Identification and Inference in Causal Mediation Analysis

Transcription

1 Identification and Inference in Causal Mediation Analysis Kosuke Imai Luke Keele Teppei Yamamoto Princeton University Ohio State University November 12, 2008 Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Causal Mediation Analysis Investigation of causal mechanisms via intermediate variables How does the treatment alter the outcome? Direct and indirect effects Mediator, M Treatment, T Outcome, Y Popular among epidemiologists, psychologists, political scientists Fast growing methodological literature Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

2 Overview 1 Identification under sequential ignorability Nonparametric identification without an additional assumption Parametric identification under the linear structural equation model 2 Estimation and inference under sequential ignorability Parametric estimation Nonparametric estimator and its asymptotic variance 3 Sensitivity analysis for the sequential ignorability assumption Nonparametric sensitivity analysis Parametric sensitivity analysis 4 Empirical illustration A randomized experiment from political psychology The treatment is randomized but the mediator is not Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Definition of Causal Mediation Effects Binary treatment: T i {0, 1} Mediator: M i M Outcome: Y i Y Observed covariates: X i X Potential mediators: M i (t) where M i = M i (T i ) Potential outcomes: Y i (t, m) where Y i = Y i (T i, M i (T i )) Total causal effect: τ i Y i (1, M i (1)) Y i (0, M i (0)) Causal mediation effects: δ i (t) Y i (t, M i (1)) Y i (t, M i (0)) Natural (pure) direct effects: ζ i (t) Y i (1, M i (t)) Y i (0, M i (t)) The relationship: τ i = δ i (t) + ζ i (1 t) Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

3 Interpretation of Causal Mediation Effects δ i (t) is the indirect causal effect of the treatment on the outcome through the mediator under treatment status t Controlled indirect effects, Y i (1, m) Y i (0, m), for the mediator that can be manipulated and/or randomized Observational studies and experiments with non-random M descriptive vs. prescriptive effects Y i (t, M i (t)) is observable but Y i (t, M i (1 t)) is not δ i (t) = 0 if M i (1) = M i (0) Quantity of interest: δ(t) E(δ i (t)) = E{Y i (t, M i (1)) Y i (t, M i (0))} Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Sequential Ignorability Assumption 1 (Sequential Ignorability) {Y i (t, m), M i (t)} T i X i, Y i (t, m) M i T i, X i for t = 0, 1 and all m M The second equation can be rewritten as, Y i (t, m) M i (t ) T i = t, X i Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

4 Nonparametric Identification Theorem 1 (Nonparametric Identification) Under Assumption 1, for t = 0, 1, { δ(t) = ( 1) t E(Y i M i, T i = t, X i ) dp(m i T i = 1 t, X i ) } E(Y i T i = t, X i ) dp(x i ) Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Proof (Discrete Mediator with No Observed Covariates): ζ(t ) = = = = 1X XJ 1 E(Y i (1, m) Y i (0, m) M i (t ) = m, T i = t) Pr(M i (t ) = m, T i = t) t=0 m=0 XJ 1 n E(Y i (1, m) Y i (0, m) T i = t ) Pr(M i (t ) = m T i = t ) Pr(T i = t ) m=0 o +E(Y i (1, m) Y i (0, m) M i (t ) = m, T i = 1 t ) Pr(M i (t ) = m, T i = 1 t ) XJ 1 E(Y i (1, m) Y i (0, m)) Pr(M i = m T i = t ) Pr(T i = t ) m=0 +E(Y i (1, M i (t )) Y i (0, M i (t )) T i = 1 t ) Pr(T i = 1 t ) XJ 1 n o E(Y i M i = m, T i = 1) E(Y i M i = m, T i = 0) Pr(M i = m T i = t ) m=0 Pr(T i = t ) + ζ(t ) Pr(T i = 1 t ). Thus, we have ζ(t ) = P J 1 m=0 {E(Y i M i = m, T i = 1) E(Y i M i = m, T i = 0)} Pr(M i = m T i = t ). Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

5 Comparison with the Existing Identification Results The literature insists that an additional assumption is required Pearl s assumption for the identification of δ(t ): Y i (t, m) M i (t ) X i, in place of Y i (t, m) M i T i, X i Robins no-interaction assumption about controlled direct effects: Y i (1, m) Y i (0, m) = B i where B i is a random variable that does not depend on m Sequential ignorability alone is sufficient Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Linear Structural Equation Model (LSEM) The Model: Y i = α 1 + β 1 T i + ɛ 1i, M i = α 2 + β 2 T i + ɛ 2i, Y i = α 3 + β 3 T i + γm i + ɛ 3i, where E(ɛ 1i T i ) = E(ɛ 2i T i ) = E(ɛ 3i M i, T i ) = 0. Baron and Kenny (1986): 1 the association between Y i and T i exists 2 the association between M i and T i exists 3 the conditional association between Y i and M i given T i exists 4 β 2 γ as the causal mediation effect One equation is redundant: Y i = (α 3 + α 2 γ) + (β 3 + β 2 γ)t i + (γɛ 2i + ɛ 3i ) where γe(ɛ 2i T i ) + E{E(ɛ 3i M i, T i ) T i } = 0. Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

