatment,
which provides the easiest intuition. Then we extend the results to structural
equations without the linearity conditions and treatment randomization. For
comparison with parametric constraints below, we note that the nonparametric
structural equations with no constraints are as follows:

$\displaystyle C$ $\displaystyle=\epsilon_{C}$ $\displaystyle A$
$\displaystyle=g_{A}(C,\epsilon_{A})$ $\displaystyle L$
$\displaystyle=g_{L}(C,A,\epsilon_{L})$ $\displaystyle M$
$\displaystyle=g_{M}(C,A,L,\epsilon_{M})$ $\displaystyle Y$
$\displaystyle=g_{Y}(C,A,L,M,\epsilon_{Y}),$

where $g_{C}$, $g_{A}$, $g_{L}$, and $g_{M}$ are arbitrary functions of their
arguments. And $\epsilon_{C},\epsilon_{A},\epsilon_{L},\epsilon_{M}$ and
$\epsilon_{Y}$ are unspecified inputs for each variable. Importantly,
throughout this section, we allow these unspecified inputs to be arbitrarily
dependent on one another and all other variables.

###  5.1 Linear Structural Equations, Randomized Treatment

Since the treatment is assumed to be randomized, $C$ is empty. We consider the
structural equations for $A$, $L$, $M$, and $Y$. In this subsection, the
notation technically only applies to one $L$ and one $M$ variables, but our
expressions can be easily extended to accommodate multiple $L$ and $M$
variables without compromising intuition.

######  Proposition 4.

Under the following linear structural equations with constant coefficients
(i.e., all $\alpha$, $\beta$, $\gamma$ terms are constants),

$\displaystyle A$ $\displaystyle=\epsilon_{A}$ $\displaystyle L$
$\displaystyle=\alpha_{0}+\alpha_{1}A+\epsilon_{L}$ $\displaystyle M$
$\displaystyle=\beta_{0}+\beta_{1}A+\beta_{2}L+\beta_{3}AL+\epsilon_{M}$
$\displaystyle Y$
$\displaystyle=\gamma_{0}+\gamma_{1}A+\gamma_{2}L+\gamma_{3}M+\gamma_{4}AL+\gamma_{5}AM+\gamma_{6}LM+\gamma_{7}ALM+\epsilon_{Y},$

we have
$\text{NIE}-\text{NIE}^{R}=(\gamma_{6}+\gamma_{7})\beta_{3}\operatorname{Var}(\epsilon_{L})$,
and
$\text{NDE}-\text{NDE}^{R}=\gamma_{7}\beta_{2}\operatorname{Var}(\epsilon_{L})+\gamma_{7}\operatorname{Cov}(\epsilon_{L},\epsilon_{M})$.888Clearly,
Proposition 4 is a special case of Proposition 2. It is also easy to show that
when $M$ is binary, Proposition 4 recovers Proposition 1.

Hence, under the linear structural equations, there are multiple sufficient
conditions for either the NIE or the NDE to coincide with their respective
RIAs. For example, the NIE and the NIER are equivalent if $\beta_{3}=0$, i.e.,
there is no $AL$ interaction in the equation for $M$. And the NDE and the NDER
are equivalent if $\gamma_{7}=0$, i.e., there is no three-way interaction
$ALM$ in the equation for $Y$. In summary, equivalences can be established by
ruling out certain interaction effects.

It is possible to have only one of the NIE and the NDE coincide with their
RIA. When only one of the natural effects equal its RIA, our test statistic in
Section 3, $\text{TE}-\text{TE}^{R}$, will capture the deviaion of the other
natural effect from its RIA. The next subsection shows that the intuitions
from the linear analysis can be extended to the settings where the structural
equations are much more unrestricted.

###  5.2 Nonlinear Structural Equations, Nonrandomized Treatment

Throughout this subsection, we focus on constraints on the structural
equations for $Y$. Thus, we maintain completely unconstrained structural
equations for $C$, $A$, $L$, and $M$. Again, the structural equations we
consider allow $L$ to affect $M$ and $Y$ in some manner. Below, we let
$g_{Y1}$ and $g_{Y2}$ denote arbitrary functions of their arguments. Thus,
within these functions, the effects of the variables are left completely
unconstrained.

######  Proposition 5.

If $Y=(1-A)g_{Y1}(C,L,M,\epsilon_{Y1})+Ag_{Y2}(C,L,\epsilon_{Y2})$, NIE=NIER;
If $Y=g_{Y1}(C,A,L,\epsilon_{Y1})+g_{Y2}(C,M,\epsilon_{Y2})$, NDE=NDER.

The first structural equation rules out any effect of $M$ when $A=1$. The
second structural equation rules out $A$-$M$ and $L$-$M$ interactions in the
equation for $Y$, in the sense that the nonparametric function containing $M$
is additively separable from the nonparametric function containing $A$ and
$L$.

In summary, in the presence of treatment-induced confounders, it is still
possible to make $\text{NIE}=\text{NIE}^{R}$ or $\text{NDE}=\text{NDE}^{R}$.
However, these equivalences require imposing constrains on relevant structural
equations by ruling out interaction effects or two-sided effects. The
structural equation constraints we present are sufficient but not necessary to
establish equivalences between the natural effects and the RIA. Nevertheless,
they are derived with the goal of being maximally flexible, in the sense that
they allow as much complexity in functional form as possible without incurring
other strong constraints.

