Causal mediation analysis: Multiple mediators

Transcription

1 Causal mediation analysis: ultiple mediators Trang Quynh guyen Seminar on Statistical ethods for ental Health Research Johns Hopkins Bloomberg School of Public Health term 4 session 4 - ay 5, / 32

2 Session overview ultiple-mediator cases and types of mediation effects ediation by all the mediators combined ediation via different pathways 2 / 32

3 There are different multiple-mediator cases. 3 / 32

4 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 1: evaluate effect mediated by and combined 4 / 32

5 causally ordered mediators (X variables are implied, left out of diagram for simplicity) placental abruption preterm delivery preeclampsia neonatal outcome potential aim 1: evaluate effect mediated by and combined 4 / 32

6 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 1: evaluate effect mediated by and combined 4 / 32

7 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 2: partition effects through different paths 4 / 32

8 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 2: partition effects through different paths 4 / 32

9 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 2: partition effects through different paths 4 / 32

10 causally ordered mediators (X variables are implied, left out of diagram for simplicity) potential aim 2: partition effects through different paths 4 / 32

11 causally unordered mediators (X variables are implied, left out of diagram for simplicity) and are generally dependent after accounting for and X 5 / 32

12 causally unordered mediators (X variables are implied, left out of diagram for simplicity) tobacco alcohol education bad outcome and are generally dependent after accounting for and X 5 / 32

13 causally unordered mediators (X variables are implied, left out of diagram for simplicity) optimal surgery optimal chemo research hospital survival and are generally dependent after accounting for and X 5 / 32

14 causally unordered mediators (X variables are implied, left out of diagram for simplicity) structure of this dependence unclear: maybe one variable causes the other 5 / 32

15 causally unordered mediators (X variables are implied, left out of diagram for simplicity) structure of this dependence unclear: maybe one variable causes the other 5 / 32

16 causally unordered mediators (X variables are implied, left out of diagram for simplicity) U structure of this dependence unclear: or both are influenced by a U 5 / 32

17 causally unordered mediators (X variables are implied, left out of diagram for simplicity) generally can t separate effects mediated by and by 5 / 32

18 causally unordered mediators (X variables are implied, left out of diagram for simplicity) but can in special case where and are independent given, X 5 / 32

19 or a mix (X variables are implied, left out of diagram for simplicity) L L before and, but and unordered 6 / 32

20 or a mix (X variables are implied, left out of diagram for simplicity) adulthood SES L tobacco alcohol childhood SES bad outcome L before and, but and unordered 6 / 32

21 or a mix (X variables are implied, left out of diagram for simplicity) L could evaluate effect mediated by L,, combined 6 / 32

22 or a mix (X variables are implied, left out of diagram for simplicity) L or consider mediation through different paths, but keep, together 6 / 32

23 or a mix (X variables are implied, left out of diagram for simplicity) L or consider mediation through different paths, but keep, together 6 / 32

24 or a mix (X variables are implied, left out of diagram for simplicity) L or consider different indirect paths, keeping, together 6 / 32

25 1. ediation by all the mediators combined 7 / 32

26 Identifying assumptions no uncontrolled exposure-outcome confounding no uncontrolled mediator-outcome confounding no uncontrolled exposure-mediator confounding no mediator-outcome confounder influenced by exposure applies to the collection of mediators, not individual mediators i.e., no variable influenced by the exposure that influences any of the mediators and the outcome 8 / 32

27 Estimation: several different strategies Using the combination of mediator and outcome models Regression with analytic results (RR) if can derive them Fit an appropriate model for each of the mediators using and X as predictors Fit an appropriate model for the outcome using, X and the mediators Predict potential outcomes and/or causal effects conditional on X for each X pattern (or for each individual in the sample) verage over distribution of X (or over sample) to get average causal effects This strategy is used by section 3 of VanderWeele & Vansteelandt (2013) and guyen et al. (2016) for causally unordered multiple mediators. 9 / 32

