Identification of causal direct-indirect effects without untestable assumptions
IIdentification of causal direct-indirect effectswithout untestable assumptions
Takahiro Hoshino
TAKAHIRO HOSHINODepartment of Economics, Keio UniversityRIKEN Center for Advanced Intelligence ProjectE-mail: [email protected] causal mediation analysis, identification of existing causal direct or indirecteffects requires untestable assumptions in which potential outcomes and potentialmediators are independent. This paper defines a new causal direct and indirect effectthat does not require the untestable assumptions. We show that the proposed mea-sure is identifiable from the observed data, even if potential outcomes and potentialmediators are dependent, while the existing natural direct or indirect effects may finda pseudo-indirect effect when the untestable assumptions are violated.Keywords: causal effect, causal mediation, mediation analysis, natural indirecteffect, potential outcome, a r X i v : . [ s t a t . M E ] S e p Introduction
In recent years, there has been considerable methodological development and appliedstudies based on potential outcome approaches (Rubin, 1974) to causal mediationanalysis to understand causal mechanisms (for example, Pearl 2001; Van der Weele2009; Imai et al. 2010a; Tchetgen Tchetgen & Shpitser 2012; Ding & Van der Weele2016; Miles et al. 2020). Let T denote the exposure or treatment of interest, Y theoutcome, M the mediator, and the baseline covariates v , which are not affected by theexposure and mediator. Following the potential outcome approach, let Y j ( m ) be thepotential outcome when T = j and M = m .Most recent studies in causal mediation analysis consider the natural direct/indirecteffects, which are defined using the expectation of the “never-observed” outcome (not“potential outcome”) Y j ( M k ) ( j (cid:54) = k ) , which is the potential outcome under treatment j for the potential mediator for treatment k , M k . To identify these effects various as-sumptions are proposed. The following assumptions (Pearl, 2001) are often made: Assumption 1 Y j ( k ) ⊥⊥ T | v , ∀ j , k , Assumption 2 M j ⊥⊥ T | v , ∀ j , Assumption 3 Y j ( k ) ⊥⊥ M | T , v , ∀ j , k , Assumption 4 Y j ( k ) ⊥⊥ M j ∗ | v , ∀ j , j ∗ , k , Another example for sufficient conditions for identifying the two natural effects in-cludes the following sequential ignorability conditions (SI1 and SI2) by Imai et al.(2010b):
Assumption SI1 { Y j ( k ) , M j ∗ } ⊥⊥ T | v , ∀ j , j ∗ , k , Assumption SI2 Y j ( k ) ⊥⊥ M j ∗ | T = j ∗ , v , ∀ 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 as-sumption SI1 are satisfied if T is randomized. Assumption 4 or SI2 is not testablein that this states independence of potential outcomes and potential mediators, some1f which we never observe simultaneously. These assumptions do not hold even ifboth T and M are randomized or ignorable given v (Pearl, 2014), while Assumption3 holds.This paper defines new causal mediation effects that are identifiable from ob-served data without the untastable assumptions when both T and M are randomizedor ignorable given v . The proposed ones are useful even if the ranzomization for themediator is not possible in that the assumptions required for identification are weakerthan 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 = M 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 