###  5.3 Summary

In this paper, we answer the question of when natural mediation estimands
differ from their randomized interventional analogues. In order to do so, we
provide tools for both empirical testing and theoretical reasoning to
researchers who wish to estimate and interpret the RIAs. Our test and theories
are complementary to one another: when the researcher empirically rejects the
null hypothesis of the test, they can conclude with confidence (subject to the
chosen significance level) that the natural effects and the RIAs _differ_ ;
when the researcher has theoretical support for specific structural equations,
they may reasonably posit that a particular natural effect and its
corresponding RIA are _equivalent_. With respect to the two theoretical
perspectives, the covariance perspective is complete, in the sense that it
provides necessary and sufficient conditions for the equivalence between the
natural effects and the RIAs; while the structural equation perspective
provides simple and intuitive conditions of equivalence even when $M$ is
vector-valued with arbitrary distributions.

##  6 Related Estimands

In causal inference, it is not unusual that a pair of competing estimands is
present, where one has a more natural interpretation and the other is easier
to identify. Apart from the natural mediation effects and their RIAs, we
discuss two other such pairs of estimands: the average treatment effect (ATE)
versus the local average treatment effect (LATE) in the IV context; and what
we call the natural Mann-Whitney estimand and its RIA. The theory we developed
for causal mediation analysis proves to be useful for unifying these three
long-standing literatures in causal inference. In particular, we establish a
formal equivalence result between estimands in the IV literature and the
mediation literature. And we reveal a striking resemblance in structure
between the Mann-Whitney estimands and the mediation estimands.

###  6.1 ATE and LATE

We first define the ATE and LATE estimands. In line with the notation we use
for causal mediation analysis, we consider three temporally ordered variables,
$A$, $M$, and $Y$. In the IV context, $A$ is the IV, $M$ is the treatment, and
$Y$ is the outcome. Here, we focus on the case where $A$ and $M$ are both
binary, and $A$ is randomized, which is a classic setting considered in the
modern IV literature (Angrist et al., 1996; Balke and Pearl, 1997). Then, the
ATE is defined as $\operatorname{E}(Y_{A=1}-Y_{A=0})$, and the LATE is defined
to be $\operatorname{E}(Y_{M=1}-Y_{M=0}\mid M_{A=1}=1,M_{A=0}=0)$, i.e., the
average effect of $M$ on $Y$ among those whose $M$ value is induced to
increase by an increase in $A$ (those who are the “compliers”). In this
subsection, we explicitly write the assignment variables in the potential
outcomes to avoid ambiguity. Also note that the labelling of the ATE and the
LATE involves a slight abuse of terminology in juxtaposition to mediation
estimands, as the “treatment” refers to $A$ in the IV context, while it refers
to $M$ in the mediation context.

In the IV context, the estimand with a more natural interpretation is ATE,
while LATE requires weaker identifying assumptions (Robins and Greenland,
1996; Aronow and Carnegie, 2013; Wang and Tchetgen Tchetgen, 2018). Just like
in the mediation context, applied researchers often interpret a LATE estimate
as if it was the ATE (Aronow and Carnegie, 2013; Sarvet et al., 2023). We show
that there exists a direct equivalence between $\text{ATE}-\text{LATE}$ and
$\text{NIE}-\text{NIE}^{R}$ under four standard identifying assumptions for
LATE: 1) Exclusion: $Y_{A=a,M=m}=Y_{M=m},\forall\\{a,m\\}$; 2) Independence:
$A\rotatebox[origin={c}]{90.0}{$\models$}\\{M_{A=1},M_{A=0},Y_{A=1},Y_{A=0}\\}$;
3) Relevance: $\operatorname{E}(M\mid A=1)-\operatorname{E}(M\mid A=0)>0$; and
4) Monotonicity: $M_{A=1}\geq M_{A=0}$. We also denote the identified
functional called the Wald estimand as
$\text{Wald}\vcentcolon=\frac{\operatorname{E}(Y\mid
A=1)-\operatorname{E}(Y\mid A=0)}{\operatorname{E}(M\mid
A=1)-\operatorname{E}(M\mid A=0)}$.

######  Proposition 6.

Under assumptions of exclusion, independence, and relevance,

$\text{Wald}-\text{ATE}=\frac{\operatorname{Cov}(M_{A=1}-M_{A=0},Y_{M=1}-Y_{M=0})}{\operatorname{E}(M_{A=1}-M_{A=0})}=\frac{\text{NIE}-\text{NIE}^{R}}{\operatorname{E}(M_{1}-M_{0})},$

which, further under monotonicity, also equals
$\text{LATE}-\text{ATE}$.999Also, by Proposition 1 and the exclusion
assumption, $\text{NIE}-\text{NIE}^{R}=\text{TE}-\text{TE}^{R}$.

Thus, under the four assumptions identifying the LATE, the difference between
the LATE and the ATE is simply the difference between the NIE and the NIER
scaled by the effect of $A$ on $M$. This means that, under these assumptions,
the LATE differs from the ATE if and only if the NIE differs the NIER.
Intuitively,
$\operatorname{Cov}(M_{A=1}-M_{A=0},Y_{M=1}-Y_{M=0})=\operatorname{Cov}[\mathbbm{1}(M_{A=1}=1,M_{A=0}=0),Y_{M=1}-Y_{M=0}]$
captures selection into the subpopulation of compliers based on the effect of
$M$ on $Y$. If there is strong selection, then the local average effect of $M$
on $Y$ among compliers must differ substantially from the corresponding global
average effect.