28 Estimation: several different strategies Using the combination of mediator and outcome models Regression-biased simulation (Imai et al. s algorithms 1 and 2) Fit appropriate models for the mediators using and X (and possibly earlier mediators) as predictors Fit an appropriate model for the outcome using, X and the mediators Estimate/simulate parameters and simulate potential mediators and potential outcomes Compute marginal causal effects using simulated potential outcomes If ordered mediators, simulate potential mediators sequentially. If unordered mediators, need a multivariate model for the mediators to capture their dependence after accounting for and X. 10 / 32

29 Estimation: several different strategies Using the combination of exposure and outcome models Weighting and imputation Fit an appropriate model for the exposure using X as predictors, and compute weights W i = P(= i ) P(= i X =X i ) Fit an appropriate model for the outcome using, X and the mediators, then combine each individual s mediator with the exposure condition he/she did not experience to impute a counterfactual (green) outcome based on this model Take weighted averages of observed potential outcomes in the exposed and unexposed groups as estimates of E[ 11 ] and E[ 00 ] Take weighted averages of imputed potential outcomes in the two groups as estimates of E[ 11 ] and E[ 01 ] Contrast these mean potential outcomes to estimate marginal causal effects This strategy was proposed by VanderWeele & Vansteelandt (2013) for the multiple-mediator case to avoid having to model the mediators. 11 / 32

30 Estimation: several different strategies Using the combination of exposure and mediator models Weighting and imputation (Lange et al., 2013) Fit an appropriate model for the exposure using X as predictors Fit an appropriate model for each of the mediators using, X Check if the mediators are independent of one another given, X If yes, proceed. If no, stop! Use a similar imputation scheme as in Lange et al. (2012): construct new dataset replicating each observation 2 k times (k is number of mediators) and adding k variables,,... (each indexing the exposure level for a [potential] mediator) and letting these them represent all 0/1 possibilities Use similar weights as in Lange et al. (2012): W i = = i P(= i = i,x =X i ) = i X =X i P(= i = i,x =X i ) P(= i = i,x =X i ) P(= i = i,x =X i )... Fit a marginal structural model regressing the outcome on,,,... and their interaction terms, using weights and adjusting for clustering pplies to the SPECIL CSE where the mediators are independent given, X. ay get extreme weights. 12 / 32

31 RR ex. 1: continuous unordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ 13 / 32

32 RR ex. 1: continuous unordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn 13 / 32

33 RR ex. 1: continuous unordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ 11 1 X ], E[ 10 0 X ] and E[ 00 0 X ] 13 / 32

34 RR ex. 1: continuous unordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ 11 1 X ], E[ 10 0 X ] and E[ 00 0 X ] DE( 0 X ) = γ 1 + γ 4 (α 0 + α 2 X ) + γ 5 (β 0 + β 2 X )+γ 7 [(α 0 + α 2 X )(β 0 + β 2 X ) + Cov(ɛ, ɛ )] IE(1 X ) = (γ 2 + γ 4 )α 1 + (γ 3 + γ 5 )β 1 + (γ 6 + γ 7 )[α 1 (β 0 + β 1 + β 2 X ) + β 1 (α 0 + α 2 X )] 13 / 32

35 RR ex. 1: continuous unordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ 11 1 X ], E[ 10 0 X ] and E[ 00 0 X ] DE( 0 X ) = γ 1 + γ 4 (α 0 + α 2 X ) + γ 5 (β 0 + β 2 X )+γ 7 [(α 0 + α 2 X )(β 0 + β 2 X ) + Cov(ɛ, ɛ )] IE(1 X ) = (γ 2 + γ 4 )α 1 + (γ 3 + γ 5 )β 1 + (γ 6 + γ 7 )[α 1 (β 0 + β 1 + β 2 X ) + β 1 (α 0 + α 2 X )] based on the following (CHECK THE LGEBR!) E[ 111 X ] = (γ0 + γ1) + (γ2 + γ4)(α0 + α1 + α2x ) + (γ3 + γ5)(β0 + β1 + β2x )+ (γ 6 + γ 7)[(α 0 + α 1 + α 2X )(β 0 + β 1 + β 2X ) + Cov(ɛ, ɛ )] + γ 8X E[ 100 X ] = (γ0 + γ1) + (γ2 + γ4)(α0 + α2x ) + (γ3 + γ5)(β0 + β2x )+ (γ 6 + γ 7)[(α 0 + α 2X )(β 0 + β 2X ) + Cov(ɛ, ɛ )] + γ 8X E[ 000 X ] = γ0 + γ2(α0 + α2x ) + γ3(β0 + β2x )+ γ 6[(α 0 + α 2X )(β 0 + β 2X ) + Cov(ɛ, ɛ )] + γ 8X 13 / 32

