Causal mediation analysis: Definition of effects and common identification assumptions

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1 Causal mediation analysis: Definition of effects and common identification assumptions Trang Quynh Nguyen Seminar on Statistical Methods for Mental Health Research Johns Hopkins Bloomberg School of Public Health term 4 session 1 - March 24, / 26

2 Term 4 seminar overview Session 1: Definition of causal effects and common identifying assumptions Sessions 2 and 3: Estimation for the single mediator case Session 4: A couple of other topics, e.g., multiple mediators, post-treatment confounder, etc. 2 / 26

3 Session overview A bit of motivation Traditional and causal approaches to mediation Definition of causal effects Identification assumptions The mediation formula 3 / 26

4 The mediation question mediator M exposure/ intervention/ treatment T Y outcome What is the effect of T on Y that goes through M? What is the effect of T on Y that does not go through M? 4 / 26

5 The mediation question bullying M sexual minority status T Y suicide attempt What is the effect of T on Y that goes through M? What is the effect of T on Y that does not go through M? 4 / 26

6 Traditional (Baron & Kenny) approach to mediation Difference-in-coefficients method Product-of-coefficients method T c M Y a M b T c b Y T c Y total effect = c direct effect = c indirect effect = c c direct effect = c indirect effect = ab total effect = c + ab Only in the linear case: c c = ab, c = c + ab 5 / 26

7 My problem with binary outcome bullying M sexual minority status T Y suicide attempt want to retain OR as effect measure want to use c c 6 / 26

8 My problem with binary outcome bullying M sexual minority status T Y suicide attempt want to retain OR as effect measure want to use c c but not appropriate to compare c to c (noncollapsibility/different scales) can rescale: c c rescaled but lose OR interpretation 6 / 26

9 My problem with binary outcome bullying M sexual minority status T Y suicide attempt want to retain OR as effect measure want to use c c but not appropriate to compare c to c (noncollapsibility/different scales) can rescale: c c rescaled but lose OR interpretation and even if could compare, what does exp(c c ) means? ratio of two ORs? yes OR associated with a difference in bullying between sexual minority and sexual majority youth? a question mark! 6 / 26

10 Mediation questions from a more causal perspective mediator M exposure/ intervention/ treatment T Y outcome If could block the path from T to Y, what would be the effect on Y? If could block effect of T on M, what would remain of effect of T on Y? If could fix M at a specific level m, what would be the effect of T on Y? 7 / 26

11 Mediation questions from a more causal perspective bullying M sexual minority status T Y suicide attempt If somehow sexual minority youth experienced the level of bullying that sexual majority youth experience (and not the level they actually experience), how would the suicide attempt rate compare with actual? 8 / 26

12 Mediation questions from a more causal perspective bullying M sexual minority status T Y suicide attempt If somehow sexual minority youth experienced the level of bullying that sexual majority youth experience (and not the level they actually experience), how would the suicide attempt rate compare with actual? If we could intervene to set the level of bullying experienced to 0 for all youth, what would be the difference is suicide attempt between being sexual minority and being sexual majority? 8 / 26

13 Some historical benchmarks Wright s path analysis (1934) the Baron and Kenny tradition (3 decades) Judd & Kenny (1981) and Baron & Kenny (1986) evolved to more complex situations wide-spread application, B&K 58,109 Google citations 2016/03/16 causal inference approach ( 1.5 decades) Robins & Greenland (1992) Pearl (2001) explosion of theoretical and methodological work since then 9 / 26

14 Criticisms of traditional mediation analysis B&K s original prescription: causal steps, ignoring confounders c c ab if model is non-linear (e.g., probit, logit) neither ab nor c c apply if allow TM interaction hidden assumptions if interpret effects as causal not clear what we are estimating 10 / 26

15 Causal inference approach to mediation Define causal mediation effects based on the potential outcome (or causal graphs) framework Clarify assumptions required to identify such effects from observed data Develop methods to estimate such effects 11 / 26

16 Definition of causal effects T : exposure/treatment/intervention status (binary for simplicity) M: mediator value Y : outcome For individual i, Y i (t): potential outcome if exposure were set to t (t = 0, 1) Total causal effect: TE i = Y i (1) Y i (0) 12 / 26

