The Supervised Learning Approach To Estimating Heterogeneous Causal Regime Effects

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1 The Supervised Learning Approach To Estimating Heterogeneous Causal Regime Effects Thai T. Pham Stanford Graduate School of Business May, 2016

2 Introduction Observations Many sequential treatment settings: patients make adjustments in medications in multiple periods; students decide whether to follow an educational honors program over multiple years; in labor market, the unemployed might participate in a set of programs (job search, subsidized job, training) sequentially. Heterogeneity in treatment sequence reactions: medication effects can be heterogeneous across patients and across time; the same for educational program and labor market. Hard to set up sequential randomized experiments in reality. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 2 / 27

3 Introduction Contributions Develop a nonparametric framework using supervised learning to estimate heterogeneous treatment regime effects from observational (or experimental) data. Treatment Regime: a set of functions of characteristics and intermediate outcomes. Propose using supervised learning approach (deep learning), which gives good estimation accuracy and which is robust to model misspecification. Propose matching based testing method for the estimation of heterogeneous treatment regime effects. Propose matching based kernel estimator for variance of heterogeneous treatment regime effects (time allows). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 3 / 27

4 Contributions (cont d) Introduction In this paper, we Focus on dynamic setting with multiple treatments applied sequentially (in contrast to a single treatment). Focus on the heterogeneous (in contrast to average) effect of a sequence of treatments, i.e. a treatment regime. Focus on observational data (in contrast to experimental data). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 4 / 27

5 An Illustrative Model Setup - Motivational Dataset The Setup The North Carolina Honors Program Dataset. There are 24, 112 observations in total. X 0 = [Y 0, d 1, d 2, d 3 ], where Y 0 is the Math test score at the end of 8th grade and d 1, d 2, d 3 are census-data dummy variables. W 0, W 1 {0, 1} are treatment variables. Y 1 : end of 9th grade Math test score. Y 2 : end of 10th grade Math test score (object of interest). Y 0, Y 1, Y 2 are pre-scaled to have zero mean and unit variance. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 5 / 27

6 Setup - Model An Illustrative Model The Setup End of eighth grade: Students initial information X 0, which includes Math test score Y 0 and other personal information (d 1, d 2, d 3 ), is observed. Decide to follow honors (W 0 = 1) or standard (W 0 = 0) program. End of ninth grade: X 0, W 0, and Math test score Y 1 are observed. Decide to switch or stay in current program (W 1 = 1 or 0). End of tenth grade: X 0, W 0, Y 1, W 1, and Math test score Y 2 are observed. Object of interest: Y 2 (It could be any functions of X 0, Y 1, Y 2 ). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 6 / 27

7 An Illustrative Model The Setup Potential Outcome (PO) Framework Treatment regime d = (d 0, d 1 ) has d 0 : X 0 W 0 {0, 1} and d 1 : X 0 W 0 Y 1 W 1 {0, 1}. Potential Outcome Y 1 (W 0 ) = Y 1. Also, the observed outcome Y 1 = W 0 Y 1 (1) + (1 W 0 ) Y 1 (0). Similarly, Y d 2 = Y 2 if the subject follows regime d. We also write Y 2 = Y d 2 = Y 2(W 0, W 1 ) when d 0 maps to W 0 and d 1 maps to W 1. We have Y 2 = W 0 W 1 Y 2 (1, 1) + W 0 (1 W 1 ) Y 2 (1, 0)+ (1 W 0 )W 1 Y 2 (0, 1) + (1 W 0 )(1 W 1 ) Y 2 (0, 0). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 7 / 27

8 An Illustrative Model Types of Treatment Regime The Setup Static Treatment Regime: subjects specify (or are specified) the whole treatment plan based only on the initial covariates (X 0 ). So d : X 0 (W 0, W 1 ) {0, 1} 2. Dynamic Treatment Regime: subjects choose (or are assigned) the initial treatment based on the initial covariates (X 0 ); then subsequently choose (or are assigned) the next treatment based on the initial covariates (X 0 ), the first period treatment (W 0 ), and the intermediate outcome (Y 1 ); and so on. This is our original setup. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 8 / 27

9 An Illustrative Model The Setup Potential Outcome (PO) Framework (Cont d) [ ] Objective: Estimate E Y2 d Y d 2 for individuals (or average), and derive heterogeneous optimal regime Difficulties: d (C) = arg max E d [ Y d 2 C ] for individual covariates C. Fundamental Problem of Causal Inference: for each subject, we never observe both Y d 2 and Y d 2. Selection Bias: students following d may fundamentally be different from those following d (e.g., students with good test scores choose the honors program in each period). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 9 / 27

