eappendix for: SAS macro for causal mediation analysis with survival data

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1 eappendix for: SAS maro for ausal mediation analysis with survival data Linda Valeri and Tyler J. VanderWeele 1 Causal effets under the ounterfatual framework and their estimators We let T a and M a denote respetively the values of the time-to-event outome and mediator that would have been observed had the exposure A been set to level a. We let T am denote the value of the time-to-event outome that would have been observed had the exposure, A, and mediator, M, been set to levels a and m, respetively. The average ontrolled diret effet omparing exposure level a to a and fixing the mediator to level m on the mean survival time ratio sale is defined by CDE a,a (m) = E[T am ]/E[T a m]. The average natural diret effet is then defined by NDE a,a (a ) = E[T ama ]/E[T a M a ]. The average natural indiret effet an be defined as NIE a,a (a) = E[T ama ]/E[T ama ], whih ompares the effet of the mediator at levels M a and M a on the survival outome when exposure A is set to a. Controlled diret effets and natural diret and indiret effets within strata of C = are then defined by: CDE a,a (m) = E[T am ]/E[T a m ], NDE a,a (a ) = E[T ama ]/E[T a M a ] and NIE a,a (a) = E[T ama ]/E[T ama ] respetively. For an arbitrary time-to-event variable V, let λ V (t) and λ V (t ) denote the hazard or hazard onditional on ovariates at time t, that is the instantaneous rate of the event onditional on V t. The ausal effets an also be defined on the hazard ratio sale, replaing E[ ] with λ( ). If we let X T Z denote that X is independent of T onditional on Z, then the identifiation assumptions for the ausal effets previously defined an be expressed formally in terms of ounterfatual independene as (i) T am A C, (ii) T am M {A, C}, (iii) M a A C, and (iv) T am M a C. Assumptions (i) and (ii) suffie to identify ontrolled diret effets; assumptions (i)-(iv) suffie to identify natural diret and indiret effets (Pearl, 2001; VanderWeele and Vansteelandt, 2009). The intuitive interpretation of these assumptions follows from the theory of ausal diagrams (Pearl, 2001). Alternative identifiation assumptions have also been proposed (Imai 2010; Hafeman and Vander- Weele, 2011). However, it has been shown that the intuitive graphial interpretation of these alternative assumptions are in fat equivalent (Shpitser and VanderWeele, 2011). Tehnial examples an be onstruted where one set of identifiation assumptions holds and another does not, but on a ausal diagram orresponding to a set of non-parametri strutural equations, whenever one set of the assumptions among those in VanderWeele and Vansteelandt (2009), Imai (2010), and Hafeman and VanderWeele (2011) holds, the

2 2 others will also. Models and Estimators for Causal Effets: Continuous Mediator and Time-to-event Outome Let M be a ontinuous mediator following a linear model, A be an exposure and C be additional ovariates. Assume that the outome T is a failure time variable following a Cox-proportional hazard model or an aelerated failure time (AFT) model. We an define the mediator regression as E(M A, C) = β 0 + β 1 a + β 2 (1) We define the outome model either as Cox proportional hazard: λ T (t a, m, ) = λ T (t 0, 0, 0)e γ 1a+γ 2 m+γ 3 am+γ 4 (2) or as aelerated failure time model: log(t ) = θ 0 + θ 1 a + θ 2 m + θ 3 am + θ 4 + νɛ (3) where ɛ follows the extreme value distribution and ν is a shape parameter taking value ν = 1 if the exponential distribution is assumed and allowed to take different values if the Weibull distribution is assumed. If the no-unmeasured onfounding assumptions hold, the model for the ontinuous mediator and for the outome are orretly speified, and the Cox regression is employed, then ontrolled diret effet, natural diret and natural indiret effets estimators on the hazard ratio sale are given by (VanderWeele, 2011): λ Tam (t )/λ Ta m (t ) = exp{(γ 1 + γ 3 m)(a a )} λ TaMa (t )/λ T a Ma (t ) exp[{γ 1 + γ 3 (β 0 + β 1 a + β 2 + γ 2 σ 2 )}(a a ) + 0.5γ 2 3σ 2 (a 2 a 2 )] λ TaMa (t )/λ TaMa (t ) exp{(γ 2β 1 + γ 3 β 1 a)(a a )} The approximations to estimate natural diret and natural indiret effets apply if the outome is rare at the end of follow-up. If the no-unmeasured onfounding assumptions hold, the model for the ontinuous mediator and for the outome are orretly speified, and the AFT model is employed, then ontrolled diret effet, natural diret and natural indiret effets estimators on the mean survival time ratio sale are given by (VanderWeele, 2011): E[T am ]/E[T a m] = exp{(θ 1 + θ 3 m)(a a )} E[T ama ]/E[T a M a ] = exp[{θ 1 + θ 3 (β 0 + β 1 a + β 2 + θ 2 σ 2 )}(a a ) + 0.5θ 2 3σ 2 (a 2 a 2 )] E[T ama ]/E[T ama ] = exp{(θ 2 β 1 + θ 3 β 1 a)(a a )}