potentialmediators 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 byImbens 2000). Using two potential mediators M (for T = M (for T = M is expressed as M = T M + ( − T ) M . (1)We assume that M is a categorical variable (i.e., M = , · · · , M ∗ ). Let M ( m ) be thebinary indicator such that M ( m ) = M = m . Similarly, let M j ( m ) be the binaryindicator under T = j treatment, M j ( m ) = j , M j , is m .The observed outcome Y is expressed by the potential outcomes, potential medi-ators, and treatment indicator as follows: Y = M ∗ ∑ m = (cid:104) T M ( m ) Y ( m )+( − T ) M ( m ) Y ( m ) (cid:105) = M ∗ ∑ m = (cid:104) T M ( m ) Y ( m )+( − T ) M ( m ) Y ( m ) (cid:105) (2)2he observed outcome can be expressed by two potential outcomes Y = Y ( M ) (for T = Y ( M ) (for T = Y = TY + ( − T ) Y = TY ( M ) + ( − T ) Y ( M ) (3)under the composition assumption (Pearl, 2009).The average treatment effect ( AT E ) is defined as the expectation of the differencebetween two potential outcomes:
AT E ≡ E [ Y − Y ] , (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 mediatorfixed at M = m , CDE ( m ) , in which the mediator is set to m uniformly over the entirepopulation: ICDE ( m ) ≡ Y ( m ) − Y ( m ) , CDE ( m ) ≡ E [ ICDE ( m )] . (5)where ICDE ( m ) is an unit-level version of the CDE ( m ) that we define here to uselater in this paper. As pointed out in existing studies, the quantity defined as AT E − CDE is not a proper measure of indirect effect in that this quantity may not be zeroeven when M = M 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 thenatural indirect effect (
NIE ), instead of using
CDE . Following Imai et al. (2010b),
NDE ( t ) ≡ E [ Y ( M t ) − Y ( M t )] , NIE ( t ) ≡ E [ Y t ( M ) − Y t ( M )] , ( t = , ) NDE ≡ [ NDE ( ) + NDE ( )] NIE ≡ [ NIE ( ) + NIE ( )] AT E = E [ Y − Y ] = E [ Y ( M ) − Y ( M )] = NDE + NIE . (6)Note that the NDE and NIE are not identified without further assumptions be-cause quantity Y ( M ) is not observable. For identification, the existing studies as-sume independence between potential outcomes and mediator (given some observ-able 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 whenthe assumption is violated while the proposed one is not.3 Definition of weighted direct effect and estimable in-direct effect
Identification of NDE and IDE requires assumption 4 or SI2 because the causal me-diation effect is defined by using the “never-observable”outcome (not “potentiallyobservable” outcome) Y j ( M k ) ( j (cid:54) = k ) . However, Y j ( k ) and M j are observed forsome portion of the units.We redefine the potential outcomes Y j ( M k ) by using the functions of potentialoutcomes and mediators as follows: Y j ( M k ) ≡ M ∗ ∑ m = M k ( m ) Y j ( m ) . (7)Under this definition, Y j ( M k ) = Y j ( m ) when M k ( m ) = AT E = E [ Y ( M ) − Y ( M )] = E [ M ∗ ∑ m = { M ( m ) Y ( m ) − M ( m ) Y ( m ) } ] (8)Then, we defined the weighted controlled direct effect ( WCDE ) WCDE ≡ E [ M ∗ ∑ m = { M ( m ) Y ( m ) − M ( m ) Y ( m ) } ]= E [ M ∗ ∑ m = M ( m )( Y ( m ) − Y ( m ))] = E [ M ∗ ∑ 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:
IIE ≡ AT E − WCDE = E [ M ∗ ∑ m = [( M ( m ) − M ( m )) Y ( m ) + ( M ( m ) − M ( m )) Y ( m )]] (10)While the quantity defined as AT E − CDE may not be zero even when M = M for all units, IIE is always zero if M = M for all units. Theorem 1.