6 Parametric Identification under Sequential Ignorability Theorem 2 (Identification under LSEM) Consider the following linear structural equation model M i = α 2 + β 2 T i + ɛ 2i, Y i = α 3 + β 3 T i + γm i + ɛ 3i. Under Assumption 1, the average causal mediation effects are identified as δ(0) = δ(1) = β 2 γ. Assumption 1 implies ɛ 2i ɛ 3i as well as ɛ 2i T i, ɛ 3i T i, and ɛ 3i M i T i. Contrary to the literature, sequential ignorability alone is sufficient β 3 is the average natural direct effect Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Identification without the No-interaction Assumption Assumption 2 (No-interaction) δ(0) = δ(1) Assumption 2 is unnecessary The LSEM with an interaction term: M i = α 2 + β 2 T i + ɛ 2i, Y i = α 3 + β 3 T i + γm i + κt i M i + ɛ 3i. Under Assumption 1, δ(t) =β 2 (γ + tκ) for t = 0, 1. Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

7 Parametric Estimation and Inference Under sequential ignorability, equation-by-equation least squares Asymptotic variance via the Delta method: 1 No-interaction: 2 With-interaction: Var(ˆδ(t)) β 2 2Var(ˆγ) + γ 2 Var( ˆβ 2 ) Var(ˆδ(t)) (γ + tκ) 2 Var( ˆβ) + β 2 2{Var(ˆγ) + tvar(ˆκ) + 2tCov(ˆγ, ˆκ)} Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Nonparametric Estimation and Inference (Discrete M) A simple nonparametric estimator ˆδ(t): ( J 1 ( 1) t m=0 n i=1 1{T i = 1 t, M i = m} n i=1 Y i1{t i = t, M i = m} n n 1 t i=1 1{T i = t, M i = m} ) 1 n 1{T i = t}y i n t where n t = n i=1 1{T i = t}. Estimate within each strata defined by X, and then aggregate i=1 Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

8 Theorem 3 (Asymptotic Variance) Under Assumption 1, the asymptotic variance of the nonparametric estimator is Var(ˆδ(t)) 1 J 1 n t m=0 { (λ1 t,m ) λ 1 t,m 2 Var(Y i M i = m, T i = t) λ tm } + n t(1 λ 1 t,m )µ 2 tm n 1 t 2 n 1 t J 1 J 2 m =m+1 m=0 + 1 n t Var(Y i T i = t) λ 1 t,m λ 1 t,m µ tm µ tm, where λ tm Pr(M i = m T i = t) and µ tm E(Y i M i = m, T i = t). Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Using Nonparametric Regressions Fit two nonparametric regressions: 1 µ tm (x) E(Y i T i = t, M i = m, X i = x) 2 λ tm (x) Pr(M i = m T i = t, X i = x) An estimator: { J 1 ( 1) t m=0 n i=1 1{T i = 1 t}ˆλ 1 t,m (X i ) n i=1 1{T i = t}ˆµ tm (X i )ˆλ tm (X i ) n n 1 t i=1 1{T i = t}ˆλ tm (X i ) ( 1 n J 1 )} 1{T i = t} ˆµ tm (X i )ˆλ tm (X i ). n t i=1 m=0 Nonparametric or parametric bootstrap for uncertainty estimates Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

9 A Simulation Study Binary mediator, lognormal outcome Y i (t, m) M i (t ) T i = t but Y i (t, m) M i (t ) T i = 1 t True values: δ(0) 0.67 and δ(1) 3.95 Estimator n Bias RMSE 90% CI 95% CI ˆδ(0) ˆδ(1) Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Need for Sensitivity Analysis The sequential ignorability assumption is often too strong! Need to assess the robustness of findings via sensitivity analysis Assumption 3 (Ignorability of Treatment Assignment) {Y i (t, m), M i (t)} T i X i Parametric and nonparametric sensitivity analysis under Assumption 3 alone Maximal degree of departure from Assumption 1 while maintaining the original conclusion Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