There is a long-standing literature on using the Wald estimand to estimate the
ATE based on exclusion, independence, relevance, and another additional
assumption (Heckman, 1997; Hernán and Robins, 2006; Wang and Tchetgen
Tchetgen, 2018).101010In fact, the identification results extend to the entire
potential outcome distributions (Mao, 2022). A weak form of the additional
assumption has recently appeared in Hernán and Robins (2020, Section 16.3) and
Hartwig et al. (2023), which can be written as
$\operatorname{Cov}(M_{A=1}-M_{A=0},Y_{M=1}-Y_{M=0})=0$. Proposition 9 shows
that this is, in fact, the weakest possible among such assumptions.

###  6.2 Natural Mann-Whitney estimand and its RIA

We define the natural Mann-Whitney estimand as
$\operatorname{E}[\mathbbm{1}(Y_{1}\geq Y_{0})]$, i.e., the probability of the
potential outcome under treatment being greater or equal to the potential
outcome under control. It is often referred to as the probability of no harm
(the probability of the treatment not worsening the outcome), given that a
larger value of $Y$ is desired. This estimand is broadly useful for rank-based
evaluation of treatment effects, especially for noncontinuous ordinal
outcomes.111111A related estimand, $\operatorname{Pr}(Y_{1}>Y_{0}\mid
A=1)/\operatorname{Pr}(Y_{1}=1\mid A=1)$, for a binary $Y$, is called the
probability of necessity (Tian and Pearl, 2000). We call this estimand a
“natural” estimand, because it is an aggregation of an individual-level
contrast of potential outcomes.

The natural Mann-Whitney estimand is difficult to identify for the same reason
that the NIE and the NDE are difficult to identify: just as
$\operatorname{E}(Y_{1,M_{0}})$, the natural Mann-Whitney estimand involves
the assignment of two different treatment values to the same individual. Due
to the fundamental problem of causal inference (Holland, 1986), the joint
distribution of two potential outcomes is impossible to identify even with a
randomized treatment. Hence, an assumption analogous to cross-world
independence (Assumption 5) can also be used to identify the natural Mann-
Whitney estimand: $Y_{1}\rotatebox[origin={c}]{90.0}{$\models$}Y_{0}$
(Greenland et al., 2020). However, this assumption is clearly unlikely to
hold.

Consequently, an alternative estimand has been used in practice:
$\operatorname{E}[\mathbbm{1}(H_{1}\geq H_{0})]$, where $H_{a}$ is a value
randomly drawn from the marginal distribution of $Y_{a}$. Clearly, this
alternative estimand has the interpretation of a RIA. In contrast to the
natural Mann-Whitney estimand, the Mann-Whitney RIA does not aggregate an
individual-level contrast. On the other hand, randomization of treatment does
enable the identification of the Mann-Whitney RIA. The Mann-Whitney RIA has a
long history in statistics, dating back to the Mann-Whitney $U$ test (Mann and
Whitney, 1947) and the Wilcoxon rank-sum test (Wilcoxon, 1945). Recent
methodological development based on the Mann-Whitney RIA includes the
probability index model (Thas et al., 2012), the win ratio (Pocock et al.,
2012; Mao, 2018), and the rank average treatment effect (Lei, 2024).

Similar to the mediation literature, conflation of the natural Mann-Whitney
estimand and its RIA is pervasive even in methodological work. For example, in
a textbook discussion on the Mann-Whitney RIA, Thas (2010) claims that “If
this conclusion is statistically significant, it is very relevant evidence to
a physician that most of his patients will be better off with the treatment.”
Wu et al. (2014) states “This allows us to make inference about the potential
outcome-based $\delta$ through the estimable quantity $\xi$…”, where $\delta$
and $\xi$ are respectively the natural Mann-Whitney estimand and its RIA. And
Demidenko (2016) names the Mann-Whitney RIA the “$D$-value” and argues that
“The $D$-value has a clear interpretation as the proportion of patients who
get worse after the treatment”, in the context where a smaller value of a
continuous $Y$ is desirable.

Interestingly, despite recurrent confusion, the literature on Mann-Whitney
estimands has been clarifying the important differences between the natural
Mann-Whitney estimand and its RIA since decades before Miles (2023) pioneered
an analogous inquiry in causal mediation analysis. The early work of Hand
(1992) already notes the possibility of sign reversal in the relationship
between the natural Mann-Whitney estimand and its RIA (when $1/2$ is
subtracted from both), which has been known as the Hand’s paradox. Multiple
work since has considered various DGPs under which the Hand’s paradox is
present or absent (Hand, 1992; Fay et al., 2018; Greenland et al., 2020). This
line of work is in the same spirit as our theoretical analysis on the
relationship between the natural mediation estimands and their RIAs.

Lastly, we note that there is also a covariance representation for the
difference between the natural Mann-Whitney estimand and its RIA.

######  Proposition 7.

$\displaystyle\operatorname{E}[\mathbbm{1}(Y_{1}\geq
Y_{0})]-\operatorname{E}[\mathbbm{1}(H_{1}\geq
H_{0})]=\sum_{t\in\mathcal{T}}\sum_{s\in\mathcal{S}}\mathbbm{1}(t\geq
s)\operatorname{Cov}[\mathbbm{1}(Y_{1}=t),\mathbbm{1}(Y_{0}=s)],$

where $\mathcal{T}$ and $\mathcal{S}$ are respectively the supports of $Y_{1}$
and $Y_{0}$. When $Y$ is binary with the support of $\\{0,1\\}$, the
expression simplifies to $\operatorname{Cov}(Y_{1},Y_{0})$.