36 RR ex. 2: continuous ordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, Var(ɛ ) = σ 2,, X = β 0 + β 1 + β 2 + β 3 + β 4 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn 14 / 32

37 RR ex. 2: continuous ordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, Var(ɛ ) = σ 2,, X = β 0 + β 1 + β 2 + β 3 + β 4 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ X ], E[ X ] and E[ X ] 14 / 32

38 RR ex. 2: continuous ordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, Var(ɛ ) = σ 2,, X = β 0 + β 1 + β 2 + β 3 + β 4 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ X ], E[ X ] and E[ X ] DE( 0 X ) = γ 1 + γ 4 (α 0 + α 2 X ) + γ 5 [β 0 + (β 2 + β 3 )(α 0 + α 2 X ) + β 4 X ]+ γ 7 {(α 0 + α 2 X )[β 0 + (β 2 + β 3 )(α 0 + α 2 X ) + β 4 X ] + (β 2 + β 3 )σ 2 } IE(1 X ) = (γ 2 + γ 4 )α 1 + (γ 3 + γ 5 )[β 1 + (β 2 + β 3 )α 1 ]+ { } α1 [β (γ 6 + γ 7 ) 0 + β 1 + (β 2 + β 3 )(α 0 + α 1 + α 2 X ) + β 4 X ]+ [β 1 + (β 2 + β 3 )α 1 ](α 0 + α 2 X ) 14 / 32

39 RR ex. 2: continuous ordered,, continuous, linear models ssume models, X = α 0 + α 1 + α 2 X + ɛ, Var(ɛ ) = σ 2,, X = β 0 + β 1 + β 2 + β 3 + β 4 X + ɛ,,, X = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 6 + γ 7 + γ 8 X + ɛ Controlled direct effect CDE(m, n) = γ 1 + γ 4 m + γ 5 n + γ 7 mn atural effects comparing E[ X ], E[ X ] and E[ X ] DE( 0 X ) = γ 1 + γ 4 (α 0 + α 2 X ) + γ 5 [β 0 + (β 2 + β 3 )(α 0 + α 2 X ) + β 4 X ]+ γ 7 {(α 0 + α 2 X )[β 0 + (β 2 + β 3 )(α 0 + α 2 X ) + β 4 X ] + (β 2 + β 3 )σ 2 } IE(1 X ) = (γ 2 + γ 4 )α 1 + (γ 3 + γ 5 )[β 1 + (β 2 + β 3 )α 1 ]+ { } α1 [β (γ 6 + γ 7 ) 0 + β 1 + (β 2 + β 3 )(α 0 + α 1 + α 2 X ) + β 4 X ]+ [β 1 + (β 2 + β 3 )α 1 ](α 0 + α 2 X ) based on the following (CHECK THE LGEBR, SERIOUSL!) E[ X ] = (γ 0 + γ 1) + (γ 2 + γ 4)(α 0 + α 1 + α 2X ) + (γ 3 + γ 5)[β 0 + β 1 + (β 2 + β 3)(α 0 + α 1 + α 2X ) + β 4X ]+ (γ 6 + γ 7){(α 0 + α 1 + α 2X )[β 0 + β 1 + (β 2 + β 3)(α 0 + α 1 + α 2X ) + β 4X ] + (β 2 + β 3)σ} 2 + γ 8X E[ X ] = (γ 0 + γ 1) + (γ 2 + γ 4)(α 0 + α 2X ) + (γ 3 + γ 5)[β 0 + (β 2 + β 3)(α 0 + α 2X ) + β 4X ]+ (γ 6 + γ 7){(α 0 + α 2X )[β 0 + (β 2 + β 3)(α 0 + α 2X ) + β 4X ] + (β 2 + β 3)σ} 2 + γ 8X E[ X ] = γ 0 + γ 2(α 0 + α 2X ) + γ 3[β 0 + (β 2 + β 3)(α 0 + α 2X ) + β 4X ]+ γ 6{(α 0 + α 2X )[β 0 + (β 2 + β 3)(α 0 + α 2X ) + β 4X ] + (β 2 + β 3)σ} 2 + γ 8X 14 / 32