3 3 Note that under the AFT the rare outome assumption is not required to estimate natural diret and indiret effets. Models and Estimators for Causal Effets: Binary Mediator and Time-to-event Outome Let T denote the time-to-event outome modeled either as in (2) or (3). Let M be a binary mediator following a logisti model. We an define the mediator regression as: logit{p (M = 1 A, C)} = β 0 + β 1 a + β 2 (4) If the no-unmeasured onfounding assumptions hold, the model for the binary mediator and for the outome are orretly speified, and the Cox regression is employed, then ontrolled diret effet, natural diret and natural indiret effets estimators on the hazard ratio sale are given by (see eappendix setion 4 for proof): λ Tam (t )/λ Ta m (t ) = exp{(γ 1 + γ 3 m)(a a )} λ TaMa (t )/λ T a Ma (t ) exp(γ 1a){1 + exp(γ 2 + γ 3 a + β 0 + β 1 a + β 2 )} exp(γ 1 a ){1 + exp(γ 2 + γ 3 a + β 0 + β 1 a + β 2 )} λ TaMa (t )/λ TaMa (t ) {1 + exp(β 0 + β 1 a + β 2 )}{1 + exp(γ 2 + γ 3 a + β 0 + β 1 a + β 2 )} {1 + exp(β 0 + β 1 a + β 2 )}{1 + exp(γ 2 + γ 3 a + β 0 + β 1 a + β 2 )} The approximations to estimate natural diret and natural indiret effets apply if the outome is rare at the end of follow-up. If the no-unmeasured onfounding assumptions hold, the model for the binary mediator and for the outome are orretly speified, and the AFT model is employed, then ontrolled diret effet, natural diret and natural indiret effets estimators on the mean survival ratio sale are given by (see eappendix setion 4): E[T am ]/E[T a m] = exp{(θ 1 + θ 3 m)(a a )} E[T ama ]/E[T a M a ] = exp(θ 1a){1 + exp(θ 2 + θ 3 a + β 0 + β 1 a + β 2 )} exp(θ 1 a ){1 + exp(θ 2 + θ 3 a + β 0 + β 1 a + β 2 )} E[T ama ]/E[T ama ] = {1 + exp(β 0 + β 1 a + β 2 )}{1 + exp(θ 2 + θ 3 a + β 0 + β 1 a + β 2 )} {1 + exp(β 0 + β 1 a + β 2 )}{1 + exp(θ 2 + θ 3 a + β 0 + β 1 a + β 2 )} Note that under the AFT the rare outome assumption is not required to estimate natural diret and indiret effets. 2 Desription of the SAS maro The present software is designed to enable the investigator to easily implement mediation analysis in the presene of exposure-mediator interation aounting for different