IIE is equivalent to zero if M = M for all units.of Theorem 1. From Equation 1, if M = M then IIE is zero because M = M = M . 4 case of a binary moderator We consider the case of a binary moderator. By Equation 2, the observed outcome Y is expressed as Y = T MY ( ) + T ( − M ) Y ( ) + ( − T ) MY ( ) + ( − T )( − M ) Y ( )= T M Y ( ) + T ( − M ) Y ( ) + ( − T ) M Y ( ) + ( − T )( − M ) Y ( ) (11)where M ( ) = M , M ( ) = M , M ( ) = − M , and M ( ) = − M .Using Equation 7, Y = Y ( M ) = M Y ( ) + ( − M ) Y ( ) , Y ( M ) = M Y ( ) + ( − M ) Y ( ) Y ( M ) = M Y ( ) + ( − M ) Y ( ) , Y = Y ( M ) = M Y ( ) + ( − M ) Y ( ) , (12)For example, the potential outcome if the unit recieves T = M = Y ( ) .Then, AT E , WCDE , and
IIE are expressed as follows:
AT E = E [ M Y ( ) − M Y ( ) + ( − M ) Y ( ) − ( − M ) Y ( ))] WCDE = E [ M × ICDE ( ) + ( − M ) × ICDE ( )]= E [ M ( Y ( ) − Y ( )) + ( − M )( Y ( ) − Y ( ))]= E [( T M + ( − T ) M )( Y ( ) − Y ( )) + ( − T M − ( − T ) M )( Y ( ) − Y ( ))] (cid:54) = NDE = [ NDE ( ) + NDE ( )]= E [ M + M ICDE ( ) + (cid:0) − M + M (cid:17) ICDE ( )] IID = E [( M − M )( Y ( ) − Y ( )) + ( M − M )( Y ( ) − Y ( ))] (cid:54) = NIE = [ NIE ( ) + NIE ( )] = E [( M − M ) { ( Y ( ) − Y ( )) + ( Y ( ) − Y ( )) } ] (13)where M ( ) = M and M ( ) = − M .It is easily shown that with randomization of T , the natural direct effect NDE = [ NDE ( ) + NDE ( )] evaluates the direct effect if the distribution of treatment isto be p ( T = ) = .
5, which is different from the “natural” observed distribution.From these equations. it is expected that the difference between WCDE and NDEwill be larger when P ( T = ) deviates from 0 .
5. Note that in the above equations theexpectation is taken over the population distribution of Y j ( k ) ( j , k = , ) , M , M and5 . Moreover, as will be mentioned in the next section, WCDE is estimable withoutAssumption 4 or SI2, while NDE is not.
Instead of Assumptions 1-4 (or SI1 and SI2), we introduce the following mean inde-pendence versions of Assumptions 1-4, SI1 and SI2:
Assumption (cid:48) E [ Y j ( k ) | T , v ] = E [ Y j ( k ) | v ] ∀ j , k , Assumption (cid:48) E [ M j | T , v ] = E [ M j | v ] ∀ j , Assumption (cid:48) E [ Y j ( k ) | M , T = j , v ] = E [ Y j ( k ) | T = j , v ] ∀ j , k , Assumption (cid:48) E [ Y j ( k ) | M j ∗ , v ] = E [ Y j ( k ) | v ] ∀ j , j ∗ , k , Assumption SI (cid:48) E [ M j ∗ Y j ( k ) | T , v ] = E [ M j ∗ Y j ( k ) | v ] ∀ j , j ∗ , k , Assumption SI (cid:48) E [ Y j ( k ) | M j ∗ , T = j , v ] = E [ Y j ( k ) | T = j , v ] ∀ j , j ∗ , k , Assumption SI (cid:48) implies the mean independence version of the ignorability assump-tion (Rosenbaum & Rubin, 1983), E [ Y j | T , v ] = E [ Y j | v ] ∀ j , which is sufficient foridentifying ATE.For identification of WCDE , we consider the following two cases: Case 1, whenboth M and T are randomized or ignorable given v , and Case 2, when M is not directlymaipulable. Case1: When both M and T can be randomized or ignorable givencovariates In this case we can identify ATE, WCDE and IIE in the following ways:
Step 1
Divide the sample into two equivalent subgroups (usually by using random-ization).
Step 2
Randomize T with v given in the first group to obtain a consistent estimatorof ATE, ˆ E [ Y − Y ] , and that of p ( M ) = p ( M | T = ) p ( T = ) + p ( M | T = ) p ( T = ) . 6 tep 3 Randomize both M and T with v given in the second group, in which thedistributions of T and M are set to be equal to those in the first group to identifyWCDE (and IID) by using Theorem 2 below.Note that in the first group by randomizing T , Assumptions 2 (cid:48) and SI (cid:48) automaticallyhold, which is sufficient to identify AT E and p ( M ) . In the second group, by random-izing both T and M , Assumptions 1 (cid:48) and 3 (cid:48) hold. From the following theorem, usingthe data from the second group WCDE and
IID = AT E − WCDE are identifiablewithout any additional assumptions such as mean independence between potentialoutcomes and potential mediators (i.e., Assumption 4 ‘ or SI ‘ ). Theorem 2.