Clearly, the natural Mann-Whitney estimand differs from its RIA to the extent
that $Y_{1}$ and $Y_{0}$ are dependent on each other. This is in parallel to
the natural mediation effects differing from their RIAs to the extent that
$M_{a}$ and $Y_{a^{\prime},m}$ are dependent. By redefining the estimands
using random draws, RIAs in both cases miss a naturally occurring dependency.
The thorny issue created by cross-world treatment assignments for
identification cannot be magically waved away by redefining the estimand.

###  6.3 Summary

The dilemma facing researchers in all these three fields of causal inference
(causal mediation analysis, instrumental variable, and Mann-Whitney estimands)
is that a natural estimand is more interpretable but hard to identify while an
alternative estimand is less interpretable but easier to identify. Going
forward, we recommend four strategies to applied researchers. First, we join
Sarvet et al. (2023) to call for more clarity in interpreting estimates of the
alternative estimands in all three areas. Second, with the addition of our two
theoretical perspectives in this paper, now researchers in all three areas are
able to reason about when the natural estimand coincides with at least does
not have the opposite sign to the alternative estimand. Third, in all three
areas, bounding methods have been developed to provide partial identification
for the natural estimands (e.g., Miles et al., 2017; Swanson et al., 2018; Lu
et al., 2020). Fourth, in causal mediation analysis, we uniquely also provide
a falsification test for interpreting the RIAs as the natural mediation
effects, which goes beyond theoretical reasoning and provides empirical
guidance.

## Acknowledgement

We are grateful for a comment from a reviewer at the Annals of Applied
Statistics for Yu and Elwert (2024), which inspired us to start this project.
We also thank Sameer Deshpande, Hyunseung Kang, Xinran Miao, Chan Park, and
Michael Sobel for helpful suggestions. An earlier version of this paper was
presented at the American Causal Inference Conference in 2024. We thank the
audience for an engaging discussion.

## Appendices

### A1. Proof of Proposition 1

The NIE and NDE are defined in terms of
$\operatorname{E}(Y_{a,M_{a^{\prime}}})$ for two treatment values
($a,a^{\prime}$). When $M$ is binary and its support is $\\{0,1\\}$, we
rewrite this quantity just using its definition:

$\displaystyle\phantom{{}={}}\operatorname{E}(Y_{a,M_{a^{\prime}}})$
$\displaystyle=\operatorname{E}[Y_{a,1}M_{a^{\prime}}+Y_{a,0}(1-M_{a^{\prime}})]$
$\displaystyle=\operatorname{E}(Y_{a,0})+\operatorname{E}[M_{a^{\prime}}(Y_{a,1}-Y_{a,0})]$
$\displaystyle=\operatorname{E}(Y_{a,0})+\operatorname{E}\\{\operatorname{E}[M_{a^{\prime}}(Y_{a,1}-Y_{a,0})\mid
C]\\}.$

The NIER and NDER are defined in terms of
$\operatorname{E}(Y_{a,G_{a^{\prime}}})$ for two treatment values
($a,a^{\prime}$). When $M$ is binary, we again rewrite this quantity using its
definition:

$\displaystyle\phantom{{}={}}\operatorname{E}(Y_{a,G_{a^{\prime}}})$
$\displaystyle=\operatorname{E}[\operatorname{E}(Y_{a,G_{a^{\prime}}}\mid C)]$
$\displaystyle=\operatorname{E}[\operatorname{E}(Y_{a,1}\mid
G_{a^{\prime}}=1,C)\operatorname{Pr}(G_{a^{\prime}}=1\mid
C)+\operatorname{E}(Y_{a,0}\mid
G_{a^{\prime}}=0,C)\operatorname{Pr}(G_{a^{\prime}}=0\mid C)]$
$\displaystyle=\operatorname{E}\\{\operatorname{E}(Y_{a,1}\mid
C)\operatorname{E}(M_{a^{\prime}}\mid C)+\operatorname{E}(Y_{a,0}\mid
C)[1-\operatorname{E}(M_{a^{\prime}}\mid C)]\\}$
$\displaystyle=\operatorname{E}(Y_{a,0})+\operatorname{E}\\{\operatorname{E}(M_{a^{\prime}}\mid
C)[\operatorname{E}(Y_{a,1}-Y_{a,0}\mid C)]\\}$
$\displaystyle=\operatorname{E}(Y_{a,M_{a^{\prime}}})-\operatorname{E}[\operatorname{Cov}(M_{a^{\prime}},Y_{a,1}-Y_{a,0}\mid
C)].$

Then using the results above, we have the following representations:

NIE
$\displaystyle=\operatorname{E}(Y_{1,M_{1}}-Y_{1,M_{0}})=\operatorname{E}[(M_{1}-M_{0})(Y_{1,1}-Y_{1,0})]$
$\displaystyle\text{NIE}^{R}$
$\displaystyle=\operatorname{E}(Y_{1,G_{1}}-Y_{1,G_{0}})=\operatorname{E}[\operatorname{E}(M_{1}-M_{0}\mid
C)\operatorname{E}(Y_{1,1}-Y_{1,0}\mid C)]$ NDE
$\displaystyle=\operatorname{E}(Y_{1,M_{0}}-Y_{0,M_{0}})=\operatorname{E}(Y_{1,0}-Y_{0,0})+\operatorname{E}\\{M_{0}[Y_{1,1}-Y_{1,0}-(Y_{0,1}-Y_{0,0})]\\}$
$\displaystyle\text{NDE}^{R}$
$\displaystyle=\operatorname{E}(Y_{1,G_{0}}-Y_{0,G_{0}})=\operatorname{E}(Y_{1,0}-Y_{0,0})+\operatorname{E}\\{\operatorname{E}(M_{0}\mid
C)\operatorname{E}[Y_{1,1}-Y_{1,0}-(Y_{0,1}-Y_{0,0})\mid C]\\}.$