10 An Illustrative Model Identification Results Identification Result - Static Treatment Regime Theorem (Identification Result - STR) Let d 0 = d 0 (X 0 ) and d 1 = d 1 (X 0 ). Then (with Assumptions) [ ] Y2 1{W 0 = d 0 } 1{W 1 = d 1 } [ ] E P(W 0 = d 0 X 0 ) P(W 1 = d 1 X 0 ) X 0 = E Y2 d X 0. Corollary: observed/estimable, Transformed Outcome [ { [ }}{ W0 W 1 E Y 2 (1 W ] ] 0)(1 W 1 ) X 0 = E [ unobserved, PO {}}{ Y 2 (1, 1) Y 2 (0, 0) ] X 0. e 0 e 1 (1 e 0 )(1 e 1 ) Here, e 0 = P(W 0 = 1 X 0 ) and e 1 = P(W 1 = 1 X 0 ). Matching Based Testing Method STR Estimation Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 10 / 27

11 An Illustrative Model Identification Results Identification Result - Dynamic Treatment Regime Theorem (Identification Result - DTR) Let d 0 = d 0 (X 0 ) and d 1 = d 1 (X 0, X 1, Y 1, W 0 ). Then (with Assumptions) In period T = 1: [ ] 1{W 1 = d 1 } E Y 2 P(W 1 = d 1 X 0, X 1, Y 1, W 0 ) X 0, X 1, Y 1, W 0 }{{} observed/estimable = E [ Y d ] 1 2 X 0, X 1, Y 1, W 0. }{{} In period T = 0: PO [ ] Y 2 1{W 1 = d 1 } 1{W 0 = d 0 } E P(W 1 = d 1 X 0, X 1, Y 1, W 0 ) P(W 0 = d 0 X 0 ) X 0 = E [ Y2 d ] X 0. DTR Estimation Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 11 / 27

12 Model Estimation Challenges In Traditional Approach Goal: Specify a relation b/w transformed outcome T and covariates C. Econometric approaches assume T = h(c; β) + ɛ for a fixed (linear) function h( ) and E[ɛ C] = 0, and estimate β by minimizing T h(c; β) 2. Problem: Linear models need not give good estimates in general. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 12 / 27

13 Model Estimation Machine Learning Approach Machine learning methods generally give much more accurate estimates than traditional econometric models. Empirical comparisons of different machine learning methods with linear regressions: Caruana and Niculescu-Mizil (2006) 1 Morton, Marzban, Giannoulis, Patel, Aparasu, and Kakadiaris (2014) 2 1 Caruana, R. and A. Niculescu-Mizil, (2006), An Empirical Comparison of Supervised Learning Algorithms, Proceedings of the 23rd International Conference on Machine Learning, Pittsburgh, PA. 2 Morton, A., E. Marzban, G. Giannoulis, A. Patel, R. Aparasu, and I. A. Kakadiaris, (2014), A Comparison of Supervised Machine Learning Techniques for Predicting Short-Term In-Hospital Length of Stay Among Diabetic Patients, 13th International Conference on Machine Learning and Applications. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 13 / 27

14 Model Estimation Machine Learning Approach (Cont d) Goal: Specify a relation b/w transformed outcome T and covariates C. Machine learning (ML) methods allow h( ) to vary in terms of complexity, and estimate β by minimizing T h(c; β) 2 + λg(β) where g penalizes complex models. Data set = (Training, Validation, Test). Use Training set (with CV) to choose the optimal h( ) in terms of complexity, validation set to choose the optimal hyperparameter λ, and test set to evaluate the performance. RMSE is the comparison criterion. Hence, ML approach is flexible and performance oriented. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 14 / 27

15 Model Estimation Estimating Model Propensity Score Estimation: Use logistic regression or other ML techniques such as Random Forest, Gradient Boosting, etc. Full Model Estimation: (Though many ML techniques would work) We use a deep learning method in machine learning literature called Multilayer Perceptron. It possesses the universal approximation property: it can approximate any continuous function on any compact subset of R n. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 15 / 27

16 Model Estimation Multilayer Perceptron (MLP) Assume we want to estimate T = h(c; β) + ɛ. MLP (with one hidden layer) considers K ( ) h(c; β) = α j σ(γj T C + θ j ) and β = K, (α j, γ j, θ j ) K j=1, j=1 where σ is a sigmoid function such as σ(x) = 1/(1 + exp( x)). Empirically, MLP (and deep learning in general) is shown to work very well. (Lecun et al. 3, Mnih et al. 4 ) 3 LeCun, Y., Y. Bengio, and G. Hinton, (2015), Deep Learning, Nature 521, (28 May). 4 Mnih, V., K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, (2015), Human-level control through deep reinforcement learning, Nature 518, (26 February). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 16 / 27

