Noncompliance in Randomized Experiments

Transcription

1 Noncompliance in Randomized Experiments Kosuke Imai Harvard University STAT186/GOV2002 CAUSAL INFERENCE Fall 2018 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

2 Encouragement Design Often, for ethical and logistical reasons, we cannot force all experimental units to follow the randomized treatment assignment 1 some in the treatment group refuse to take the treatment 2 some in the control group manage to receive the treatment Intention-to-Treat (ITT) analysis: ITT effect can be estimated without bias ITT analysis does not yield the treatment effect As-Treated analysis comparison of the treated and untreated subjects no benefit of randomization selection bias Can we estimate the treatment effect somehow? Encouragement design: randomize the encouragement to receive the treatment rather than the receipt of the treatment itself attractive to policy makers Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

3 Potential Outcomes Notation Randomized encouragement: Z i {0, 1} Potential treatment variables: (T i (1), T i (0)) 1 T i (z) = 1: would receive the treatment if Z i = z 2 T i (z) = 0: would not receive the treatment if Z i = z Observed treatment receipt indicator: T i = T i (Z i ) Observed and potential outcomes: Y i = Y i (Z i, T i (Z i )) Can be written as Y i = Y i (Z i ) No interference assumption for T i (Z i ) and Y i (Z i, T i ) Randomization of encouragement: But (Y i (1), Y i (0)) T i Z i = z (Y i (1), Y i (0), T i (1), T i (0)) Z i Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

4 Principal Stratification (Angrist, et al J. Am. Stat. Assoc) Four principal strata (latent types): compliers (T i (1), T i (0)) = (1, 0), always takers (T i (1), T i (0)) = (1, 1), non-compliers never takers (T i (1), T i (0)) = (0, 0), defiers (T i (1), T i (0)) = (0, 1) Observed and principal strata: Z i = 1 Z i = 0 T i = 1 Complier/Always-taker Defier/Always-taker T i = 0 Defier/Never-taker Complier/Never-taker Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

5 Instrumental Variables Assumptions: 1 Randomized encouragement as an instrument for the treatment 2 Monotonicity: No defiers T i (1) T i (0) for all i. 3 Exclusion restriction: Instrument (encouragement) affects outcome only through treatment Y i (1, t) = Y i (0, t) for t = 0, 1 Zero ITT effect for always-takers and never-takers ITT effect decomposition: ITT = ITT c Pr(compliers) + ITT a Pr(always takers) +ITT n Pr(never takers) = ITT c Pr(compliers) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

6 IV Estimand and Interpretation IV estimand: ITT c = ITT Pr(compliers) = E(Y i Z i = 1) E(Y i Z i = 0) E(T i Z i = 1) E(T i Z i = 0) = Cov(Y i, Z i ) Cov(T i, Z i ) ITT c = Complier Average Treatment Effect (CATE) Local Average Treatment Effect (LATE) CATE ATE unless ATE for noncompliers equals CATE Different encouragement (instrument) yields different compliers Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

7 Asymptotic Inference Wald estimator: IV Wald = Cov(Y i,z i ) Cov(T = ITT Y i,z i ) ITT T Identical to the two-stage least squares estimator: 1 Regress T i on Z i and obtain fitted values T i 2 Regress Y i on T i Consistency: IV p Wald CATE = ITT c Asymptotic variance via the Delta method: V( IV Wald ) 1 { ITT 4 ITT 2 T V(ÎTT Y ) + ITT 2 Y V(ÎTT T ) T } 2 ITT Y ITT T Cov(ÎTT Y, ÎTT T ). Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

8 Testing Habitual Voting (Coppock and Green Am. J. Political Sci.) Settings (Revisit the Social Pressure Experiment): Randomized encouragement to vote in the 2006 August primary Treatment: turnout in the 2007 November municipal election Outcome: turnout in the 2008 January party primary and subsequent elections Assumptions: 1 Monotonicity: Being contacted by a canvasser would never discourage anyone from voting 2 Exclusion restriction: being contacted by a canvasser in this election has no effect on turnout in the next election other than through turnout in this election CATE: Habitual voting for those who would vote if and only if they are contacted by a canvasser in this election Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

