Covariate Balancing Propensity Score for General Treatment Regimes

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Covariate Balancing Propensity Score for General Treatment Regimes Kosuke Imai Princeton University October 14, 2014 Talk at the Department of Psychiatry, Columbia University Joint work with Christian Fong Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 1 / 30

Motivation Central role of propensity score in causal inference Adjusting for observed confounding in observational studies Matching and inverse-probability weighting methods Extensions of propensity score to general treatment regimes Weighting (e.g., Imbens, 2000; Robins et al., 2000) Subclassification (e.g., Imai & van Dyk, 2004) Regression (e.g., Hirano & Imbens, 2004) But, propensity score is mostly applied to binary treatment All existing methods assume correctly estimated propensity score No reliable methods to estimate generalized propensity score Harder to check balance across a non-binary treatment Many researchers dichotomize the treatment Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 2 / 30

Contributions of the Paper Results are often sensitive to misspecification of propensity score Solution: Estimate the generalized propensity score such that covariates are balanced Generalize the covariate balancing propensity score (CBPS; Imai & Ratkovic, 2014, JRSSB) 1 Multi-valued treatment (3 and 4 categories) 2 Continuous treatment Useful especially because checking covariate balance is harder for non-binary treatment Facilitates the use of generalized propensity score methods Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 3 / 30

Propensity Score for a Binary Treatment Notation: T i {0, 1}: binary treatment X i : pre-treatment covariates Dual characteristics of propensity score: 1 Predicts treatment assignment: π(x i ) = Pr(T i = 1 X i ) 2 Balances covariates (Rosenbaum and Rubin, 1983): T i X i π(x i ) Use of propensity score Strong ignorability: Y i (t) T i X i and 0 < Pr(T i = 1 X i ) < 1 Propensity score matching: Y i (t) T i π(x i ) Propensity score (inverse probability) weighting Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 4 / 30

Propensity Score Tautology Propensity score is unknown and must be estimated Dimension reduction is purely theoretical: must model T i given X i Diagnostics: covariate balance checking In theory: ellipsoidal covariate distributions = equal percent bias reduction In practice: skewed covariates and adhoc specification searches Propensity score methods are sensitive to model misspecification Propensity score tautology (Ho et al. 2007 Political Analysis): it works when it works, and when it does not work, it does not work (and when it does not work, keep working at it). Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 5 / 30

Kang and Schafer (2007, Statistical Science) Simulation study: the deteriorating performance of propensity score weighting methods when the model is misspecified 4 covariates X i : 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(x 1i ) + 10) X i3 = (Xi1 X i3 /25 + 0.6)3 X i4 = (Xi1 + X i4 + 20)2 Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 6 / 30

Weighting Estimators Evaluated 1 Horvitz-Thompson (HT): 1 n n i=1 { Ti Y i ˆπ(X i ) (1 T } i)y i 1 ˆπ(X i ) 2 Inverse-probability weighting with normalized weights (IPW): HT with normalized weights (Hirano, Imbens, and Ridder) 3 Weighted least squares regression (WLS): linear regression with HT weights 4 Doubly-robust least squares regression (DR): consistently estimates the ATE if either the outcome or propensity score model is correct (Robins, Rotnitzky, and Zhao) Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 7 / 30

Weighting Estimators Do Fine If the Model is Correct Bias RMSE Sample size Estimator GLM True GLM True (1) Both models correct HT 0.33 1.19 12.61 23.93 n = 200 IPW 0.13 0.13 3.98 5.03 WLS 0.04 0.04 2.58 2.58 DR 0.04 0.04 2.58 2.58 HT 0.01 0.18 4.92 10.47 n = 1000 IPW 0.01 0.05 1.75 2.22 WLS 0.01 0.01 1.14 1.14 DR 0.01 0.01 1.14 1.14 (2) Propensity score model correct HT 0.05 0.14 14.39 24.28 n = 200 IPW 0.13 0.18 4.08 4.97 WLS 0.04 0.04 2.51 2.51 DR 0.04 0.04 2.51 2.51 HT 0.02 0.29 4.85 10.62 n = 1000 IPW 0.02 0.03 1.75 2.27 WLS 0.04 0.04 1.14 1.14 DR 0.04 0.04 1.14 1.14 Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 8 / 30

