Statistical Analysis of Causal Mechanisms
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1 Statistical Analysis of Causal Mechanisms Kosuke Imai Princeton University April 13, 2009 Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Papers and Software Collaborators: Luke Keele, Dustin Tingley, and Teppei Yamamoto Identification, Inference, and Sensitivity Analysis for Causal Mediation Effects. available at A General Approach to Causal Mediation Analysis Causal Mediation Analysis in R R package mediation Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
2 Statistics and Causal Mechanisms Causal inference is a central goal of social science and public policy research Randomized experiments are seen as gold standard Design and analyze observational studies to replicate experiments But, experiments are a black box Can only tell whether the treatment causally affects the outcome Not how and why the treatment affects the outcome Qualitative research uses process tracing How can quantitative research be used to identify causal mechanisms? Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Overview of the Talk Goal: Convince you that statistics can play a role in identifying causal mechanisms Method: Causal Mediation Analysis Mediator, M Treatment, T Outcome, Y Direct and indirect effects; intermediate and intervening variables Path analysis, structural equation modeling Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
3 Causal Mediation Analysis in American Politics The political psychology literature on media framing Nelson et al. (APSR, 1998) Popular in social psychology Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Causal Mediation Analysis in Comparative Politics Resource curse thesis Authoritarian government civil war Natural resources Slow growth Causes of civil war: Fearon and Laitin (APSR, 2003) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
4 Causal Mediation Analysis in International Relations The literature on international regimes and institutions Krasner (International Organization, 1982) Power and interests are mediated by regimes Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Current Practice in the Discipline Regression Y i = α + βt i + γm i + δx i + ɛ i Each coefficient is interpreted as a causal effect Sometimes, it s called marginal effect Idea: increase T i by one unit while holding M i and X i constant The Problem: Post-treatment bias If you change T i, that may also change M i Usual advice: only include causally prior (or pre-treatment) variables But, then you lose causal mechanisms! Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
5 Statistical Framework of Causal Inference Units: i = 1,..., n Treatment : T i = 1 if treated, T i = 0 otherwise Observed outcome: Y i Pre-treatment covariates: X i Potential outcomes: Y i (1) and Y i (0) where Y i = Y i (T i ) Voters Contact Turnout Age Party ID i T i Y i (1) Y i (0) X i X i 1 1 1? 20 D 2 0? 0 55 R 3 0? 1 40 R n 1 0? 62 D Causal effect: Y i (1) Y i (0) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Notation for Causal Mediation Analysis Binary treatment (can be generalized): T i {0, 1} Mediator: M i Outcome: Y i Observed covariates: X i 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 )) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
6 Defining and Interpreting Causal Mediation Effects Total causal effect: Causal mediation effects: τ i Y i (1, M i (1)) Y i (0, M i (0)) δ i (t) Y i (t, M i (1)) Y i (t, M i (0)) Change the mediator from M i (0) to M i (1) while holding the treatment constant at t Indirect effect of the treatment on the outcome through the mediator under treatment status t Y i (t, M i (t)) is observable but Y i (t, M i (1 t)) is not Different from controlled direct effects: Y i (t, m) Y i (t, m ) Not applicable if the mediator is manipulated Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Direct effects: ζ i (t) Y i (1, M i (t)) Y i (0, M i (t)) Change the treatment from 0 to 1 while holding the mediator constant at M i (t) Total effect = mediation (indirect) effect + direct effect: τ i = δ i (t) + ζ i (1 t) = {δ i (t) + ζ i (t)} t=0 Quantities of interest: Average Causal Mediation Effects, δ(t) E(δ i (t)) = E{Y i (t, M i (1)) Y i (t, M i (0))} Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
7 The Proposed Identification Assumption Assumption 1 (Sequential Ignorability) {Y i (t, m), M i (t)} T i X i = x, Y i (t, m) M i T i = t, X i = x {Y i (t, m), M i (t)} T i = t X i = x is not sufficient Y i (t, m) M i T i = t, X i = x is not sufficient Weaker than Pearl (2001) if the treatment is randomized Cannot condition on post-treatment confounders that are causally prior to the mediator If such confounders are exist, an additional assumption, e.g., no-interaction assumption, is necessary (Robins) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Nonparametric Identification and Inference Theorem 1 (Nonparametric Identification) Under Assumption 1, Z Z δ(t) = E(Y i M i, T i = t, X i ) {dp(m i T i = 1, X i ) dp(m i T i = 0, X i )} dp(x i ), Z Z ζ(t) = {E(Y i M i, T i = 1, X i ) E(Y i M i, T i = 0, X i )} dp(m i T i = t, X i ) dp(x i ). Two regressions: µ tm (x) E(Y i T i = t, M i = m, X i = x), λ t (x) f (M i T i = t, X i = x). When M i is discrete, λ tm (x) Pr(M i = m T i = t, X i = x), and { n ˆδ(t) = 1 J 1 ˆµ tm (X i ) (ˆλ 1m (X i ) ˆλ 0m (X i )) }. n i=1 m=0 Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