40 RR ex. 3: rare binary (logit model), continuous unordered, (linear normal model) (VanderWeele & V, 2013) ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ, ( ) (( ) ɛ 0, Σ ɛ 0) logit[p( = 1,,, X )] = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 8 X Log controlled direct effect Log conditional natural effects (not allowing mediator-mediator interaction) log[cde OR (m, n)] = γ 1 + γ 4 m + γ 5 n log[de OR ( 0 X )] γ 1 + γ 4 (α 0 + α 2 X + γ 2 σ 2 ) + γ 5(β 0 + β 2 X + γ 3 σ 2 )+ ( ) ( ) ( ) ( ) γ2 + γ γ2 + γ Σ 4 γ2 γ2 0.5 Σ γ 3 + γ 5 γ 3 + γ 5 γ 3 γ 3 log[ie OR (1 X )] (γ 2 + γ 4 )α 1 + (γ 3 + γ 5 )β 1 15 / 32

41 RR ex. 4: binary (probit model), continuous/ordinal unordered, (linear normal/probit model) (guyen et al. 2016) ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ, ( ) (( ) ɛ 0, Σ ɛ 0) (slight changes if ordinal, ) probit[p( = 1,,, X )] = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 8 X (not allowing mediator-mediator interaction) 16 / 32

42 RR ex. 4: binary (probit model), continuous/ordinal unordered, (linear normal/probit model) (guyen et al. 2016) ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ, ( ) (( ) ɛ 0, Σ ɛ 0) (slight changes if ordinal, ) probit[p( = 1,,, X )] = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 8 X atural effects (not allowing mediator-mediator interaction) DE( 0) = P( 10 0 = 1) P( 00 0 = 1) IE(1 ) = P( 11 1 = 1) P( 10 0 = 1) 16 / 32

43 RR ex. 4: binary (probit model), continuous/ordinal unordered, (linear normal/probit model) (guyen et al. 2016) ssume models, X = α 0 + α 1 + α 2 X + ɛ, X = β 0 + β 1 + β 2 X + ɛ, ( ) (( ) ɛ 0, Σ ɛ 0) (slight changes if ordinal, ) probit[p( = 1,,, X )] = γ 0 + γ 1 + γ 2 + γ 3 + γ 4 + γ 5 + γ 8 X atural effects (not allowing mediator-mediator interaction) DE( 0) = P( 10 0 = 1) P( 00 0 = 1) IE(1 ) = P( 11 1 = 1) P( 10 0 = 1) based on potential outcome probabilties P(111 = 1) = P(111 = 1 X = x)p(x = x) = x x P( = 1) = P( = 1 X = x)p(x = x) = x x P(000 = 1) = P(000 = 1 X = x)p(x = x) = x x Φ (γ0 + γ1) + (γ2 + γ4)(α0 + α1 + α2x) + (γ3 + γ5)(β0 + β1 + β2x) + γ8x (γ2 ) ( ) P(X = x) + γ4 γ2 + γ4 Σ + 1 γ3 + γ5 γ3 + γ5 Φ (γ0 + γ1) + (γ2 + γ4)(α0 + α2x) + (γ3 + γ5)(β0 + β2x) + γ8x (γ2 ) ( ) P(X = x) + γ4 γ2 + γ4 Σ + 1 γ3 + γ5 γ3 + γ5 Φ γ0 + γ2(α0 + α2x) + γ3(β0 + β2x) + γ8x (γ2 ) ( ) γ2 P(X = x) Σ + 1 γ3 γ3 16 / 32