Hence,

NIE
$\displaystyle=\text{NIE}^{R}+\operatorname{E}[\operatorname{Cov}(M_{1}-M_{0},Y_{1,1}-Y_{1,0}\mid
C)]$ NDE
$\displaystyle=\text{NDE}^{R}+\operatorname{E}\\{\operatorname{Cov}[M_{0},Y_{1,1}-Y_{1,0}-(Y_{0,1}-Y_{0,0})\mid
C]\\}$ TE
$\displaystyle=\text{TE}^{R}+\operatorname{E}[\operatorname{Cov}(M_{1},Y_{1,1}-Y_{1,0}\mid
C)-\operatorname{Cov}(M_{0},Y_{0,1}-Y_{0,0}\mid C)].$

When the treatment is randomized, $C$ becomes an empty set, and we obtain the
results shown in Proposition 1.

### A2. Proof of Propositions 2 and 3

The NIE and NDE are still defined in terms of
$\operatorname{E}(Y_{a,M_{a^{\prime}}})$ for two treatment values
($a,a^{\prime}$). Treating $M$ as a vector of continuous variable, we rewrite
this quantity using its definition:

$\displaystyle\phantom{{}={}}\operatorname{E}(Y_{a,M_{a^{\prime}}})$
$\displaystyle=\operatorname{E}\left[\int
Y_{a,m}\mathbbm{1}(M_{a^{\prime}}=m)\differential m\right]$
$\displaystyle=\int\operatorname{E}[Y_{a,m}\mathbbm{1}(M_{a^{\prime}}=m)]\differential
m$
$\displaystyle=\int\operatorname{E}\\{\operatorname{E}[Y_{a,m}\mathbbm{1}(M_{a^{\prime}}=m)\mid
C]\\}\differential m,$

where the first equality holds by treating the Dirac delta function
$\mathbbm{1}(M_{a^{\prime}}=m)$ as a limiting case of a probability density
function concentrated at $M_{a^{\prime}}=m$. This allows us to express a
function of $M_{a^{\prime}}$ as an integral over the support of
$M_{a^{\prime}}$.

The NIER and NDER are defined in terms of
$\operatorname{E}(Y_{a,G_{a^{\prime}}})$ for two treatment values
($a,a^{\prime}$). We rewrite this quantity as follows:

$\displaystyle\phantom{{}={}}\operatorname{E}(Y_{a,G_{a^{\prime}}})$
$\displaystyle=\operatorname{E}[\operatorname{E}(Y_{a,G_{a^{\prime}}}\mid C)]$
$\displaystyle=\iint\operatorname{E}(Y_{a,m}\mid
G_{a^{\prime}}=m,C=c)f_{G_{a^{\prime}}\mid c}(m)f_{C}(c)\differential
m\differential c$ $\displaystyle=\iint\operatorname{E}(Y_{a,m}\mid
C=c)f_{M_{a^{\prime}}\mid c}(m)f_{C}(c)\differential m\differential c$
$\displaystyle=\iint\operatorname{E}(Y_{a,m}\mid
C=c)\operatorname{E}[\mathbbm{1}(M_{a^{\prime}}=m)\mid
C=c]f_{C}(c)\differential m\differential c,$

where the last equality is by the property of the Dirac delta function
$\mathbbm{1}(M_{a^{\prime}}=m)$.

Therefore,

NIE
$\displaystyle=\operatorname{E}(Y_{1,M_{1}}-Y_{1,M_{0}})=\int\operatorname{E}\\{\operatorname{E}\\{[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m)]Y_{1,m}\\}\mid
C\\}\differential m$ $\displaystyle\text{NIE}^{R}$
$\displaystyle=\operatorname{E}(Y_{1,G_{1}}-Y_{1,G_{0}})=\int\operatorname{E}\\{\operatorname{E}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m)\mid
C]\operatorname{E}(Y_{1,m}\mid C)\\}\differential m$ NDE
$\displaystyle=\operatorname{E}(Y_{1,M_{0}}-Y_{0,M_{0}})=\int\operatorname{E}\\{\operatorname{E}[(Y_{1,m}-Y_{0,m})\mathbbm{1}(M_{0}=m)\mid
C]\\}\differential m$ $\displaystyle\text{NDE}^{R}$
$\displaystyle=\operatorname{E}(Y_{1,G_{0}}-Y_{0,G_{0}})=\int\operatorname{E}\\{\operatorname{E}[(Y_{1,m}-Y_{0,m})\mid
C]\operatorname{E}[\mathbbm{1}(M_{0}=m)\mid C]\\}\differential m.$

And

NIE
$\displaystyle=\text{NIE}^{R}+\int\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}\differential m$ NDE
$\displaystyle=\text{NDE}^{R}+\int\operatorname{E}\\{\operatorname{Cov}[Y_{1,m}-Y_{0,m},\mathbbm{1}(M_{0}=m)\mid
C]\\}\differential m$ TE
$\displaystyle=\text{TE}^{R}+\int\operatorname{E}\\{\operatorname{Cov}[Y_{1,m},\mathbbm{1}(M_{1}=m)\mid
C]\\}-\operatorname{E}\\{\operatorname{Cov}[Y_{0,m},\mathbbm{1}(M_{0}=m)\mid
C]\\}\differential m.$

When $M$ is a vector of discrete variables, we replace the integrals with
summations to obtain the results in Proposition 2.