4 4 types of outomes (normal, dihotomous (with logit or log link), poisson, negative binomial, failure time (with Cox PH model or AFT Weibull or AFT exponential), an be assumed) and mediators of interest (normal or dihotomous with logit link). The logit link for dihotomous outomes and Cox model for survival outome should only be used if the outome is rare. If the outome is not rare and binary the log link an be used (though the outome model may not always onverge) and failure times ould be modeled using AFT. For binary outomes, in the ase of logit link the diret and indiret effets are on the odds ratio sale and in the ase of using the log link the diret and indiret effets are on the risk ratio sale. For survival outomes, in the ase of Cox model the diret and indiret effets are on the hazard ratio sale and in the ase of using the AFT model the diret and indiret effets are on the mean survival ratio sale. This SAS maro, provides estimates, and onfidene intervals for the diret and indiret effets defined in Valeri and VanderWeele (2013). The estimates assume the model assumptions are orret and the identifiability assumptions disussed in Valeri and VanderWeele (2013) hold. In order to implement mediation analysis via the mediation maro in SAS the investigator first inputs the data whih has to inlude the outome, treatment and mediator variables. The investigator inputs also the ovariates to be adjusted for in the model. Maro ativation requires the investigator to save the maro sript and input information in the statement %mediation(data=,yvar=,avar=,mvar=,var=,a0=,a1=,m=,yreg=, mreg=,interation=,ens=) run; First one inputs the name of the dataset, then the name of the outome variable (yvar =), the treatment variable (avar =), the mediator variable (mvar =), the other ovariates (var =). Then the investigator needs to speify the baseline level of the exposure a (a0 =), the new exposure level a (a1 =) and the level of mediator m at whih the ontrolled diret effet is to be estimated. The user must also speify whih types of regression have to be implemented. In partiular either linear, logisti, loglinear, poisson, negbin, survcox, survaf T exp, or survaf T weibull an be speified in the option yreg. If survcox, survaf T exp, or survaf T weibull for failure time outome are speified, the user need to speify the ensoring variable (ens =) whih has to take value 1 if the event is ensored and 0 if the failure time is observed. For the option mreg either linear or logisti regressions are allowed. Finally, the analyst needs to speify whether an exposure-mediator interation is present (true or false). The software provides the following output: first the outome and mediator regression output are provided. The output in the SAS maro is derived from the proedures of pro reg when the variable is ontinuous and pro logisti when the variable is binary. When the outome is speified as poisson, negative binomial or log-linear the proedure pro genmod is employed. When the outome is failure time and the Cox model is speified, the proedure phreg is employed while if aelerated failure time model is speified, the proedure lifereg is employed. The SAS maro is ase-sensitive and the options speified should be given in lower-ase letters, unless otherwise speified.

5 5 A table with diret and indiret effets together with total effets and proportion mediated follows. The total effet is omputed as the sum of the natural diret effet and the natural indiret effet when the outome is ontinuous and the produt of the natural diret and indiret effets in the other ases. The proportion mediated an be defined as the ratio of the natural indiret effet over the total effet when the outome is ontinuous; the proportion mediated on outome differene sale is given in the other ases using a transformation of the ratio sale (VanderWeele and Vansteelandt, 2010). The effets are reported for the mean level of the ovariate C. The table ontains p- values, and onfidene intervals for eah effet. The redued output is the default option. The table will just display ontrolled diret effet, natural diret effet, natural indiret effet, total effet and proportion mediated. When the option output=full is used, both onditional effets and effets evaluated at the mean ovariate levels are shown. When output equal to full is hosen as an option, the investigator must enter fixed values for the ovariates C at whih to ompute onditional effets. The maro statement is then as follows: %mediation(data=,yvar=,avar=,mvar=,var=,a0=,a1=,m=,yreg=,mreg=,interation=,ens=,output=,=) run; When these ommands are added, in addition to ontrolled diret effet, and the natural diret and indiret effet desribed above, two other effets are displayed. The natural diret and indiret effets we have been onsidering are sometimes alled the pure natural diret effet and the total natural diret effet (Robins, Greenland, 1992). We an also onsider the total natural diret effet and the pure natural indiret effet. The total natural diret effet expresses how muh the outome would hange on average if the exposure hanged from level a = 0 to level a = 1, but the mediator for eah individual was fixed at the natural level whih would have taken at exposure level a = 1. The pure natural indiret effet expresses how muh the outome would hange on average if the exposure were ontrolled at level a = 0 but the mediator were hanged from the natural level it would take if a = 0 to the level that would have taken at exposure level a = 1. These effets are also reported if the user selets output = f ull. For an in-depth desription of these ausal effets the interested reader an refer to VanderWeele and Vansteelandt (2009). Finally, the investigator has the option of implementing mediation analysis when data arise from a ase-ontrol design, provided the outome in the population is rare. The formulas for the effets remain the same, however the mediator regression will be run only for ontrols, in order to minimize the bias due to the design, and sine with a rare outome Y the ontrols will approximate the distribution of M in the population. To do so the option aseontrol=true an be used. In this ase the maro statement hanges to %mediation(data=,yvar=,avar=,mvar=,var=,a0=,a1=,m=,yreg=,mreg=,interation=,ens=,aseontrol=) run; Finally, the investigator an hoose whether to obtain standard errors and onfidene intervals via the delta method or a bootstrapping tehnique. The default is the delta method. To use bootstrapping the option boot true an be given. In this ase the maro