10 Parametric Sensitivity Analysis Assumption 3 implies ɛ 2i T i and ɛ 3i T i but not ɛ 2i ɛ 3i Sensitivity parameter: ρ Corr(ɛ 2i, ɛ 3i ) Theorem 4 (Identification with a Known Error Correlation) Under Assumption 3, δ(0) = δ(1) = β 2 σ 23 σ 2 2 ρ ( 1 σ 2 1 ρ 2 σ3 2 σ 23 σ ), where σj 2 Var(ɛ ji ) for j = 2, 3, σ 32 Var(ɛ 3i ), σ 23 Cov(ɛ 2i, ɛ 3i ), and ɛ 3i = γɛ 2i + ɛ 3i. Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Fit the following LSEM via eq.-by-eq. least squares or SUR M i = α 2 + β 2 T i + ɛ 2i Y i = α 3 + β 3 T i + ɛ 3i Monotone function of ρ ( ρ δ(t) β 2 = 1 σ 2 (1 ρ 2 ) 1 ρ 2 σ3 2 σ 23 δ(t) = 0 if and only if ρ = Corr(ɛ 2i, ɛ 3i ) (easy to compute!) For confidence intervals, apply the iterative FGLS algorithm to σ ) M i = α 2 + β 2 T i + ɛ 2i Y i = α 3 + β 3 T i + γm i + ɛ 3i Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

11 Large Sample Nonparametric Bounds Balke and Pearl (1997) s strategy: discrete outcome and mediator Binary case: population probabilities of 64 types π m 1m 0 Pr(Y i (1, 1) = y 11, Y i (1, 0) = y 10, Y i (0, 1) = y 01, Y i (0, 0) = y 00, M i (1) = m 1, M i (0) = m 0 ) Mediation effects as a linear function of π δ(t) = π mm 0 m=0 y 1 t,m =0 y 1,1 m =0 y 0,1 m =0 m 0 =0 Assumption 3 implies linear restrictions 1 π m 1m m 1 =0 Pr(Y i = y, M i = m T i = t) = y 1 t,m =0 y t,1 m =0 y 1 t,1 m =0 m 1 t =0 π m 1m 0, where m t = m and y tm = y. Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Symbolic linear programming Theorem 5 (Sharp Large Sample Bounds) Under Assumption 3 the sharp large sample bounds of the average causal mediation effects are given by, Pr(Y i = 1 t T i = t) max Pr(M i = 1 t T i = 1 t) Pr(Y i = M i = 1 t T i = t) Pr(M i = t T i = 1 t) Pr(Y i = 1 t, M i = t T i = t) Pr(Y i = t T i = t) δ(t) min Pr(M i = 1 t T i = 1 t) + Pr(Y i = t, M i = 1 t T i = t), Pr(M i = t T i = 1 t) + Pr(Y i = M i = t T i = t) for t = 0, 1. [α, β] always improves upon [ 1, 1]; β α 1 Not very informative 1 α 0 β 1 Possible to impose the no-interaction assumption δ(1) = δ(0) Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

12 Nonparametric Sensitivity Analysis Bounds are not informative even under additional assumptions Ignorability of the mediator implies Pr(Y i (1, 1) = y 11, Y i (1, 0) = y 10, Y i (0, 1) = y 01, Y i (0, 0) = y 00 M i = 1, T i = t ) = Pr(Y i (1, 1) = y 11, Y i (1, 0) = y 10, Y i (0, 1) = y 01, Y i (0, 0) = y 00 M i = 0, T i = t ) Sensitivity parameter: 1 m 0 =0 π1m 0 Pr(M i = 1 T i = 1) 1 m 1 =0 πm 11 Pr(M i = 1 T i = 0) 1 m 0 =0 π0m 0 Pr(M i = 0 T i = 1) 1 m 1 =0 πm 10 Pr(M i = 0 T i = 0) ρ, ρ, where 0 ρ 1 Compute the sharp bounds for various values of ρ Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Political Psychology Experiment: Nelson et al. (APSR) How does media framing affect citizens political opinions? News stories about the Ku Klux Klan rally in Ohio Free speech frame (T i = 0) and public order frame (T i = 1) Randomized experiment with the sample size = 136 Mediators: general attitudes (12 point scale) about the importance of free speech and public order Outcome: tolerance (7 point scale) for the Klan rally Expected findings: negative mediation effects Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

13 Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Analysis under Sequential Ignorability Mediator Estimator Public Order Free Speech Parametric No-interaction [ 0.969, 0.051] [ 0.388, 0.135] ˆδ(0) [ 0.871, 0.031] [ 0.404, 0.143] ˆδ(1) [ 1.081, 0.050] [ 0.380, 0.136] Nonparametric ˆδ(0) [ 0.823, 0.074] [ 0.434, 0.246] ˆδ(1) [ 1.168, 0.024] [ 0.662, 0.219] Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28

14 Sensitivity Analysis Parametric Analysis Nonparametric Analysis Average Mediation Effect: δ Mediation Effect: δ Sensitivity Parameter: ρ Sensitivity Parameter: ρ Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28 Concluding Remarks and Future Work Nonparametric identification under sequential ignorability Parametric identification under LSEM Nonparametric estimator and its asymptotic variance Nonparametric and parametric sensitivity analysis Nonparametric sensitivity analysis in a more general setting Nonparametric estimation under the no-interaction assumption Use of parametric/nonparametric regressions in practical causal mediation analysis Extension to multiple mediators Kosuke Imai (Princeton) Causal Mediation Analysis November 12, / 28