17 Definition of causal effects T : exposure/treatment/intervention status (binary for simplicity) M: mediator value Y : outcome For individual i, Y i (t): potential outcome if exposure were set to t (t = 0, 1) Total causal effect: TE i = Y i (1) Y i (0) Y i (t, m): potential outcome if exposure were set to t and mediator to m 12 / 26

18 Definition of causal effects T : exposure/treatment/intervention status (binary for simplicity) M: mediator value Y : outcome For individual i, Y i (t): potential outcome if exposure were set to t (t = 0, 1) Total causal effect: TE i = Y i (1) Y i (0) Y i (t, m): potential outcome if exposure were set to t and mediator to m Controlled direct effect: CDE i (m) = Y i (1, m) Y i (0, m) 12 / 26

19 Definition of causal effects T : exposure/treatment/intervention status (binary for simplicity) M: mediator value Y : outcome For individual i, Y i (t): potential outcome if exposure were set to t (t = 0, 1) Total causal effect: TE i = Y i (1) Y i (0) Y i (t, m): potential outcome if exposure were set to t and mediator to m Controlled direct effect: CDE i (m) = Y i (1, m) Y i (0, m) M i (t): potential mediator value if exposure were set to t (t = 0, 1) Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) 12 / 26

20 Definition of causal effects Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) 13 / 26

21 Definition of causal effects Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) With Y i (1, M i (1)), Y i (0, M i (0)), re-express TE: TE i = Y i (1) Y i (0) = Y i (1, M i (1)) Y i (0, M i (0)) (composition assumption) 13 / 26

22 Definition of causal effects Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) With Y i (1, M i (1)), Y i (0, M i (0)), re-express TE: TE i = Y i (1) Y i (0) = Y i (1, M i (1)) Y i (0, M i (0)) (composition assumption) Natural direct & indirect effects using Y i (1, M i (0)): 13 / 26

23 Definition of causal effects Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) With Y i (1, M i (1)), Y i (0, M i (0)), re-express TE: TE i = Y i (1) Y i (0) = Y i (1, M i (1)) Y i (0, M i (0)) (composition assumption) Natural direct & indirect effects using Y i (1, M i (0)): TE i = [Y i (1, M i (1)) Y i (1, M i (0))] + [Y i (1, M i (0)) Y i (0, M i (0))] = NIE i (1) + NDE i (0) 13 / 26

24 Definition of causal effects Y i (t, M(t )): potential outcome if exposure were set to t and mediator were set to its natural level under exposure t (t, t = 0, 1) With Y i (1, M i (1)), Y i (0, M i (0)), re-express TE: TE i = Y i (1) Y i (0) = Y i (1, M i (1)) Y i (0, M i (0)) (composition assumption) Natural direct & indirect effects using Y i (1, M i (0)): TE i = [Y i (1, M i (1)) Y i (1, M i (0))] + [Y i (1, M i (0)) Y i (0, M i (0))] = NIE i (1) + NDE i (0) Natural direct & indirect effects using Y i (0, M i (1)): TE i = [Y i (1, M i (1)) Y i (0, M i (1))] + [Y i (0, M i (1)) Y i (0, M i (0))] = NDE i (1) + NIE i (0) 13 / 26

25 Definition of causal effects Individual causal effects are not identified, because for each individual i, we observe only one potential outcome Y i (T i ) = Y i (T i, M i (T i )) = Y i. Average causal effects are of interest. 14 / 26

26 Definition of causal effects Individual causal effects are not identified, because for each individual i, we observe only one potential outcome Y i (T i ) = Y i (T i, M i (T i )) = Y i. Average causal effects are of interest. abbreviate: E[Y tmt ] = E[Y (t, M(t ))], E[Y tm ] = E[Y (t, m)] 14 / 26