Weighting Estimators are Sensitive to Misspecification Bias RMSE Sample size Estimator GLM True GLM True (3) Outcome model correct HT 24.25 0.18 194.58 23.24 n = 200 IPW 1.70 0.26 9.75 4.93 WLS 2.29 0.41 4.03 3.31 DR 0.08 0.10 2.67 2.58 HT 41.14 0.23 238.14 10.42 n = 1000 IPW 4.93 0.02 11.44 2.21 WLS 2.94 0.20 3.29 1.47 DR 0.02 0.01 1.89 1.13 (4) Both models incorrect HT 30.32 0.38 266.30 23.86 n = 200 IPW 1.93 0.09 10.50 5.08 WLS 2.13 0.55 3.87 3.29 DR 7.46 0.37 50.30 3.74 HT 101.47 0.01 2371.18 10.53 n = 1000 IPW 5.16 0.02 12.71 2.25 WLS 2.95 0.37 3.30 1.47 DR 48.66 0.08 1370.91 1.81 Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 9 / 30

Covariate Balancing Propensity Score (CBPS) Idea: Estimate propensity score such that covariates are balanced Goal: Robust estimation of parametric propensity score model Covariate balancing conditions: { Ti X i E π β (X i ) (1 T } i)x i 1 π β (X i ) = 0 Over-identification via score conditions: { Ti π β E (X i) π β (X i ) (1 T i)π β (X } i) 1 π β (X i ) = 0 Can be interpreted as another covariate balancing condition Combine them with the Generalized Method of Moments or Empirical Likelihood Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 10 / 30

CBPS Makes Weighting Methods Work Better Bias RMSE Estimator GLM CBPS1 CBPS2 True GLM CBPS1 CBPS2 True (3) Outcome model correct HT 24.25 1.09 5.42 0.18 194.58 5.04 10.71 23.24 n = 200 IPW 1.70 1.37 2.84 0.26 9.75 3.42 4.74 4.93 WLS 2.29 2.37 2.19 0.41 4.03 4.06 3.96 3.31 DR 0.08 0.10 0.10 0.10 2.67 2.58 2.58 2.58 HT 41.14 2.02 2.08 0.23 238.14 2.97 6.65 10.42 n = 1000 IPW 4.93 1.39 0.82 0.02 11.44 2.01 2.26 2.21 WLS 2.94 2.99 2.95 0.20 3.29 3.37 3.33 1.47 DR 0.02 0.01 0.01 0.01 1.89 1.13 1.13 1.13 (4) Both models incorrect HT 30.32 1.27 5.31 0.38 266.30 5.20 10.62 23.86 n = 200 IPW 1.93 1.26 2.77 0.09 10.50 3.37 4.67 5.08 WLS 2.13 2.20 2.04 0.55 3.87 3.91 3.81 3.29 DR 7.46 2.59 2.13 0.37 50.30 4.27 3.99 3.74 HT 101.47 2.05 1.90 0.01 2371.18 3.02 6.75 10.53 n = 1000 IPW 5.16 1.44 0.92 0.02 12.71 2.06 2.39 2.25 WLS 2.95 3.01 2.98 0.19 3.30 3.40 3.36 1.47 DR 48.66 3.59 3.79 0.08 1370.91 4.02 4.25 1.81 Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 11 / 30

The Setup for a General Treatment Regime T i T : non-binary treatment X i : pre-treatment covariates Y i (t): potential outcomes Strong ignorability: T i Y i (t) X i and p(t i = t X i ) > 0 for all t T p(t i X i ): generalized propensity score T i : dichotomized treatment T i = 1 if T i T 1 T i = 0 if T i T 0 T 0 T1 = and T 0 T1 = T What is the problem of dichotomizing a non-binary treatment? Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 12 / 30