44 Estimation: comments Regression w/ analytic result requires deriving analytic results for each new situation. lso, the multiple models are multiple chances of misspecification. Regression-based simulation does not require deriving things, but could be more computationally intensive. Same risk of model misspecification. V & V s imputation and weighting method is simplest. It requires that the exposure model is correctly specified, so need a good collection of X variables. Lange et al. s imputation and weighting method should be reserved to situations with few binary/discrete mediators and requires that the mediators are independent given, X (check this!). 17 / 32

45 2. Different mediation pathways (w/ causally ordered mediators) 18 / 32

46 Definition of effects With a single mediator, the potential outcomes relevant to natural direct and indirect effects are of the form aa [a, (a )] where a and a can be the same or different. With two causally ordered mediators, such potential outcomes are aa a a {a, (a ), [a, (a )]} where a, a, a and a can be the same or different. 19 / 32

47 Definition of effects With a single mediator, there are two ways to decompose total causal effect into natural direct and indirect effects. Let s take one decomposition TE = DE( 0) + IE(1 ) and extend the idea to the case with causally ordered,. 20 / 32

48 00 [0, (0)] 21 / 32

49 00 [0, (0)] 10 [1,(0)] 21 / 32

50 00 [0, (0)] 10 [1,(0)] 11 [1, (1)] 21 / 32

51 00 10 [0, (0)] [1,(0)] DE( 0) 11 [1, (1)] IE(1 ) 21 / 32

52 00 00 {0, (0), [0, (0)]} {1,(0), [0, (0)]} {1, (1),[0, (0)]} {1, (1), [1,(0)]} {1, (1), [1, (1)]} 22 / 32

53 00 00 {0, (0), [0, (0)]} DE( 000) {1,(0), [0, (0)]} IE(1 00) {1, (1),[0, (0)]} IE(11 0) {1, (1), [1,(0)]} IE(111 ) {1, (1), [1, (1)]} 22 / 32

54 That was only one way to decompose TE. The number of possible decompositions is k! where k is the number of a, a, a,... indexes (Daniel et al. 2015). Single mediator: aa k = 2 2! = 2 decompositions Two mediators: aa a a k = 4 4! = 24 decompositions 23 / 32

55 n decomposition intuition: There are four paths for information to flow from to : but they are blocked. We need to turn them on, one at a time. We have just turned them on in the above order and got the decomposition TE = DE( 000) + IE(1 00) + IE(11 0) + IE(111 ). There are 4!=24 ways to order these 4 paths, hence 24 decompositions. 24 / 32

56 Identifying assumptions Same as the combined case no uncontrolled exposure-outcome confounding no uncontrolled mediator-outcome confounding no uncontrolled exposure-mediator confounding no mediator-outcome confounder influenced by exposure applies to the collection of mediators, not individual mediators i.e., no variable influenced by the exposure that influences any of the mediators and the outcome and more 25 / 32

57 Estimation: cannot avoid modeling the mediators Strategies: Regression with analytic results Regression-based simulation Details need to be worked out for specific cases depending on the models used. Generally requires additional assumption about cross-world correlation between the errors of, Corr(ɛ 0, ɛ 1 ). 26 / 32