27 Definition of causal effects Individual causal effects are not identified, because for each individual i, we observe only one potential outcome Y i (T i ) = Y i (T i, M i (T i )) = Y i. Average causal effects are of interest. abbreviate: E[Y tmt ] = E[Y (t, M(t ))], E[Y tm ] = E[Y (t, m)] TE = E[Y 1M1 ] E[Y 0M0 ] { NDE(0) = E[Y1M0 ] E[Y 0M0 ] NIE(1) = E[Y 1M1 ] E[Y 1M0 ] or { NIE(0) = E[Y0M1 ] E[Y 0M0 ] NDE(1) = E[Y 1M1 ] E[Y 0M1 ] CDE(m) = E[Y 1m ] E[Y 0m ] (note different terms used by Robins, Pearl and others) 14 / 26

28 In what situations would CDE be of interest? 15 / 26

29 In what situations would NDE or NIE be of interest? 16 / 26

30 Identification assumptions no uncontrolled confounding of relationships ignorability of variables exchangeability of units/potential outcomes 17 / 26

31 Identification assumptions no uncontrolled confounding of relationships ignorability of variables exchangeability of units/potential outcomes Conditional sequential ignorability assumptions (Imai et al., 2010) X : a set of measured pre-treatment covariates i1. ignorability of treatment assigned: {Y t m, M t} T X i2. ignorability of potential mediators: Y t m M t T, X 17 / 26

32 Identification assumptions no uncontrolled confounding of relationships ignorability of variables exchangeability of units/potential outcomes Conditional sequential ignorability assumptions (Imai et al., 2010) X : a set of measured pre-treatment covariates i1. ignorability of treatment assigned: {Y t m, M t} T X i2. ignorability of potential mediators: Y t m M t T, X Confounding assumptions (VanderWeele & Vansteelandt s version) C: a set of measured covariates c1. no uncontrolled exposure-outcome confounding: Y tm T C c2. no uncontrolled mediator-outcome confounding: Y tm M T, C c3. no uncontrolled exposure-mediator confounding: M t T C c4. no mediator-outcome confounder affected by exposure (L) c1 + c3 = i1 c2 + c4 i2 17 / 26

33 Identification assumptions no uncontrolled confounding of relationships ignorability of variables exchangeability of units/potential outcomes Conditional sequential ignorability assumptions (Imai et al., 2010) X : a set of measured pre-treatment covariates i1. ignorability of treatment assigned: {Y t m, M t} T X i2. ignorability of potential mediators: Y t m M t T, X Confounding assumptions (VanderWeele & Vansteelandt s version) C: a set of measured covariates c1. no uncontrolled exposure-outcome confounding: Y tm T C c2. no uncontrolled mediator-outcome confounding: Y tm M T, C c3. no uncontrolled exposure-mediator confounding: M t T C c4. no mediator-outcome confounder affected by exposure (L) c1 + c3 = i1 c2 + c4 i2 NDE and NIE require all c1-c4 (or i1 and i2); CDE requires c1 and c2. 17 / 26

34 Identification assumptions X 3 i1 M T Y X 1 18 / 26

35 Identification assumptions L X 2 i2 M T Y 18 / 26

36 Identification assumptions c1 M T Y X 1 18 / 26

37 Identification assumptions L X 2 c2 M T Y 18 / 26

38 Identification assumptions X 3 c3 M T Y 18 / 26

39 Identification assumptions L c4 M T Y 18 / 26

40 Identification assumptions X 3 L X 2 i1-2 c1-4 M T Y X 1 18 / 26

41 Identification assumptions Pearl points out (Pearl, 2014) X (1), X (2), X (3) do not need to be pre-exposure, just not influenced by exposure X (1), X (2), X (3) deconfound can replace i1 (or c1+c3) by other ways of identifying M t, Y t m 19 / 26

42 Identification intuition Consider person i with T i = 1. Observe: M i (1) = M i = m and Y i (1, m ) = Y i (1, M i (1)) = Y i = y For NDE(0) and NIE(1), need Y i (0, M i (0)) and Y i (1, M i (0)) 20 / 26

43 Identification intuition Consider person i with T i = 1. Observe: M i (1) = M i = m and Y i (1, m ) = Y i (1, M i (1)) = Y i = y For NDE(0) and NIE(1), need Y i (0, M i (0)) and Y i (1, M i (0)) under c1 (no uncontrolled T -Y confounding), Y i (0, M i (0)) is informed by the outcomes of persons j with the same X (1) values who are in the control condition: Y j = Y j (0, M j (0)) Ŷ i (0, M i (0)) = y 20 / 26