Proposition 3 similarly follows from the expressions of
$\operatorname{E}(Y_{a,M_{a^{\prime}}})$ and
$\operatorname{E}(Y_{a,G_{a^{\prime}}})$ derived above.

### A3. Proof of Proposition 4

We let $L_{a}$ denote the potential values of $L$ under treatment assignment
$a$. Under the structural equations of Proposition 4,

$\displaystyle Y_{1M_{1}}$
$\displaystyle=\gamma_{0}+\gamma_{1}+(\gamma_{2}+\gamma_{4})L_{1}+(\gamma_{3}+\gamma_{5})M_{1}+(\gamma_{6}+\gamma_{7})L_{1}M_{1}+\epsilon_{Y}$
$\displaystyle Y_{1M_{0}}$
$\displaystyle=\gamma_{0}+\gamma_{1}+(\gamma_{2}+\gamma_{4})L_{1}+(\gamma_{3}+\gamma_{5})M_{0}+(\gamma_{6}+\gamma_{7})L_{1}M_{0}+\epsilon_{Y}$
$\displaystyle Y_{0M_{0}}$
$\displaystyle=\gamma_{0}+\gamma_{2}L_{0}+\gamma_{3}M_{0}+\gamma_{6}L_{0}M_{0}+\epsilon_{Y}$
$\displaystyle Y_{1G_{1}}$
$\displaystyle=\gamma_{0}+\gamma_{1}+(\gamma_{2}+\gamma_{4})L_{1}+(\gamma_{3}+\gamma_{5})G_{1}+(\gamma_{6}+\gamma_{7})L_{1}G_{1}+\epsilon_{Y}$
$\displaystyle Y_{1G_{0}}$
$\displaystyle=\gamma_{0}+\gamma_{1}+(\gamma_{2}+\gamma_{4})L_{1}+(\gamma_{3}+\gamma_{5})G_{0}+(\gamma_{6}+\gamma_{7})L_{1}G_{0}+\epsilon_{Y}$
$\displaystyle Y_{0G_{0}}$
$\displaystyle=\gamma_{0}+\gamma_{2}L_{0}+\gamma_{3}G_{0}+\gamma_{6}L_{0}G_{0}+\epsilon_{Y}.$

Hence,

NDE
$\displaystyle=\gamma_{1}+(\gamma_{2}+\gamma_{4})\operatorname{E}(L_{1})-\gamma_{2}\operatorname{E}(L_{0})+\gamma_{5}\operatorname{E}(M_{0})+(\gamma_{6}+\gamma_{7})\operatorname{E}(L_{1}M_{0})-\gamma_{6}\operatorname{E}(L_{0}M_{0})$
$\displaystyle\text{NDE}^{R}$
$\displaystyle=\gamma_{1}+(\gamma_{2}+\gamma_{4})\operatorname{E}(L_{1})-\gamma_{2}\operatorname{E}(L_{0})+\gamma_{5}\operatorname{E}(G_{0})+(\gamma_{6}+\gamma_{7})\operatorname{E}(L_{1}G_{0})-\gamma_{6}\operatorname{E}(L_{0}G_{0})$
NIE
$\displaystyle=(\gamma_{3}+\gamma_{5})\operatorname{E}(M_{1}-M_{0})+(\gamma_{6}+\gamma_{7})\operatorname{E}(L_{1}M_{1}-L_{1}M_{0})$
$\displaystyle\text{NIE}^{R}$
$\displaystyle=(\gamma_{3}+\gamma_{5})\operatorname{E}(G_{1}-G_{0})+(\gamma_{6}+\gamma_{7})\operatorname{E}(L_{1}G_{1}-L_{1}G_{0}).$

Noting that $\operatorname{E}(M_{a})=\operatorname{E}(G_{a})$, and

$\displaystyle\phantom{{}={}}\operatorname{E}(L_{a}M_{a^{\prime}})-\operatorname{E}(L_{a}G_{a^{\prime}})$
$\displaystyle=\operatorname{E}(L_{a}M_{a^{\prime}})-\operatorname{E}(L_{a})\operatorname{E}(G_{a^{\prime}})$
$\displaystyle=\operatorname{Cov}(L_{a},M_{a^{\prime}})$
$\displaystyle=\operatorname{Cov}[\alpha_{0}+\alpha_{1}a+\epsilon_{L},\beta_{0}+\beta_{1}a^{\prime}+\beta_{2}(\alpha_{0}+\alpha_{1}a^{\prime}+\epsilon_{L})+\beta_{3}a^{\prime}(\alpha_{0}+\alpha_{1}a^{\prime}+\epsilon_{L})+\epsilon_{M}]$
$\displaystyle=(\beta_{2}+\beta_{3}a^{\prime})\operatorname{Var}(\epsilon_{L})+\operatorname{Cov}(\epsilon_{L},\epsilon_{M}).$

we have

$\displaystyle\text{NDE}-\text{NDE}^{R}$
$\displaystyle=(\gamma_{6}+\gamma_{7})\operatorname{Cov}(L_{1},M_{0})-\gamma_{6}\operatorname{Cov}(L_{0},M_{0})$
$\displaystyle=\gamma_{7}\beta_{2}\operatorname{Var}(\epsilon_{L})+\gamma_{7}\operatorname{Cov}(\epsilon_{L},\epsilon_{M})$
$\displaystyle\text{NIE}-\text{NIE}^{R}$
$\displaystyle=(\gamma_{6}+\gamma_{7})\\{\operatorname{Cov}(L_{1},M_{1})-\operatorname{Cov}(L_{1},M_{0})\\}$
$\displaystyle=(\gamma_{6}+\gamma_{7})\beta_{3}\operatorname{Var}(\epsilon_{L}).$