58 RR ex. 5: continuous ordered,, continuous, linear models ssume models (a) = α 0 + α 1 a + α 2 X + ɛ a, Var(ɛ a ) = σ 2 (a, m) = β 0 + β 1 a + β 2 m + β 3 am + β 4 X + ɛ am (a, m, n) = γ 0 + γ 1 a + γ 2 m + γ 3 n + γ 4 am + γ 5 an + γ 6 mn + γ 7 amn + γ 8 X + ɛ amn atural effects (CHECK DERIVTIO!) DE( 000 X ) = γ 1 + γ 4 α 0 + (γ 5 + γ 6 α 0 )(β 0 + β 2 α 0 ) + γ 6 β 2 σ+ 2 (γ 4 + γ 5 β 2 + 2γ 6 α 0 β 2 + γ 6 β 0 )(α 2 X ) + (γ 5 + γ 6 α 0 )(β 4 X )+ γ 6 β 2 (α 2 X ) 2 + γ 6 (α 2 X )(β 4 X ) IE(1 00 X ) = (γ 2 + γ 4 )α 1 + (γ 6 + γ 7 )α 1 (β 0 + β 2 α 0 ) (γ 6 + γ 7 )β 2 σ[1 2 Corr(ɛ 1, ɛ 0 )]+ (γ 6 + γ 7 )α 1 β 2 (α 2 X ) + (γ 6 + γ 7 )α 1 (β 4 X ) IE(11 0 X ) = (γ 3 + γ 5 )(β 1 + β 3 α 0 ) + (γ 6 + γ 7 )[(α 0 + α 1 )(β 1 + β 3 α 0 ) + β 3 σcorr(ɛ 2 1, ɛ 0 )]+ [(γ 3 + γ 5 )β 3 + (γ 6 + γ 7 )(β 1 + 2β 3 α 0 + β 3 α 1 )](α 2 X )+ (γ 6 + γ 7 )β 3 (α 2 X ) 2 IE(111 X ) = (γ 3 + γ 5 )(β 2 + β 3 )α 1 + (γ 6 + γ 7 )(β 2 + β 3 ) { (α 0 + α 1 )α 1 + σ[1 2 Corr(ɛ 1, ɛ 0 )] } + (γ 6 + γ 7 )(β 2 + β 3 )α 1 (α 2 X ) If interaction, need to make assumption about cross-world error correlation for. 27 / 32

59 Estimation: my suggestions Check the methods literature for new results/software that are relevant. If not, first, fit appropriate models for the mediators and the outcome then evaluate the situation: based on these models can I derive formulas for potential outcomes and causal effects? or can I adapt/develop an algorithm for simulation of potential mediators and potential outcomes? if unsure, get help be patient and willing to work hard to figure things out lso, things get complicated fast. If you have a complex structure be sure to draw it out clearly then consider which mediators you are willing to consider jointly 28 / 32

60 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Which paths do you want to turn on? 29 / 32

61 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Which paths do you want to turn on? 29 / 32

62 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Let s turn on the red paths first. Which potential outcomes are we comparing? IE(00 ) = E[ ] E[ ] 29 / 32

63 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Let s turn on the red paths first. Which potential outcomes are we comparing? IE(00 ) = E[ ] E[ ] 29 / 32

64 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths after turning on the black path. IE(10 ) = E[ ] E[ ] 29 / 32

65 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths after turning on the black path. IE(10 ) = E[ ] E[ ] 29 / 32

66 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths after turning on the path. IE(01 ) = E[ ] E[ ] 29 / 32

67 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths after turning on the path. IE(01 ) = E[ ] E[ ] 29 / 32

68 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths last. IE(11 ) = E[ ] E[ ] 29 / 32

69 back to effect definition and decomposition: a little bonus quiz How do you define effect mediated by? Say we turn on the red paths last. IE(11 ) = E[ ] E[ ] 29 / 32

70 back to effect definition and decomposition: a little bonus quiz Comments the effect mediated by one mediator may be a combination of more than one pathway can also think about effect that goes through only, through only, and through both if interested in a single mediator but there is a post-exposure confounder, you can treat this as a two mediator system, define IE for that mediator accordingly, and it is identified 30 / 32

71 Session overview ultiple-mediator cases and types of mediation effects ediation by all the mediators combined ediation via different pathways 31 / 32

72 References cited Daniel R, De Stavola BL, Cousens S, Vansteelandt S. (2015). Causal mediation analysis with multiple mediators. Biometrics. 71:1-14. Lange T, Rasmussen, Thygesen LC. ssessing natural direct and indirect effects through multiple pathways. (2014). merican Journal of Epidemiology. 2014;179(4): Lange T, Vansteelandt S, Bekaert. (2012). simple unified approach for estimating natural direct and indirect effects. merican Journal of Epidemiology. 176(3): guyen TQ, Webb-Vargas, Koning IH, Stuart E. (2016). Causal mediation analysis with a binary outcome and multiple continuous or ordinal mediators: Simulations and application to an alcohol intervention. Structural Equation odeling: ultidisciplinary Journal. 23(3): VanderWeele TJ, Vansteelandt S. (2013). ediation analysis with multiple mediators. Epidemiologic ethods. 2(1): / 32