WCDE is identifiable by observed data under Assumptions (cid:48) and (cid:48) .of Theorem 2. Under these assumptions, the
WCDE is expressed as
WCDE = E v (cid:104) M ∗ ∑ m = E [ M ( m ) ICDE ( m ) | v ] (cid:105) = E v (cid:104) M ∗ ∑ m = E T [ E [ M ( m ) | T ] E [ ICDE ( m ) | T ] | v ] (cid:105) = E v M ∗ ∑ m = (cid:104) p ( T = | v ) p ( M ( m ) = | T = , v ) E [ Y ( m ) − Y ( m ) | T = , v ]+ p ( T = | v ) p ( M ( m ) = | T = , v ) E [ Y ( m ) − Y ( m ) | T = , v ] (cid:105) = E v M ∗ ∑ m = (cid:104) p ( M ( m ) = | v )( E [ Y ( m ) | v ] − E [ Y ( m ) | v ]) (cid:105) = E v M ∗ ∑ m = (cid:104) p ( M ( m ) = | v ) × ( E [ Y ( m ) | T = , M = m , v ] − E [ Y ( m ) | T = , M = m , v ]) (cid:105) (14)Considering that p ( M ) , E [ Y j ( k ) | T = j , M = k , v ] ( ∀ j , k ) are observable, the WCDE is identifiable.It should be noted again that the identification of NDE and NIE requires Assump-tion 4 (cid:48) or SI (cid:48) even after randomizing both T and M (Pearl, 2014). Case2: When M is not directly manipulable If randomization of M is not feasible, it is inevitable to accept some untestable as-sumptions to identify causal direct/indirect effects. As stated in Theorem 2, it issufficient to assume Assumptions 1 (cid:48) , (cid:48) , (cid:48) and SI (cid:48) hold given abundant covariates to7dentify WCDE and IID. In this case, M is not directly manipulatable, then Assump-tion 3 (cid:48) (and the other assumptions when T is also not manipulatable) is untestable,but as mentioned earlier, these assumptions are weaker than Assumption 4 (cid:48) or SI (cid:48) . Estimation
For simplicity, we consider the case without covariates. For Case 1, the estimator ofWCDE is expressed by the observed quantities: (cid:92)
WCDE = M ∗ ∑ m = ˆ p ( M ( m ) = )( ¯ y | M = m − ¯ y | M = m ) (15)where ˆ p ( M ( m ) = ) is the sample proportion with M = m in the first group and¯ y j | M = m is the average of y for units with T = j and M = m in the second group.By simple application of the Delta method the asymptotic variance of (cid:92) WCDE isexpressed as1 N (cid:110) d t ( diag ( p ) − pp t ) d + M ∗ ∑ m = p m [ V ( ¯ y | M = m ) + V ( ¯ y | M = m )] (cid:111) (16)where p m = P ( M ( m ) = ) , p = ( p , · · · , p M ∗ ) t , d m = E ( y ( m )) − E ( y ( m )) and d =( d , · · · , d M ∗ ) t .The unbiased ATE is ¯ y − ¯ y , where ¯ y j is the average of y for units with T = j in the whole sample, because even in the second group Assumptions 1 (cid:48) and 3 (cid:48) hold,then the difference of averages of outcomes is an unbiased estimator of ATE in thesecond group.For Case 2, under ignorability given covariate v , ATE and WCDE are expressedas: AT E = E v (cid:104) E [ Y | T = , v ] − E [ Y | T = , v ] } (cid:105) WCDE = E v M ∗ ∑ m = (cid:104) P ( M ( m ) = | v ) { E [ Y | T = , M = m , v ] − E [ Y | T = , M = m , v ] } (cid:105) (17)Then various methods such as inverse probability weighting estimator or doubly ro-bust type estimator can be used to estimate AT E and