### A4. Proof of Proposition 5

For the NDE part, our proof leverages an assumption in Robins (2003):
$Y_{1,m}-Y_{0,m}$ is a random variable not dependent on $m$. Originally, this
assumption was proposed to identify NDE in the presence of treatment-induced
confounding. We first prove that this assumption is sufficient for
$\text{NDE}=\text{NDE}^{R}$. Then we prove that the structural equation in
Proposition 5 is, in turn, sufficient for this assumption to hold.

According to our Proposition 2, we just need to show that under the assumption
of Robins (2003),
$\int\operatorname{E}\\{\operatorname{Cov}[Y_{1,m}-Y_{0,m},\mathbbm{1}(M_{0}=m)\mid
C]\\}\differential m=0$. Let $Y_{1,m}-Y_{0,m}=B$, then,

$\displaystyle\phantom{{}={}}\int\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{0}=m),Y_{1,m}-Y_{0,m}\mid
C]\\}\differential m$
$\displaystyle=\int\operatorname{E}\\{\operatorname{E}[\mathbbm{1}(M_{0}=m)B\mid
C]-\operatorname{E}[\mathbbm{1}(M_{0}=m)\mid C]\operatorname{E}(B\mid
C)\\}\differential m$
$\displaystyle=\operatorname{E}\left\\{\operatorname{E}\left[\int\mathbbm{1}(M_{0}=m)\differential
mB\mid C\right]-\int f_{M_{0}}(m\mid C)\differential m\operatorname{E}(B\mid
C)\right\\}$ $\displaystyle=\operatorname{E}[\operatorname{E}(B\mid
C)-\operatorname{E}(B\mid C)]=0.$

Next, we show that, if
$Y=g_{Y1}(C,A,L,\epsilon_{Y1})+g_{Y2}(C,M,\epsilon_{Y2})$, the assumption of
Robins (2003) is satisfied. Under this structural equation for $Y$,

$\displaystyle\phantom{{}={}}Y_{1,m}-Y_{0,m}$
$\displaystyle=g_{Y1}(C,1,g_{L}(C,1,\epsilon_{L}),\epsilon_{Y1})+g_{Y2}(C,m,\epsilon_{Y2})-g_{Y1}(C,0,g_{L}(C,0,\epsilon_{L}),\epsilon_{Y1})-g_{Y2}(C,m,\epsilon_{Y2})$
$\displaystyle=g_{Y1}(C,1,g_{L}(C,1,\epsilon_{L}),\epsilon_{Y1})-g_{Y1}(C,0,g_{L}(C,0,\epsilon_{L}),\epsilon_{Y1}),$

which is not dependent on $m$.

For the NIE part, we propose a novel condition that is analogous to the
assumption of Robins (2003) used above: $Y_{1,m}$ is a random variable not
dependent on $m$. We refer to this condition as the analogous assumption. We
first show that the analogous assumption is sufficient for NIE to be equal to
NIER. According to Proposition 2, it suffices to show
$\int\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}\differential m=0$. Let $Y_{1,m}=B$, then under the analogous assumption,

$\displaystyle\phantom{{}={}}\int\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m),Y_{1,m}\mid
C]\\}\differential m$
$\displaystyle=\int\operatorname{E}\\{\operatorname{Cov}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m),B\mid
C]\\}\differential m$
$\displaystyle=\int\operatorname{E}\\{\operatorname{E}[\mathbbm{1}(M_{1}=m)B\mid
C]-\operatorname{E}[\mathbbm{1}(M_{0}=m)B\mid C]$
$\displaystyle\phantom{{}={}}-\operatorname{E}[\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m)\mid
C]\operatorname{E}(B\ \mid C)\\}\differential m$
$\displaystyle=\operatorname{E}\\{\operatorname{E}[\int\mathbbm{1}(M_{1}=m)\differential
mB\mid C]-\operatorname{E}[\int\mathbbm{1}(M_{0}=m)\differential mB\mid C]$
$\displaystyle\phantom{{}={}}-\operatorname{E}[\int\mathbbm{1}(M_{1}=m)-\mathbbm{1}(M_{0}=m)\differential
m\mid C]\operatorname{E}(B\ \mid C)\\}$
$\displaystyle=\operatorname{E}\\{\operatorname{E}[B\mid
C]-\operatorname{E}[B\mid C]\\}$ $\displaystyle=0.$

Then, we show that if
$Y=(1-A)g_{Y1}(C,L,M,\epsilon_{Y1})+Ag_{Y2}(C,L,\epsilon_{Y2})$, the analogous
assumption is satisfied. Under this structural equation,
$Y_{1,m}=g_{Y2}(C,g_{L}(C,1,\epsilon_{L}),\epsilon_{Y2})$, which clearly does
not depend on $m$.