44 Identification intuition Consider person i with T i = 1. Observe: M i (1) = M i = m and Y i (1, m ) = Y i (1, M i (1)) = Y i = y For NDE(0) and NIE(1), need Y i (0, M i (0)) and Y i (1, M i (0)) under c1 (no uncontrolled T -Y confounding), Y i (0, M i (0)) is informed by the outcomes of persons j with the same X (1) values who are in the control condition: Y j = Y j (0, M j (0)) Ŷ i (0, M i (0)) = y under c3 (no uncontrolled T -M confounding), M i (0) is informed by the mediator values of persons k with the same X (3) values who are in the control condition: M k = M k (0) ˆM i (0) = m 20 / 26

45 Identification intuition Consider person i with T i = 1. Observe: M i (1) = M i = m and Y i (1, m ) = Y i (1, M i (1)) = Y i = y For NDE(0) and NIE(1), need Y i (0, M i (0)) and Y i (1, M i (0)) under c1 (no uncontrolled T -Y confounding), Y i (0, M i (0)) is informed by the outcomes of persons j with the same X (1) values who are in the control condition: Y j = Y j (0, M j (0)) Ŷ i (0, M i (0)) = y under c3 (no uncontrolled T -M confounding), M i (0) is informed by the mediator values of persons k with the same X (3) values who are in the control condition: M k = M k (0) ˆM i (0) = m under c2+c4 (no uncontrolled M-Y confounding & no M-Y confounder affected by T ), Y i (1, M i (0)) = Y i (1, m ) is informed by the outcomes of persons l with the same X (2) values who are in the intervention/exposed condition, whose mediator value is m : Y l = Y l (1, M l (1)) = Y l (1, m ) Ŷ i (1, M i (0)) = y 20 / 26

46 Identification of NDE and NIE Consider TE = E[Y 1M1 ] E[Y 0M0 ] NDE(0) = E[Y 1M0 ] E[Y 0M0 ] NIE(1) = E[Y 1M1 ] E[Y 1M0 ] Imai et al. s theorem about identification of E[Y tmt ]: in the simplified case with no X : E[Y 1M1 ] = E[Y T = 1] = m E[Y T = 1, M = m]p(m = m T = 1) E[Y 0M0 ]= E[Y T = 0] = m E[Y T = 0, M = m]p(m = m T = 0) E[Y 1M0 ] = m E[Y T = 1, M = m]p(m = m T = 0) 21 / 26

47 Identification of NDE and NIE Consider TE = E[Y 1M1 ] E[Y 0M0 ] NDE(0) = E[Y 1M0 ] E[Y 0M0 ] NIE(1) = E[Y 1M1 ] E[Y 1M0 ] Imai et al. s theorem about identification of E[Y tmt ]: in the simplified case with no X : E[Y 1M1 ] = E[Y T = 1] = m E[Y T = 1, M = m]p(m = m T = 1) E[Y 0M0 ] = E[Y T = 0] = m E[Y T = 0, M = m]p(m = m T = 0) E[Y 1M0 ] = m E[Y T = 1, M = m]p(m = m T = 0) 21 / 26

48 Identification of NDE and NIE Consider TE = E[Y 1M1 ] E[Y 0M0 ] NDE(0) = E[Y 1M0 ] E[Y 0M0 ] NIE(1) = E[Y 1M1 ] E[Y 1M0 ] Imai et al. s theorem about identification of E[Y tmt ]: in the simplified case with no X : E[Y 1M1 ] = E[Y T = 1] = m E[Y T = 1, M = m]p(m = m T = 1) E[Y 0M0 ] = E[Y T = 0] = m E[Y T = 0, M = m]p(m = m T = 0) E[Y 1M0 ] = m E[Y T = 1, M = m]p(m = m T = 0) 21 / 26