WCDE .8 ypothetical ratio adjustment for treatment As stated in the previous section, “WCDE” evaluates the direct effect with the ob-served proportion ( ˆ p ( T = ) ) of the treatment group. If the researcher needs to con-sider the direct effect with a hypothetical proportion (say p ∗ ) of the treatment group,use a weight of p ∗ ˆ p ( T = ) for treatment individual and use a weight of − p ∗ − ˆ p ( T = ) forcontrol individual. Generalization
We can address the case where m is continuous. Under continuous mediator m ,WCDE is expressed as follows: WCDE = (cid:90) (cid:90) { E [ Y ( m ) | m , v ] − E [ Y ( m ) | m , v ] } p ( m | v ) p ( v ) dmdv (18)Under Assumptions 1 (cid:48) and 3 (cid:48) , WCDE is expressed as the following quantities identi-fiable by observed data: WCDE = (cid:90) (cid:90) { E [ Y ( m ) | T = , m , v ] − E [ Y ( m ) | T = , m , v ] } p ( m | v ) p ( v ) dmdv (19)ATE is identifiable by Assumption SI (cid:48) , then IID is also identified as IID = AT E − WCDE . For illustrative purposes, we present a simulation study that compares the definedeffects with the previously proposed ones. We numerically evaluated bias from thetrue values (population WCDE for the proposed method in Case 1 and populationNDE 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 continuouspotential mediators M L and M L so that M j = M Lj > M j = ( j = , ) .We generated 10,000 samples of size n = ,
000 from the joint vector of potentialmediators and potential , W = ( M L , M L , Y ( ) , Y ( ) , Y ( ) , Y ( )) t which follows a9nite scale mixture of multivariate-normal distributions; W ∼ . × N ( µ , Σ ) + . × N ( µ , Σ ) (20)where µ = ( − , , , . , . , ) , Σ = Σ , Σ = × Σ and the diagonal elements of Σ is set to be 1. Off-diagonal elements are set such that Cov ( Y j ( k )) , Y l ( m )) = . Cov ( M , M ) = .
6. For simplicity, the covariances between potential medi-ators and potential outcomes are set as follows:
Cov ( M , Y ( )) = Cov ( M , Y ( ) = Cov ( M , Y ( )) = Cov ( M , Y ( )) = φ , and Cov ( M , Y ( )) = Cov ( M , Y ( )) = Cov ( M , Y ( )) = Cov ( M Y ( )) = − φ . In this setup, p ( M = ) ≈ .
809 and p ( M = ) ≈ . T is generated from the Bernoulli distribution with Pr ( T = ) = p , independently of W . We consider four cases with p = .
01, 0 .
1, 0 . . φ varies from − .
15 to 0 .
15 in 0 .