6 6 will ompute 1,000 bootstrap samples from whih ausal effets are obtained along with their standard errors (SE) and perentile onfidene intervals. If the investigator wishes to use a higher number of bootstrap samples, instead of true he or she inputs the number of bootstrap samples desired (e.g., boot = 5000 would estimate standard errors and onfidene intervals using 5,000 bootstrap samples). When using the bootstrap the maro statement hanges to %mediation(data=,yvar=,avar=,mvar=,var=,a0=,a1=,m=,yreg=,mreg=,interation=,ens=,boot=) run; Note that if the investigator wants to add a ategorial variable as ovariate, this must be reoded as a series of indiator variables. 3 Example We present in this setion an example of using the mediation maro when the outome is failure time. The example is to be onsidered for illustration purposes only. Suppose interest lies in the effets of soio-eonomi position (SEP) on survival of oloretal aner patients. SEP is measured by perentage of people living below the poverty line in the patients ounty of residene. In this example, stage at diagnosis, measured as advaned vs non-advaned is investigated as a potential mediator of the relationship between residing in poor ounties and the survival outome. Data for Surveillane Epidemiology End Results (SEER) linked to Amerian Community Survey (ACS) data for patients diagnosed in and followed up to 2010 is employed to address this question. In the present analysis we allow for interation between the SEP measure and stage at diagnosis. Furthermore we adjust for potential onfounders: gender, age at diagnosis, year at diagnosis and state of residene. Sine the outome is failure time we define a ensoring variable taking value 1 if the individual is ensored or value 0 if the event is observed. After having saved the dataset and inserted maro sript we run the following ommand %mediation(data=dat,yvar=survival,avar=new_poverty,mvar=grade,var=rae date_ sex age_ state_2 state_3 state_4 state_5 state_6 state_7 state_8, a0=0,a1=0.3,m=0,yreg=survaft_exp,mreg=logisti,interation=true, ens=ensor) run; The first output provided is the results of the outome and mediator regressions. Then the diret and indiret effets follow. We give the redued output whih provides estimates for the ontrolled diret effet, the natural diret and indiret effet, total effet and perentage mediated. We fit the survival outome with aelerated failure time model sine the event is non rare by the end of follow-up in this population. By inspeting Kaplan-Meier survival urves by quartiles of the exposure, the assumption of proportional hazard appears satisfied and we therefore fit the AFT model assuming exponential distribution. The survival analysis yields a negative effet of poverty on survival, marginally signifiant adjusting for grade. A positive, signifiant interation between tumor stage at diagnosis and poverty is deteted. The logisti regression analysis shows that the SEP measure is positively assoiated with stage at diagnosis. We

7 7 The SAS System The LIFEREG Proedure Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error 95% Confidene Limits Chi-Square Pr > ChiSq Interept <.0001 new_poverty grade <.0001 int <.0001 RACE <.0001 date_ <.0001 SEX <.0001 age_ <.0001 state_ state_ state_ <.0001 state_ state_ <.0001 state_ state_ Sale Weibull Shape The LOGISTIC Proedure Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Interept <.0001 new_poverty <.0001 RACE <.0001 date_ <.0001 SEX age_ <.0001 state_ state_ <.0001 state_ <.0001 state_ state_ state_ state_ <.0001 Figure 1: Output of aelerated failure time outome regression and mediator logisti regression

8 8 Obs Effet Estimate p_value _95 CI_lower _95 CI_upper 1 de nde nie total effet Obs Effet Estimate 1 proportion mediated Figure 2: Output of mediation analysis with ausal effets estimated for a hange in the exposure from 0 to 0.3 and at the mean level of the ovariates