9 Downstream Effects Estimated proportion of principal strata: compliers: est. = 0.083, s.e. = always-takers: est. = 0.311, s.e. = never-takers: est. = 0.606, s.e. = CATE: Downstream effects of turnout in the August 2006 Primary Election Estimated CACE Aug 2008 Aug 2010 Nov 2006 Feb 2012 Nov 2012 Jan 2008 Nov 2008 Nov 2010 Aug 2012 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

10 Likelihood Inference Observed data = {Y i, T i, Z i, X i } n i=1 Complete data = {Y i, T i, Z i, X i, C i } n i=1 The model: 1 Compliance submodel: f θ (C i X i ) e.g., Pr(C i = c X i ) = exp(x i γ c ) 1 + exp(x i γ c ) + exp(x i γ a ) 2 Outcome submodel: g ψ (Y i C i, Z i, X i ) e.g., Y i indep. N (α Ci + β1{c i = c}z i + X i δ, σ 2 ) Given C i and Z i, T i is redundant Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

11 The observed-data likelihood function: n {f θ (C i = n X i ) g ψ (Y i C i = n, Z i, X i )} (1 T i )Z i i=1 {f θ (C i = a X i ) g ψ (Y i C i = a, Z i, X i )} T i (1 Z i ) f θ (C i = c X i ) g ψ (Y i C i = c, Z i, X i ) c {a,c} c {c,n} f θ (C i = c X i ) g ψ (Y i C i = c, Z i, X i ) The complete-data likelihood function: n {f θ (C i = c X i )g ψ (Y i C i = c, Z i, X i )} C i =c i=1 c {a,c,n} T i Z i (1 T i )(1 Z i ) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

12 Optimization Using the EM Algorithm The Expectation and Maximization algorithm by Dempster, Laird, and Rubin Useful for maximizing the likelihood function with missing data Goal: maximize the observed-data log-likelihood, l n (θ Y obs ) The EM algorithm: Repeat the following steps until convergence 1 E-step: Compute Q(θ θ (t) ) E{l n (θ Y obs, Y mis ) Y obs, θ (t) } where l n (θ Y obs, Y mis ) is the complete-data log-likelihood 2 M-step: Find θ (t+1) = argmax Q(θ θ (t) ) θ Θ The ECM algorithm: M-step replaced with multiple conditional maximization steps Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

13 Monotone Convergence Property The observed-data likelihood increases each step: Sketch of Proof: l n (θ (t+1) Y obs ) l n (θ (t) Y obs ) 1 l n (θ Y obs ) = log f (Y obs, Y mis θ) log f (Y mis Y obs, θ) 2 Taking the expectation w.r.t. f (Y mis Y obs, θ (t) ) l n (θ Y obs ) = Q(θ θ (t) ) log f (Y mis Y obs, θ)f (Y mis Y obs, θ (t) )dy mis 3 Finally, l n (θ (t+1) Y obs ) l n (θ (t) Y obs ) = Q(θ (t+1) θ (t) ) Q(θ (t) θ (t) ) + log f (Y mis Y obs, θ (t) ) f (Y mis Y obs, θ (t+1) ) f (Y mis Y obs, θ (t) )dy mis 0 Stable, no derivative required Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

14 Application to the Complier Average Causal Effect 1 E-step: w i (c ) (t+1) = Pr(C i = c Y, Z, X) = f θ (t)(c i = c X i )g ψ (t)(y i C i = c, Z i, X i ) c {a,c,n} f (C i = c X i )g ψ (t)(y i C i = c, Z i, X i ) for c {c, a, n}. 2 M-step: θ (t+1) = argmax θ ψ (t+1) = argmax ψ n i=1 c {a,c,n} n i=1 c {a,c,n} w(c ) (t+1) log f θ (C i = c X i ) w(c ) (t+1) log g ψ (Y i C i = c, Z i, X i ) Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15

15 Summary Noncompliance in randomized experiments ITT vs. CATE (LATE) additional assumptions are required 1 randomization of instrument 2 monotonicity 3 exclusion restriction Traditional instrumental variables ignoring heterogeneity Tradeoff between internal and external validity: compliers vs. noncompliers compliers as latent group defined by an instrument Suggested readings: IMBENS AND RUBIN, Chapters 23 and 24 ANGRIST AND PISCKE, Chapter 4 Kosuke Imai (Harvard) Noncompliance in Experiments Stat186/Gov2002 Fall / 15