8 Linear Structural Equation Model 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. Run two regressions and multiply two coefficients (Baron-Kenny)! No need to run: Y i = α 1 + β 1 T i + ɛ 1i Direct effect: β 3 Total effect: β 2 γ + β 3 Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Relaxing the no-interaction assumption: Then, δ(t) = β 2 (γ + tκ) Y i = α 3 + β 3 T i + γm i + κt i M i + ɛ 2i The product formula applies to the nonparametric identification with a binary mediator δ(t) = {E(Y i M i = 1, T i = t, X i ) E(Y i M i = 0, T i = t, X i )} {Pr(M i = 1 T i = 1, X i ) Pr(M i = 1 T i = 0, X i )} Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
9 Need for Sensitivity Analysis The sequential ignorability assumption is often too strong Need to assess the robustness of findings via sensitivity analysis Question: How large a departure from the key assumption must occur for the conclusions to no longer hold? Parametric sensitivity analysis by assuming but not {Y i (t, m), M i (t)} T i X i = x Y i (t, m) M i T i = t, X i = x Possible existence of unobserved pre-treatment confounder Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Parametric Sensitivity Analysis Sensitivity parameter: ρ Corr(ɛ 2i, ɛ 3i ) Sequential ignorability implies ρ = 0 Set ρ to different values and see how mediation effects change Theorem 3 (Identification with a Given Error Correlation) δ(0) = δ(1) = β 2 σ 12 σ2 2 ρ ( ) 1 σ 2 1 ρ 2 σ1 2 σ2 12 σ2 2, where σ 2 j var(ɛ ji ) for j = 1, 2 and σ 12 cov(ɛ 1i, ɛ 2i ). When do my results go away completely? δ(t) = 0 if and only if ρ = Corr(ɛ 1i, ɛ 2i ) (easy to compute!) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
10 Facilitating Interpretation How big is ρ? An unobserved (pre-treatment) confounder formulation: ɛ 2i = λ 2 U i + ɛ 2i and ɛ 3i = λ 3 U i + ɛ 3i, Assume Y i (t, m) M i T i = t, U i = u Assume also ɛ 2i U i and ɛ 3i U i Proportion of previously unexplained variance explained by the unobserved confounder R 2 M var(ɛ 2i) var(ɛ 2i ) var(ɛ 2i ) and R 2 Y var(ɛ 3i) var(ɛ 3i ) var(ɛ 3i ) Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Proportion of original variance explained by the unobserved confounder R 2 M var(ɛ 2i) var(ɛ 2i ) var(m i ) and R2 Y var(ɛ 3i) var(ɛ 3i ) var(y i ) Specify sgn(λ 2 λ 2 ) and RM 2, RY 2 (or R M 2, R Y 2 ) ρ = sgn(λ 2 λ 3 )R M R Y = sgn(λ 2λ 3 ) R M RY (1 R 2M )(1 R2Y ), where R 2 M and R2 Y are based on 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 Mechanisms April 13, / 26
11 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 Mediator: a scale measuring general attitudes about the importance of public order Outcome: a scale measuring tolerance for the Klan rally Expected findings: negative mediation effects Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Analysis under Sequential Ignorability Parametric Nonparametric Average Mediation Effects Free speech frame ˆδ(0) [ 0.871, 0.031] [ 0.823, 0.074] Public order frame ˆδ(1) [ 1.081, 0.050] [ 1.168, 0.024] Average Total Effect ˆτ [ 1.207, 0.127] [ 1.153, 0.099] With the no-interaction assumption Average Mediation Effect ˆδ(0) = ˆδ(1) [ 0.969, 0.051] Average Total Effect ˆτ [ 1.206, 0.126] Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
12 Parametric Sensitivity Analysis Unobserved pre-treatment confounder (e.g., political ideology) Average Mediation Effect: δ Sensitivity Parameter: ρ Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Proportion of unexplained variance explained by an unobserved confounder R M 2 * R Y 2 * sgn(λ 2 λ 3 ) = sgn(λ 2 λ 3 ) = R 2 M * R Y 2 * Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
13 Proportion of original variance explained by an unobserved confounder sgn(λ 2 λ 3 ) = 1 R ~ 2 M R ~ Y sgn(λ 2 λ 3 ) = R ~ Y 2 R ~ 2 M Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26 Concluding Remarks and Work in Progress Quantitative analysis can be used to identify causal mechanisms! Estimate causal mediation effects rather than marginal effects Wide applications in social science disciplines Generalization: identification, inference, and sensitivity analysis linear and nonlinear relationships parametric and nonparametric models continuous and discrete mediators various outcome data types multiple mediators development of easy-to-use statistical software Kosuke Imai (Princeton) Causal Mechanisms April 13, / 26
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