The Problems of Dichotomization Under strong ignorability, = E(Y i T i = 1, X i ) E(Y i T i = 0, X i ) E(Y i (t) X i )p(t i = t T i = 1, X i )dt T 1 E(Y i (t) X i )p(t i = t T i = 0, X i )dt T 0 Aggregation via p(t i T i, X i ) 1 some substantive insights get lost 2 external validity issue Checking covariate balance: T i X i does not imply T i X i Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 13 / 30

Two Motivating Examples 1 Effect of education on political participation Education is assumed to play a key role in political participation T i : 3 education levels (graduated from college, attended college but not graduated, no college) Original analysis dichotomization (some college vs. no college) Propensity score matching Critics employ different matching methods 2 Effect of advertisements on campaign contributions Do TV advertisements increase campaign contributions? T i : Number of advertisements aired in each zip code ranges from 0 to 22,379 advertisements Original analysis dichotomization (over 1000 vs. less than 1000) Propensity score matching followed by linear regression with an original treatment variable Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 14 / 30

Balancing Covariates for a Dichotomized Treatment Kam and Palmer Some vs. No College Graduated vs. No College Graduated vs. Some College Original Propensity Score Matching Genetic Matching 0.0 0.2 0.4 0.6 0.8 1.0 Absolute Difference in Standardized Means Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 15 / 30

May Not Balance Covariates for the Original Treatment Urban and Niebler Main Variables Original Propensity Score Matching Fixed Effects 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Absolute Pearson Correlations Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 16 / 30

Propensity Score for a Multi-valued Treatment Consider a multi-valued treatment: T = {0, 1,..., J 1} Standard approach: MLE with multinomial logistic regression π j exp ( ) Xi β j (X i ) = Pr(T i = j X i ) = ( J ) 1 + exp j =1 X i β j where β 0 = 0 and J 1 j=0 πj (X i ) = 1 Covariate balancing conditions with inverse-probability weighting: ( ) ( ) ( ) 1{T i = 0}X i 1{T i = 1}X i 1{T i = J 1}X i E πβ 0(X = E i) πβ 1(X = = E i) π J 1 β (X i ) which equals E(X i ) Idea: estimate π j (X i ) to optimize the balancing conditions Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 17 / 30

CBPS for a Multi-valued Treatment Consider a 3 treatment value case as in our motivating example Sample balance conditions with orthogonalized contrasts: ḡ β (T, X) = 1 N 2 1{T i =0} 1{T i =1} πβ 0 (X i ) πβ 1 (X 1{T i =2} i ) πβ 2 (X i ) X i N i=1 1{T i =1} π 1 β (X i ) 1{T i =2} π 2 β (X i ) Generalized method of moments (GMM) estimation: ˆβ CBPS = argmin β ḡ β (T, X) Σ β (T, X) 1 ḡ β (T, X) where Σ β (T, X) is the covariance of sample moments Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 18 / 30

Score Conditions as Covariate Balancing Conditions Balancing the first derivative across treatment values: 1 N = 1 N = 1 N N s β (T i, X i ) i=1 N ( i=1 N i=1 ( 1{T i =1} π 1 β (X i ) 1{T i =1} π 1 β (X i ) ) 1{T i =0} πβ 0 (X i ) ) 1{T i =0} π 0 β (X i ) ( 1{Ti = 1} πβ 1(X ) i) 1{T i = 2} πβ 2(X X i i) β 1 π 1 β (X i) + β 2 π 1 β (X i) + ( 1{T i =2} πβ 2 (X i ) ) 1{T i =0} πβ 0 (X i ) ) ( 1{T i =2} 1{T i =0} πβ 2 (X i ) πβ 0 (X i ) Can be added to CBPS as over-identifying restrictions β 1 πβ 2(X i) β 2 πβ 2(X i) Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 19 / 30