05 increments. Note that for allthe models assumption 4 or SI2 is violated, except φ = WCDE and
NDE of the model is difficult to calculate analyti-cally. 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 inapplied studies. The results are shown in Table 1 and those with p = . φ and the vertical axis is the bias from thetrue values.As shown in this figure and Table1, the bias of the previously proposed estimatorcan be large for large deviations from Assumption SI2 (i.e., large φ ) as mentioned inthe sensitivity analysis in Imai et al. (2010b), while the proposed method can find truevalues 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 estimatorsis large because the sample size with T = AT E , WCDE and
NDE ineach 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 = ) = . φ = − ,
15 where
AT E is almost the same as
NDE (i.e.,
NIE is almostzero), the existing method wrongly finds a “(pseudo-)mediation” effect under thesetup when the true model does not contain a mediation effect, but the mediator andpotential outcomes are correlated.Tab.1. Simulation results φ -0.15 -0.1 -0.05 0 0.05 0.1 0.15ATE 0.6925 0.7567 0.8209 0.8851 0.9493 1.0135 1.0776WCDE 0.6393 0.6392 0.6392 0.6392 0.6392 0.6392 0.6393NDE 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997Bias of Prop -0.0008 -0.0011 -0.0011 -0.0011 -0.0010 -0.0007 -0.0002Bias of Existing -0.1936 -0.1308 -0.0668 -0.0025 0.0622 0.1272 0.1933S.d. of Prop 0.2813 0.2809 0.2808 0.2807 0.2805 0.2803 0.2802S.d. of Exsisting 0.2744 0.2736 0.2735 0.2735 0.2735 0.2736 0.2744ATE 0.6925 0.7567 0.8209 0.8851 0.9493 1.0135 1.0776WCDE 0.6504 0.6503 0.6503 0.6503 0.6503 0.6503 0.6504NDE 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997Bias of Prop -0.0006 -0.0006 -0.0005 -0.0004 -0.0003 -0.0002 0.0000Bias of Existing -0.1927 -0.1289 -0.0649 -0.0008 0.0634 0.1277 0.1924S.d. of Prop 0.0880 0.0883 0.0885 0.0886 0.0888 0.0889 0.0890S.d. of Exsisting 0.0876 0.0875 0.0875 0.0875 0.0876 0.0877 0.0878ATE 0.6925 0.7567 0.8209 0.8851 0.9493 1.0135 1.0776WCDE 0.6750 0.6750 0.6750 0.6750 0.6750 0.6750 0.6750NDE 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997Bias of Prop -0.0004 -0.0004 -0.0004 -0.0004 -0.0004 -0.0004 -0.0004Bias of Existing -0.1920 -0.1277 -0.0635 0.0007 0.0648 0.1289 0.1929S.d. of Prop 0.0578 0.0578 0.0579 0.0580 0.0580 0.0581 0.0581S.d. of Exsisting 0.0561 0.0562 0.0563 0.0564 0.0564 0.0564 0.0565ATE 0.6925 0.7567 0.8209 0.8851 0.9493 1.0135 1.0776WCDE 0.6997 0.6996 0.6996 0.6996 0.6996 0.6996 0.6997NDE 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997 0.6997Bias of Prop 0.0002 0.0002 0.0001 0.0000 0.0000 0.0000 -0.0001Bias of Existing -0.1923 -0.1281 -0.0639 0.0003 0.0644 0.1286 0.1928S.d. of Prop 0.0530 0.0530 0.0530 0.0530 0.0531 0.0531 0.0532S.d. of Exsisting 0.0517 0.0516 0.0516 0.0516 0.0517 0.0518 0.0519p=0.01p=0.1p=0.3p=0.5 In this paper, we proposed a new definition of causal direct and indirect effects incausal mediation analysis.Identification of the previously proposed quantities, natural direct effect, and nat-ural indirect effect is not possible even when both treatment and mediator are ran-domized. Therefore, it is unavoidable to employ untestable assumption of the inde-pendence of potential outcomes and potential mediators, some of which we neverobserve simultaneously. 11igure 1: Result for p = . ‐ ‐ ‐ ‐ ‐ ‐ p=0.5 Proposed Existing
The error bar indicates one standard deviation calculated from the 10,000 estimates.The proposed quantities are identifiable without any assumption when both treat-ment and mediator are randomized. Even when randomization is not possible, theproposed quantities require weaker assumptions than those for the identification oftraditional quantities.When it is difficult to directly manipulate M , Assumption 3 (cid:48) is required for iden-tification of the proposed effects. Another approach for identification such as princi-pal stratification approach (Frangakis & Rubin , 2002; Forastiere et al., 2018) is anpromising strategy that we will investigate in a future study.In this paper, we focused on the case with binary treatment, but the definition ofWCDE 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 orvariance of the expectations, instead of the traditional “difference of the expectations”formulation in the case of binary treatment, which will be considered elsewhere.12 eferences D ING , P., & V
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