49 Identification of NDE and NIE Consider TE = E[Y 1M1 ] E[Y 0M0 ] NDE(0) = E[Y 1M0 ] E[Y 0M0 ] NIE(1) = E[Y 1M1 ] E[Y 1M0 ] Imai et al. s theorem about identification of E[Y tmt ]: in the realistic case with X : E[Y 1M1 ] = E[Y T = 1, M = m, X = x]p(m = m T = 1, X = x)p(x = x) x m E[Y 0M0 ] = E[Y T = 0, M = m, X = x]p(m = m T = 0, X = x)p(x = x) x m E[Y 1M0 ] = E[Y T = 1, M = m, X = x]p(m = m T = 0, X = x)p(x = x) x m 22 / 26

50 Identification of NDE and NIE E[Y 1M1 ] = x E[Y 0M0 ] = x E[Y 1M0 ] = x E[Y T = 1, M = m, X = x]p(m = m T = 1, X = x)p(x = x) m E[Y T = 0, M = m, X = x]p(m = m T = 0, X = x)p(x = x) m E[Y T = 1, M = m, X = x]p(m = m T = 0, X = x)p(x = x) m Pearl s (2011, 2012) causal mediation formula follows NDE(0) = E[Y 1M0 ] E[Y 0M0 ] = x m { E[Y T = 1, M = m, X = x] E[Y T = 0, M = m, X = x] } P(M = m T = 0, X = x)p(x = x) NIE(1) = E[Y 1M1 ] E[Y 1M0 ] = [ ] P(M = m T = 1, X = x) E[Y T = 1, M = m, X = x] P(X = x) P(M = m T = 0, X = x) x m 23 / 26

51 Assumptions in addition to i1-2 and c1-4 General causal inference positivity SUTVA consistency: Y i (T i ) = Y i Mediation-specific composition: Y i (T i, M i (T i )) = Y i (T i ) 24 / 26

52 More complex decompositions of TE (so you know) 3-way (VanderWeele 2013) TE = NDE(0) + NIE(1) = NDE(0) + NIE(0) + [NIE(1) NIE(0)] = NDE(0) + NIE(0) + INT = PDE + PIE + INT (just switching names) 4-way using a reference mediator value (VanderWeele 2014) TE = NDE(0) + NIE(0) + [NIE(1) NIE(0)] = CDE(m ref ) + [NDE(0) CDE(m ref )] + NIE(0) + [NIE(1) NIE(0)] = CDE(m ref ) + INT ref + NIE(0) + INT med = CDE(m ref ) + INT ref + PIE + INT med (just switching names) m ref should be a meaningful zero value, e.g., no bullying. 25 / 26

53 References cited Baron RM, Kenny DA. (1986) The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51(6): Imai K, Keele L, Tingley D. (2010). A general approach to causal mediation analysis. Psychological Methods, 15(4): Judd CM, Kenny DA. (1981). Process analysis: Estimating mediation in treatment evaluations. Evaluation Review, 5(5): Pearl J. (2001). Direct and Indirect Effects. In: Proceedings of the Seventeenth Conference on Uncertainty and Artificial Intelligence. San Francisco: Morgan Kaufmann, Pearl J. (2011). The mediation formula: A guide to the as- sessment of causal pathways in non-linear models. In C Berzuini, P Dawid, L Bernardinelli (Eds.), Causal inference: Statistical perspectives and applications. Chichester, England: Wiley. Pearl J. (2012) The causal mediation formula a guide to the assessment of pathways and mechanisms. Prevention Science, 13(4): Pearl J. (2014). Interpretation and Identification of Causal Mediation. Psychological Methods, 19(4): Robins JM, Greenland S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2): VanderWeele TJ. (2013). A three-way decomposition of a total effect into direct, indirect, and interactive effects. Epidemiology, 24(2): VanderWeele TJ. (2014). A unification of mediation and interaction: A 4-way decomposition. Epidemiology, 25(5): VanderWeele TJ, Vansteelandt S. (2009). Conceptual issues concerning mediation, interventions and composition. Statistics and Its Interface, 2: Wright S. (1934). The method of path coefficients. The Annals of Mathematical Statistics, 5(3): / 26