### A5. Proof of Proposition 6

Wald
$\displaystyle=\frac{\operatorname{E}(Y_{A=1}-Y_{A=0})}{\operatorname{E}(M_{A=1}-M_{A=0})}$
$\displaystyle=\frac{\operatorname{E}[(M_{A=1}-M_{A=0})(Y_{M=1}-Y_{M=0})]}{\operatorname{E}(M_{A=1}-M_{A=0})}$
$\displaystyle=\frac{\operatorname{E}(M_{A=1}-M_{A=0})\operatorname{E}(Y_{M=1}-Y_{M=0})+\operatorname{Cov}(M_{A=1}-M_{A=0},Y_{M=1}-Y_{M=0})}{\operatorname{E}(M_{A=1}-M_{A=0})}$
$\displaystyle=\text{ATE}+\frac{\text{NIE}-\text{NIE}^{R}}{\operatorname{E}(M_{A=1}-M_{A=0})}.$

The first equality is by the independence assumption, the second is by the
exclusion assumption (equation 9 in Angrist et al. (1996)), the third is by
the definition of covariance, the fourth is by Proposition 1 and the exclusion
assumption. The relevance assumption makes sure the denominator is nonzero.
Finally, under assumptions of exclusion, independence, relevance, and
monotonicity, the classic result of Angrist et al. (1996) equates Wald with
LATE.

### A6. Proof of Proposition 7

$\displaystyle\phantom{{}={}}\operatorname{E}[\mathbbm{1}(Y_{1}\geq
Y_{0})]-\operatorname{E}[\mathbbm{1}(H_{1}\geq H_{0})]$
$\displaystyle=\iint\mathbbm{1}(t\geq s)f_{Y_{1},Y_{0}}(t,s)\differential
t\differential s-\iint\mathbbm{1}(t\geq s)f_{H_{1},H_{0}}(t,s)\differential
t\differential s$ $\displaystyle=\iint\mathbbm{1}(t\geq
s)f_{Y_{1},Y_{0}}(t,s)\differential t\differential s-\iint\mathbbm{1}(t\geq
s)f_{H_{1}}(t)f_{H_{0}}(s)\differential t\differential s$
$\displaystyle=\iint\mathbbm{1}(t\geq s)f_{Y_{1},Y_{0}}(t,s)\differential
t\differential s-\iint\mathbbm{1}(t\geq
s)f_{Y_{1}}(t)f_{Y_{0}}(s)\differential t\differential s$
$\displaystyle=\iint\mathbbm{1}(t\geq
s)\operatorname{E}[\mathbbm{1}(Y_{1}=t)\mathbbm{1}(Y_{0}=s)]\differential
t\differential s-\iint\mathbbm{1}(t\geq
s)\operatorname{E}[\mathbbm{1}(Y_{1}=t)]\operatorname{E}[\mathbbm{1}(Y_{0}=s)]\differential
t\differential s$ $\displaystyle=\iint\mathbbm{1}(t\geq
s)\operatorname{Cov}[\mathbbm{1}(Y_{1}=t),\mathbbm{1}(Y_{0}=s)]\differential
t\differential s.$

When $Y$ is discrete, this becomes the expression in Proposition 7.
Furthermore, when the support of $Y$ is $\\{0,1\\}$,

$\displaystyle\phantom{{}={}}\sum_{t\in\mathcal{T}}\sum_{s\in\mathcal{S}}\mathbbm{1}(t\geq
s)\operatorname{Cov}[\mathbbm{1}(Y_{1}=t),\mathbbm{1}(Y_{0}=s)]$
$\displaystyle=\operatorname{Cov}[\mathbbm{1}(Y_{1}=1),\mathbbm{1}(Y_{0}=1)]+\operatorname{Cov}[\mathbbm{1}(Y_{1}=1),\mathbbm{1}(Y_{0}=0)]+\operatorname{Cov}[\mathbbm{1}(Y_{1}=0),\mathbbm{1}(Y_{0}=0)]$
$\displaystyle=\operatorname{E}[\mathbbm{1}(Y_{1}=1)\mathbbm{1}(Y_{0}=1)]-\operatorname{E}[\mathbbm{1}(Y_{1}=1)]\operatorname{E}[\mathbbm{1}(Y_{0}=1)]$
$\displaystyle\phantom{{}={}}+\operatorname{E}[\mathbbm{1}(Y_{1}=1)\mathbbm{1}(Y_{0}=0)]-\operatorname{E}[\mathbbm{1}(Y_{1}=1)]\operatorname{E}[\mathbbm{1}(Y_{0}=0)]$
$\displaystyle\phantom{{}={}}+\operatorname{E}[\mathbbm{1}(Y_{1}=0)\mathbbm{1}(Y_{0}=0)]-\operatorname{E}[\mathbbm{1}(Y_{1}=0)]\operatorname{E}[\mathbbm{1}(Y_{0}=0)]$
$\displaystyle=\operatorname{E}(Y_{1}Y_{0})-\operatorname{E}(Y_{1})\operatorname{E}(Y_{0})$
$\displaystyle\phantom{{}={}}+\operatorname{E}[Y_{1}(1-Y_{0})]-\operatorname{E}(Y_{1})[1-\operatorname{E}(Y_{0})]+\operatorname{E}[(1-Y_{1})(1-Y_{0})]-\operatorname{E}[(1-Y_{1})]\operatorname{E}[(1-Y_{0})]$
$\displaystyle=\operatorname{E}(Y_{1}Y_{0})-\operatorname{E}(Y_{1})\operatorname{E}(Y_{0})=\operatorname{Cov}(Y_{1},Y_{0}).$

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