9 9 estimate a total effet of 0.89 indiating that the mean survival time of individuals living in ounties with 30% of the population living below the poverty level is 89% that of individuals living in ounties with no individuals living below the poverty line. The ontrolled diret effet, ontrolling the mediator at level m = 0, reveals that, had we intervened setting stage at diagnosis to be non-advaned for all individuals, the mean survival time of individuals living in ounties with 30% of the population living below the poverty level would be 94% that of individuals living in ounties with no individuals living below the poverty line. Natural diret and natural indiret effets take values 93% and 95%, respetively and stage at diagnosis is found to mediate 42% of the effet of poverty on survival. The results of this example should be interpreted with aution as several biases might be present. First, an aggregate, rather than an individual-level measure of soio-eonomi position is employed, potentially introduing eologi bias. Seond, the no-unmeasured onfounding assumptions are likely violated in this example. 4 Estimators for diret and indiret when the outome is failure time and the mediator is binary In this setion we derive estimators of diret and indiret ausal effets when the outome is failure time modeled using either Cox proportional hazard model or aelerated failure time models (AFT) and the binary mediator is modeled using logisti regression. The results oinide with the ones obtained for diret and indiret effets when outome and mediator are binary as derived in Valeri and VanderWeele (2013). The ausal effets have hazard ratio interpretation if the Cox model is employed and mean survival ratio interpretation under the AFT model. Notably, the derivations under the Cox model assume that the failure event is rare while this assumption is not required if AFT is employed. Models Let M be a binary mediator following a logisti model, A be an exposure and C be additional ovariates. Assume that the outome T is a failure time variable following a ox-proportional hazard model or an aelerated failure time model. We an define the mediator regression as We define the outome model either as or as logit{p (M = 1 A, C)} = β 0 + β 1 a + β 2 (5) λ T (t a, m, ) = λ T (t 0, 0, 0)e γ 1a+γ 2 m+γ 3 am+γ 4 (6) log(t ) = θ 0 + θ 1 a + θ 2 m + θ 3 am + θ 4 + νɛ (7) where ɛ follows the extreme value distribution and ν is a shape parameter taking value ν = 1 if the exponential distribution is assumed and allowed to take different

10 10 values if the Weibull distribution is assumed. Cox-proportional hazard model for the outome where, Consider λ TaMa (t ) = f T ama (t ) S TaMa (t ) f TaMa (t ) = f Tam (t M a = m, )dp Ma (m ) = f Tam (t )dp Ma (m ) by assumption (iv) = f T (t a, m, )dp M (m a, ) by assumptions (i) (iii) = λ T (t 0, 0, 0)e γ 1a+γ 2 m+γ 3 am+γ 4 exp{ Λ T (t 0, 0, 0)e γ 1a+γ 2 m+γ 3 am+γ 4 }dp M (m a, ) where Λ T (t 0, 0, 0) = t 0 λ T (t 0, 0, 0)dt, and S TaMa (t ) = exp{ Λ T (t 0, 0, 0)e γ 1a+γ 2 m+γ 3 am+γ 4 }dp M (m a, ). Thus, λ TaMa (t ) = λt (t 0,0,0)e γ 1 a+γ 2 m+γ 3 am+γ 4 exp{ Λ T (t 0,0,0)e γ 1 a+γ 2 m+γ 3 am+γ 4 }dp M (m a,) exp{ ΛT (t 0,0,0)e γ 1 a+γ 2 m+γ 3 am+γ 4 }dp M (m a,) λ T (t 0, 0, 0)e γ 1a+γ 4 e (γ 2+γ 3 a)m dp M (m a, ) = λ T (t 0, 0, 0)e γ 1a+γ 4 E[e (γ 2+γ 3 a)m ] The approximation holds if we assume rare outome so that Λ T (t 0, 0, 0) 0 whih implies exp{ Λ T ( )e ( ) } 1. Now, onsidering that M Be(p) with p = exp(β 0+β 1 a+β 2 ) 1+exp(β 0 +β 1 a+β 2 ), by the properties of the moment generating funtions and we finally obtain E[e tm ] = 1 p + pe t λ TaMa (t ) = λ T (t 0, 0, 0)e γ 1a+γ 4 (1 exp(β 0 + β 1 a + β 2 ) 1 + exp(β 0 + β 1 a + β 2 )+exp(β 0 + β 1 a + β 2 + γ 2 + γ 3 a) 1 + exp(β 0 + β 1 a + β 2 ) ) Define the onditional total hazard ratio by λ T E = λ T ama λ Ta. Likewise, we define the Ma pure diret effet hazard ratio by λ NDE = λ T ama λ Ta Ma and the total indiret effet hazard

