Comparison of Three Approaches to Causal Mediation Analysis Donna L. Coffman David P. MacKinnon Yeying Zhu Debashis Ghosh
Introduction Mediation defined using the potential outcomes framework natural effects controlled effects principal strata effects. Different assumptions proposed for identifying and estimating these effects 2
Objectives 1. Compare the three definitions and the assumptions typically used to identify each. 2. Examine robustness to violations of assumptions. 3
Natural Effects Natural direct effects NDE M(1) =E[Y i (1,M i (1))-Y i (0,M i (1))] NDE M(0) =E[Y i (1,M i (0))-Y i (0,M i (0))] Natural indirect effects NIE 1 =E[Y i (1,M i (1))-Y i (1,M i (0))] NIE 0 =E[Y i (0,M i (1))-Y i (0,M i (0))] 4
Controlled Effects Controlled direct effect CDE m = E[Y i (1,m)-Y i (0,m)] Causal effect of M on Y b = E[Y i (1,m)-Y i (1,m )] 5
Principal Strata Effects Initially developed to handle non-compliance. Based on cross-classification of the potential values for the mediator. AACE=E[Y i (1)-Y i (0) M i (1)=M i (0)=1] NACE=E[Y i (1)-Y i (0) M i (1)=M i (0)=0] CACE=E[Y i (1)-Y i (0) M i (1)=1, M i (0)=0] DACE=E[Y i (1)-Y i (0) M i (1)=0,M i (0)=1] 6
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation 3 IPW 4 RPM 5 TSLS 1 Bayesian 2 No unmeasured confounders of No interactions between T & M T & Y M & Y T & M on Y T & X on Y M & X on Y Interactions of T & X on M No post-treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 1 Angrist JD, Imbens GW, Rubin DB. 1996. Identification of causal effects using instrumental variables. JASA 91:444-472. 2 Gallop R, Small DS, Lin JY, Elliott MR, Joffe MM, Ten Have TR. 2009. Mediation analysis with principal stratification. Stat Med 28:1108 1130. 3 Imai K, Keele L, Tingley D. 2010. A general approach to causal mediation analysis. Psych Methods 15:309-334. 4 Robins JM, Hernan MA. Brumback BA. 2000. Marginal structural models and causal inference in epidemiology. Epidemiology 11:550-560. 5 Ten Have TR, Joffe MM, Lynch KG, Brown GK, Maisto SA, Beck AT. 2007. Causal mediation analyses with rank preserving models. Biometrics 36:926-934. 7
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation 3 IPW 4 RPM 5 TSLS 1 Bayesian 2 No unmeasured confounders of No interactions between T & M T & Y M & Y T & M on Y T & X on Y M & X on Y Interactions between T & X on M No post-treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 1 Angrist JD, Imbens GW, Rubin DB. 1996. Identification of causal effects using instrumental variables. JASA 91:444-472. 2 Gallop R, Small DS, Lin JY, Elliott MR, Joffe MM, Ten Have TR. 2009. Mediation analysis with principal stratification. Stat Med 28:1108 1130. 3 Imai K, Keele L, Tingley D. 2010. A general approach to causal mediation analysis. Psych Methods 15:309-334. 4 Robins JM, Hernan MA. Brumback BA. 2000. Marginal structural models and causal inference in epidemiology. Epidemiology 11:550-560. 5 Ten Have TR, Joffe MM, Lynch KG, Brown GK, Maisto SA, Beck AT. 2007. Causal mediation analyses with rank preserving models. Biometrics 36:926-934. 8
Research Questions How biased are the estimates if there is an unmeasured confounder of M and Y? if there is an interaction between T and M? if there is a post-treatment confounder? Principal strata effects: how biased if the monotonicity assumption or exclusion restriction is violated? 9
Confounding Scenarios no confounders pre-treatment confounder, c 0, of M and Y pre-treatment confounder, c 0, and posttreatment confounder, c 1, of M and Y 10
Data Generation Exposure T~Bernoulli (0.5) Mediator Confounders c 0 ~N(0,1) c 1 ~T+N(0,1) Outcome 11
1000 replications N= 500 and 100 Method No adjustment for confounder in pretreatment confounding condition: unmeasured confounding of M and Y. Adjustment for both pre- and post-treatment confounders in post-treatment confounding condition. 12
Results How biased are the estimates if there is an unmeasured pre-treatment confounder of M and Y? TSLS-IV unbiased NIE, NDE, CDE slightly biased (.02) b for IPW and RPM biased (.06) Bayesian PS biased (.07) 13
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation IPW RPM TSLS Bayesian No unmeasured confounders of No interactions between T & M T & Y M & Y T & M on Y T & X on Y M & X on Y Interactions between T & X on M No post -treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 14
Results How biased are the estimates if there is a post-treatment confounder? TSLS-IV, NDE, RPM CDE, and Bayesian PS biased (-.2) NIE slightly biased (.02) IPW CDE and b unbiased 15
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation IPW RPM TSLS Bayesian No unmeasured confounders of T & M T & Y No interactions between M & Y T & M on Y T & X on Y M & X on Y Interactions between T & X on M No post-treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 16
Results How biased are the estimates if there is an interaction between T and M? TSLS-IV biased with 2% coverage all other effects unbiased with both a post-treatment confounder of M and Y and an interaction between T and M CDE 0, CDE 1, b t=0, and b t=1 biased (approx..1 ) 17
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation IPW RPM TSLS Bayesian No unmeasured confounders of T & M T & Y No interactions between M & Y T & M on Y T & X on Y M & X on Y Interactions of T & X on M No post -treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 18
Results For principal strata effects, how biased are the estimates if the monotonicity assumption is violated? TSLS-IV biased (1.1) Bayesian PS slightly biased (-.03) Coverage > 99% for both if the exclusion restriction assumption is violated? TSLS-IV biased (.8), 24% coverage Bayesian PS unbiased, 99.9% coverage 19
Effects Natural Effects Controlled Effects Principal Strata Effects Assumption Methods Mediation IPW RPM TSLS Bayesian No unmeasured confounders of T & M T & Y No interactions between M & Y T & M on Y T & X on Y M & X on Y Interactions between T & X on M No post -treatment confounders Monotonicity (no defiers) Exclusion restriction (full mediation) 20
Results Generally, mean squared error for RPM was much larger than mean squared error for IPW. Generally, coverage for Bayesian PS too large and coverage for TSLS-IV too small. 21
Conclusions When assumptions are violated, all estimators are biased to some degree; some seem less robust than others (e.g., TSLS-IV). Many of the assumptions not testable need further development of sensitivity analysis. 22
Recommendations Define the causal estimand based on the scientific question to be addressed. Justify the assumptions that are plausible given the data. Try several methods for that estimand (e.g., IPW, RPM). For natural effects, estimate natural direct & indirect effects on the exposed (Vansteelandt & VanderWeele, 2012) or impose parametric assumptions 23
Acknowledgment This project was supported by grants P50-DA010075 and R01 DA09757 from the National Institute on Drug Abuse (NIDA) and R21-DK082858 from the National Institute on Diabetes and Digestive and Kidney Diseases (NIDDK). The content is solely the responsibility of the authors and does not necessarily represent the official views of NIDA, NIDDK, or the National Institutes of Health. 24