Extension to More Treatment Values The same idea extends to a treatment with more values For example, consider a four-category treatment Sample moment conditions based on orthogonalized contrasts: ḡ β (T i, X i ) = 1 N N i=1 1{T i =0} πβ 0 (X i ) 1{T i =0} πβ 0 (X i ) 1{T i =0} πβ 0 (X i ) + 1{T i =1} π 1 β (X i ) 1{T i =1} πβ 1 (X i ) + 1{T i =1} πβ 1 (X i ) 1{T i =2} πβ 2 (X i ) 1{T i =2} πβ 2 (X i ) 1{T i =2} π 2 β (X i ) 1{T i =3} π 3 β (X i ) + 1{T i =3} πβ 3 (X i ) + 1{T i =3} πβ 3 (X i ) X i A similar orthogonalization strategy can be applied to the longitudinal setting with marginal structural models (Imai & Ratkovic, JASA, in-press) Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 20 / 30

Propensity Score for a Continuous Treatment The stabilized weights: f (T i ) f (T i X i ) Covariate balancing condition: ( ) f (T { E i ) f (Ti Xi ) T i Xi = where T i and X i f (T i = E(T i )E(X f (T i ) Xi ) T i df (Ti i ) = 0. are centered versions of T i and X i Again, estimate the generalized propensity score such that covariate balance is optimized } Xi ) Xi df(xi ) Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 21 / 30

CBPS for a Continuous Treatment Standard approach (e.g., Robins et al. 2000): T i X i T i indep. N (X i β, σ 2 ) i.i.d. N (0, σ 2 ) where further transformation of T i can make these distributional assumptions more credible Sample covariate balancing conditions: ḡ θ (T, X) = ( ) sθ (T, X) w θ (T, X) = 1 N N [ exp i=1 { 1 2σ{ 2 1 2σ 2 1 (T σ 2 i Xi β)xi 1 1 σ 2 (T 2X i i X i β) 2 } }] β + (Xi β) 2 Ti Xi GMM estimation: covariance matrix can be analytically calculated Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 22 / 30

Back to the Education Example: CBPS vs. ML CBPS achieves better covariate balance Some College vs. No College Graduated vs. No College Graduated vs. Some College CBPS 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 ML CBPS 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 ML CBPS 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 ML Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 23 / 30

CBPS Avoids Extremely Large Weights No College Some College Graduated Share of Total Weight 0.0 0.2 0.4 0.6 0.8 1.0 ML CBPS 0.0 0.2 0.4 0.6 0.8 1.0 ML CBPS 0.0 0.2 0.4 0.6 0.8 1.0 ML CBPS 0 100 200 300 0 100 300 500 0 50 100 150 Number of Observations Number of Observations Number of Observations Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 24 / 30

CBPS Balances Well for a Dichotomized Treatment CBPS 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Propensity Score Matching (Kam and Palmer) CBPS 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Genetic Matching (Henderson and Chatfield) CBPS 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 ML Propensity Score Weighting Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 25 / 30

Empirical Results: Graduation Matters, Efficiency Gain Effect on Political Participation 4 2 0 2 4 ML ML CBPS ML CBPS CBPS Some College Graduated Dichotomized Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 26 / 30

Onto the Advertisement Example Original ML CBPS Main Variables Fixed Effects 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Absolute Pearson Correlations Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 27 / 30

Empirical Finding: Some Effect of Advertisement Effect on Contributions 0 500 1500 2500 1 5 50 500 2500 Ads (on log scale) Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 28 / 30

Concluding Remarks Numerous advances in generalizing propensity score methods to non-binary treatments Yet, many applied researchers don t use these methods and dichotomize non-binary treatments We offer a simple method to improve the estimation of propensity score for general treatment regimes Open-source R package: CBPS: Covariate Balancing Propensity Score available at CRAN Ongoing extensions: 1 nonparametric estimation via empirical likelihood 2 generalizing instrumental variables estimates 3 spatial treatments Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 29 / 30

References Covariate Balancing Propensity Score. Journal of the Royal Statistical Society, Series B, Vol. 76, pp. 243 263. Robust Estimation of Inverse Probability Weights for Marginal Structural Models. Journal of the American Statistical Association, Forthcoming. Covariate Balancing Propensity Score for General Treatment Regimes. Paper available at http://imai.princeton.edu Send comments and questions to kimai@princeton.edu Kosuke Imai (Princeton) Covariate Balancing Propensity Score Columbia (October 14, 2014) 30 / 30

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