Causal Mediation Analysis in R. Quantitative Methodology and Causal Mechanisms
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1 Causal Mediation Analysis in R Kosuke Imai Princeton University June 18, 2009 Joint work with Luke Keele (Ohio State) Dustin Tingley and Teppei Yamamoto (Princeton) Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Quantitative Methodology and Causal Mechanisms Investigation of causal mechanisms via intermediate variables Randomized experiments can only determine whether the treatment causes changes in the outcome Not how and why the treatment affects the outcome Social scientists use qualitative methods (e.g. process tracing) to answer these questions How can quantitative research be used to identify causal mechanisms? Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
2 Causal Mediation Analysis Mediator, M Treatment, T Outcome, Y Quantities of interest: Direct and indirect effects Traditional tools: Path analysis, structural equation modeling Fast growing methodological literature Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Common Practice 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 Mediation Analysis June 18, / 22
3 Defining Causal Mediation Effects Binary treatment (can be generalized): 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)) Direct effects: ζ i (t) Y i (1, M i (t)) Y i (0, M i (t)) Total effect = Mediation (indirect) effect + Direct effect: τ i = δ i (t) + ζ i (1 t) = t=0 {δ i(t) + ζ i (t)} Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Interpreting Causal Mediation Effects δ i (t): Causal effect of the change in M i on Y i that would be induced by T i, holding actual treatment constant at t ζ i (t): Causal effect of T i on Y i, holding mediator constant at its potential value that would realize when T i = t Different from controlled direct effects: Y i (t, m) Y i (t, m ) Mediation effects identify causal paths from T i to Y i Controlled effects study how T i moderates the effect of M i on Y i 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 Mediation Analysis June 18, / 22
4 Nonparametric Identification Problem: Y i (t, M i (t)) is observed but Y i (t, M i (1 t)) can never be observed Proposed identification assumption: 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 Theorem 1 (Imai, Keele, and Yamamoto (2008)) Under sequential ignorability, δ(t) = R R 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 ), ζ(t) = R R {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 ). Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Inference Under Sequential Ignorability Model outcome and mediator Outcome model: p(y i T i, M i, X i ) Mediator model: p(m i T i, X i ) Can use parametric or nonparametric regressions; probit, logit, GAM, quantile regression etc. Two new algorithms for statistical inference: 1 Quasi-Bayesian approximation: approximating the posterior by the sampling distribution of MLE 2 Bootstrap: works for nonparametric models as well as parametric ones The details and examples are in Imai, Keele and Tingley (2009) Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
5 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 {Y i (t, m), M i (t)} T i X i = x but not Y i (t, m) M i T i = t, X i = x Possible existence of unobserved pre-treatment confounder Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Parametric Sensitivity Analysis Consider LSEM (aka Baron-Kenny procedure): M i = α 2 + β 2 T i + ɛ 2i, Y i = α 3 + β 3 T i + γm i + ɛ 3i. Sensitivity parameter: ρ Corr(ɛ 2i, ɛ 3i ) Sequential ignorability implies ρ = 0 Set ρ to different values and see how mediation effects change An alternative explanation of ρ based on R 2 Work for probit models binary outcome, binary mediator, etc. Difficult to construct a more general sensitivity analysis Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
6 An Example Sensitivity Analysis Plot Average Mediation Effect: δ Sensitivity Parameter: ρ Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 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 1 var(ɛ 2i ) var(ɛ 2i ) and R 2 Y 1 var(ɛ 3i ) var(ɛ 3i ) Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
7 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 Mediation Analysis June 18, / 22 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 Mediation Analysis June 18, / 22
8 Overview of R Package mediation Object-oriented nature of R made this design possible Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 An Illustrative Example Job Search Intervention Study (JOBS II) A randomized evaluation of a job training program Treatment: Job-skills workshop Mediator: a continuous measure of job-search self-efficacy Outcome: a binary measure of employment Question: Does the workshop improve the prospect of future employment by increasing the level of job-search self-efficacy? Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
9 Step 1: Fitting the Outcome and Mediator Models > # load the library > library(mediation) > # load the data set > data(jobs) > > # fit the mediator model > model.m <- lm(job_seek ~ treat + depress1 + econ_hard + sex + age + occp + marital + nonwhite + educ + income, data = jobs) > > # fit the outcome model > model.y <- glm(work1 ~ treat + job_seek + depress1 + econ_hard + sex + age + occp + marital + nonwhite + educ + income, family = binomial(link="probit"), data = jobs) Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Step 2: Conducting Causal Mediation Analysis > # mediation analysis > m.out <- mediate(model.m, model.y, sims = 1000, T = "treat", M = "job_seek") > # summary of the analysis > summary(m.out) Causal Mediation Analysis Quasi-Bayesian Confidence Intervals Mediation Effect: % CI Direct Effect: % CI Total Effect: % CI Proportion of Total Effect via Mediation: % CI Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
10 Step 3: Conducting Sensitivity Analysis > s.out <- medsens(model.m, model.y, sims = 1000, T = "treat", M = "job_seek", INT = FALSE, DETAIL=FALSE) > summary(s.out) Mediation Sensitivity Analysis Sensitivity Region Rho Med. Eff. 95% CI Lower 95% CI Upper [1,] [2,] [3,] output truncated > plot(s.cont) Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Average Mediation Effect: δ Sensitivity Parameter: ρ Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
11 Concluding Remarks Quantitative analysis can be used to identify causal mechanisms! Wide applications in many social scientific disciplines Sensitivity analysis is critical Development of easy-to-use software mediation Object-oriented nature of R facilitated this development Future extensions: multiple mediators, sensitivity analysis for other models Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22 Papers and Software Keele, Tingley, Yamamoto, and Imai. (2009). mediation: R Package for Causal Mediation Analysis. available at CRAN Imai, Keele, Tingley, and Yamamoto. (2009). Causal Mediation Analysis in R. Imai, Keele, and Yamamoto. (2008). Identification, Inference, and Sensitivity Analysis for Causal Mediation Effects. Imai, Keele, and Tingley. (2009). A General Approach to Causal Mediation Analysis. All are available at Kosuke Imai (Princeton) Causal Mediation Analysis June 18, / 22
Statistical Analysis of Causal Mechanisms
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