11 11 ratio by λ NIE and = λ T ama λ TaMa. Then we have the following deomposition an be given: λ T E = λ NDE λ NIE λ NDE = λ T ama λ Ta Ma = e 1 exp(β 0+β 1a +β 2 ) γ 1(a a ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+β 1 a +β 2 +γ 2+γ 3 a) 1+exp(β 0 +β 1 a +β 2 ) 1 exp(β 0+β 1 a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+(β 1 +γ 3 )a +β 2 +γ 2) 1+exp(β 0 +β 1 a +β 2 ) λ NIE = λ T ama λ TaMa = 1 exp(β 0+β 1a+β 2 ) 1+exp(β 0 +β 1 a+β 2 ) + exp(β 0+β 1 a+β 2 +γ 2+γ 3 a) 1+exp(β 0 +β 1 a+β 2 ) 1 exp(β 0+β 1 a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+β 1 a +γ 3 a+β 2 +γ 2) 1+exp(β 0 +β 1 a +β 2 ) Aelerated failure time model for the outome Consider the ounterfatual T ama. E[T ama ] = E[T am M a, ]dp Ma (m ) = E[T am ]dp Ma (m ) by assumption (iv) = E[T a, m, ]dp M (m a, ) by assumptions (i) (iii) = E[e θ 0+θ 1 a+θ 2 m+θ 3 am+θ 4 +νɛ ]dp M (m a, ) = e θ 0+θ 1 a+θ 4 E[e νɛ ]E[e (θ 2+θ 3 a)m ] = e θ 0+θ 1 a+θ 4 E[e νɛ ](1 exp(β 0+β 1 a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+β 1 a +β 2 +θ 2+θ 3 a) 1+exp(β 0 +β 1 a +β 2 ) ) If we onsider the mean survival time onditional on the vetor of ovariates C as our measure of interest, the following deomposition an be given E(T ama )}/{E(T a M a ) = [{E(T ama )}/{E(T a M a )}] [{E(T ama )}/{E(T a M a )}] where the total effet, given by {E(T ama )}/{E(T a M a )}, deomposes in pure diret effet {E(T ama )}/{E(T a M a )} = e θ 1(a a ) 1 times total indiret effet exp(β 0+β 1a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+β 1 a +β 2 +θ 2+θ 3 a) 1+exp(β 0 +β 1 a +β 2 ) 1 exp(β 0+β 1 a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+(β 1 +θ 3 )a +β 2 +θ 2) 1+exp(β 0 +β 1 a +β 2 )

12 12 {E(T ama )}/{E[(T ama )} = 1 exp(β 0+β 1a+β 2 ) 1+exp(β 0 +β 1 a+β 2 ) + exp(β 0+β 1 a+β 2 +θ 2+θ 3 a) 1+exp(β 0 +β 1 a+β 2 ) 1 exp(β 0+β 1 a +β 2 ) 1+exp(β 0 +β 1 a +β 2 ) + exp(β 0+β 1 a +θ 3 a+β 2 +θ 2) 1+exp(β 0 +β 1 a +β 2 ) These estimators oinide with the ones obtained when a rare binary outome is modeled using logisti regression and a binary mediator is modeled using logisti regression (Valeri and VanderWeele, 2013). Note that when a survival outome is modeled using Cox proportional hazard model the event is assumed to be rare. This assumption is not required if an aelerated failure time model is employed instead. Referenes - Hafeman, D. M., VanderWeele, T. J. (2011). Alternative assumptions for the identifiation of diret and indiret effets. Epidemiology, 22: Imai, K., Keele, L., Tingley, D. (2010). A general approah to ausal mediation analysis. Psyhologial Methods, 15: Pearl J. Diret and indiret effets. In Proeedings of the 17th Conferene in Unertainty in Artifiial Intelligene, 2001; , San Franiso, CA, USA, Morgan Kaufmann Publishers In. - Shpitser, I., VanderWeele, T. J. (2011). A omplete graphial riterion for the adjustment formula in mediation analysis. International Journal of Biostatistis, 7: Surveillane, Epidemiology, and End Results (SEER) Program Researh Data ( ), National Caner Institute, DCCPS, Surveillane Researh Program, Surveillane Systems Branh, released April Valeri, L. and VanderWeele, T.J. (2013). Mediation analysis allowing for exposuremediator interation and ausal interpretation: SAS and SPSS maros. Psyhologial Methods, 18: VanderWeele, T. J., Vansteelandt S. (2009). Coneptual issues onerning mediation, interventions and omposition. Statistis and Its Interfae, 2: VanderWeele, T.J.(2011). Causal mediation analysis with survival data. Epidemiology, 22: