Weighting Methods. Harvard University STAT186/GOV2002 CAUSAL INFERENCE. Fall Kosuke Imai

Transcription

1 Weighting Methods Kosuke Imai Harvard University STAT186/GOV2002 CAUSAL INFERENCE Fall 2018 Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

2 Motivation Matching methods for improving covariate balance Potential limitations of matching methods: 1 inefficient it may throw away data 2 ineffective it may not be able to balance covariates Recall that matching is a special case of weighting: ˆτ match = 1 n T i Y i 1 Y i n 1 M i i M i = 1 Y i 1 n 0 1{i M i } n 1 n 0 n 1 M i:t i =1 i:t i =0 i i :T i } =1 {{} W i Idea: weight each observation in the control group such that it looks like the treatment group (i.e., good covariate balance) Y i Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

3 Inverse Propensity Score Weighting Weighting for surveys: down-weight over-sampled respondents Sampling weights inversely proportional to samplig probability Horvitz-Thompson estimator (1952. J. Am. Stat. Assoc.): Ê(Y i ) = 1 N N S i Y i Pr(S i = 1) Inverse probability-of-treatment weighting estimators (IPW): ÂTE = 1 n { Ti Y i n ˆπ(X i ) (1 T } i)y i 1 ˆπ(X i ) ÂTT = 1 n { T i Y i ˆπ(X } i)(1 T i )Y i n 1 1 ˆπ(X i ) ÂTC = 1 n { } (1 ˆπ(Xi ))T i Y i (1 T i )Y i n 0 π(x i ) Identical propensity score difference-in-means Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

4 Normalized Weights Survey sampling when the population size is unknown Hajek Estimator: Ê(Y i ) = i S iy i / Pr(S i = 1) i S i/ Pr(S i = 1) Weights are normalized but no longer unbiased Normalization of weights may be important when propensity score is estimated ÂTE = n T iy i /ˆπ(X i ) n T i/ˆπ(x i ) n (1 T i)y i /{1 ˆπ(X i )} n (1 T i)/{1 ˆπ(X i )} Weighted least squares: n (ˆα wls, ˆβ wls ) = argmin α,β T i (1 ˆπ(X i )) + (1 T i )ˆπ(X i ) (Y i α βt i ) 2 ˆπ(X i ){1 ˆπ(X i )} Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

5 Variance IPW estimator as the method of moments estimator: 1 moment condition from the propensity score model (e.g., score) 2 moment conditions from the weighting estimator Horvitz/Thompson : Hajek : 1 n n 1 n n T i (Y i µ 1 ) ˆπ(X i ) T i Y i ˆπ(X i ) µ 1 = 1 n = 1 n n n (1 T i )(Y i µ 0 ) 1 ˆπ(X i ) large sample variances are readily available (1 T i )Y i 1 ˆπ(X i ) µ 0 = 0 = 0 If the propensity score model is correctly specified, these variances are smaller than those with the true propensity score Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

6 Doubly Robust Estimator (Robins et al J. Am. Stat. Assoc.) Augmented IPW (AIPW) estimator: { ˆτ DR = 1 n T i Y i n ˆπ(X i ) T i ˆπ(X i ) ˆµ(1, X i ) ˆπ(X i ) = 1 n (1 T i)y i 1 ˆπ(X i ) + T i ˆπ(X i ) 1 ˆπ(X i ) ˆµ(0, X i) { n ˆµ(1, X i ) + T i(y i ˆµ(1, X i )) ˆπ(X i ) ˆµ(0, X i ) (1 T i)(y i ˆµ(0, X i )) 1 ˆπ(X i ) Consistent if either the propensity score model or the outcome model is correct you get two chances to be correct Efficient: smallest asymptotic variance among estimators that are consistent when the propensity score model is correct Estimator may not behave well when both models are incorrect Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13 (especially if weights are highly variable) }

7 A Simulation Study (Kang and Schafer Statistical Science) The deteriorating performance of propensity score weighting methods when the model is misspecified Led to improvements of doubly robust estimators Cao et al. (2009), Tan (2010), Rotnitzky et al. (2012), Han and Wang (2013) Biometrika. etc. Setup: 4 covariates Xi : all are i.i.d. standard normal Outcome model: linear model Propensity score model: logistic model with linear predictors Misspecification induced by measurement error: X i1 = exp(xi1/2) X i2 = Xi2/(1 + exp(x1i) + 10) X i3 = (Xi1X i3/ ) 3 X i4 = (Xi1 + Xi4 + 20) 2 Weighting estimators to be evaluated: 1 Horvitz-Thompson 2 Inverse-probability weighting with normalized weights 3 Weighted least squares regression with covariates 4 Doubly-robust least squares regression with covariates Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

8 Weighting Estimators Do Fine If the Model is Correct Bias RMSE Sample size Estimator logit True logit True (1) Both models correct HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR (2) Propensity score model correct HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

9 Weighting Estimators are Sensitive to Misspecification Bias RMSE Sample size Estimator logit True logit True (3) Outcome model correct HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR (4) Both models incorrect HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

10 Covariate Balancing Propensity Score (CBPS) (Imai and Ratkovic J. Royal Stat. Soc. B.) How can we improve the estimation of propensity score? Estimate the propensity score such that covariates are balanced Covariate balance conditions: { Ti f (X i ) E π β (X i ) (1 T } i) f (X i ) 1 π β (X i ) Usual score condition: f (X i ) = π β (X i) Optimal choice (Fan et al Working Paper): = 0 f (X i ) = π(x i )µ(0, X i ) + (1 π(x i ))µ(1, X i ) 1 double robustness 2 smallest asymptotic variance when the propensity score is correct Estimation via the (generalized) method of moments Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

11 More Robust Weighting Methods Bias RMSE Sample size Estimator GLM CBPS1 CBPS2 True GLM CBPS1 CBPS2 True (3) Outcome model correct HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR (4) Both models incorrect HT n = 200 IPW WLS DR HT n = 1000 IPW WLS DR Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

12 Calibration Methods Forget about the propensity score just balance covariates avoid modeling assumptions and balance certain moments in theory, propensity score balances the entire distributions validation and interpretation are more difficult Entropy balancing (Hainmueller Political Anal.) {w 1, w 2,..., w n 0 } = argmin w i:t i =0 w i log(w i /q i ) where w i 0, i:t i =0 w i = 1, i:t i =0 w if (X i ) = 1 n 1 i:t i =1 f (X i) exact balance in moments extreme weights Stable weights (Zubizarreta J. Am. Stat. Assoc.) Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13

13 Summary Weighting methods as a generalization of matching methods Propensity score weighting Doubly robust estimation Robust estimation of propensity score for balancing covariates Calibration methods Recommended readings: Imbens and Rubin. Chapter 17 (Section 8) Lunceford and Davidian Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Statistics in Medicine Kosuke Imai (Harvard) Weighting Methods Stat186/Gov2002 Fall / 13