17 Model Estimation Testing Method Matching Based Testing Method The identification results relate unobserved difference of potential outcomes Z to observed (or estimable) transformed outcome T : E[T C] = E[Z C]. For example in STR : Z = Y d 2 Y d 2, C = X 0. Randomly draw M units with treatment regime d. Denote by x d,m 0 s and y d,m 2 s the covariates and corresponding outcomes. For each m, determine x d,m 0 = arg min x i 0 regime = d x i 0 x d,m 0 2. Let τ m = y d,m 2 y d,m 2. Here, τ is a proxy for the unobserved Z. Let τ be the estimator which fits x 0 to T. Define τ m = 1 d,m 2 ( τ(x 0 ) + τ(x d,m 0 )). Define the matching loss M : 1 M M m=1 ( τ m τ m ) 2. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 17 / 27

18 Simulation Setup Simulations Setup Test the ability of our method in adapting to heterogeneity in the treatment regime effect. 50, 000 obs for training; 5, 000 obs for validation; 5, 000 for testing. X 0 U([0, 1] 10 ); W 0 {0, 1}; Y 1 R with standard normal noise; W 1 {0, 1}; Y 2 R with standard normal noise. e 0 (X 0 ) = e 1 (X 0 ) = e 1 (X 0, W 0, Y 1 ) = 0.5. [ τ 1 (X 0 ) = E Y W 0=1 1 Y W 0=0 1 τ 2 (X 0, W 0, Y 1 ) = E where ξ(x) = ] X 0 = ξ(x 0 [1])ξ(X 0 [2]); and ] X 0, W 0, Y 1 [ Y W 1=1 2 Y W 1=0 2 = ρ(y 1 )ρ(w 0 )ξ(x 0 [1]) e 12(x 1/2) ; ρ(x) = e 20(x 1/3). Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 18 / 27

19 Simulations Results Simulation Results Table: Performance In Terms of Root Mean Squared Error (RMSE) Method Linear Regression (LR) Multilayer Perceptron (MLP) STR DTR: T = DTR: T = sdv (TO: T = 0) = 2.41; sdv (true effect: T = 0) = sdv (TO: T = 1) = 3.21; sdv (true effect: T = 1) = Comments: MLP returns really good results, and it outperforms LR. Static setting does not fit here as the RMSEs on STR are bad. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 19 / 27

20 Simulations Results Simulation Results (cont d) 5 MLP: T = 0 True Effect: T = 0 LR: T = 0 MLP: T = 1 True Effect: T = 1 LR: T = 1 Figure: Heterogeneous Treatment Regime Effect Using Validation and Test Data. The first row corresponds to period T = 0 and the second row corresponds to period T = 1. In each period: the middle picture visualizes the true treatment effect; the left one is the estimated effect by using Multilayer Perceptron; and the right one is the estimated effect by using Linear Regression. 5 We thank Wager and Athey (2015) for sharing their visualization code. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 20 / 27

21 Empirical Application Propensity Score Estimation Estimation of Propensity Scores Use the North Carolina Honors Program data (in illustrative model). Estimate and P(W 0 = 1 X 0 ), P(W 1 = 1 X 0 ) P(W 1 = 1 X 0, Y 1, W 0 ). Use Random Forest as a probabilistic classification problem. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 21 / 27

22 Empirical Application Static Treatment Regime Estimation Model Estimation - Static Treatment Regime Use three methods: Linear Regression, Gradient Boosting, and Multilayer Perceptron. Method Validation Matching Loss Test Matching Loss Linear Regression Gradient Boosting Multilayer Perceptron Comments: MLP outperforms other methods. All results are bad, which signals the dynamic nature of the data. STR Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 22 / 27

23 Empirical Application Dynamic Treatment Regime Estimation Model Estimation - Dynamic Treatment Regime Period T = 1 - Method Validation Matching Loss Test Matching Loss Linear Regression Gradient Boosting Multilayer Perceptron *sdv(to: val) = 4.06; sdv(est. true effect: val) = *sdv(to: test) = 4.03; sdv(est. true effect: test) = Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 23 / 27

24 Empirical Application Dynamic Treatment Regime Estimation Model Estimation - Dynamic Treatment Regime (Cont d) Period T = 0 - Method Validation Matching Loss Test Matching Loss Linear Regression Gradient Boosting Multilayer Perceptron DTR - Use only students who follow the optimal treatment in T = 1. *sdv(to: val) = 6.94; sdv(est. true effect: val) = *sdv(to: test) = 7.45; sdv(est. true effect: test) = Remark: The results are worse than that in simulations due to unobserved heterogeneity. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 24 / 27

25 Empirical Application Heterogeneous Optimal Regime Estimation Heterogeneous Optimal Regime Static Treatment Regime Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 25 / 27

26 Empirical Application Heterogeneous Optimal Regime Estimation Heterogeneous Optimal Regime (cont d) STR: Estimated gain per student from heterogeneous optimal regime over homogeneous optimal ( regime (0, 0): 1 [Ŷ2 (1, 1) 0)] Ŷ2(0, + (1,1) opt (0,0) not opt [Ŷ2 (1, 0) 0)] Ŷ2(0, + 1 ] [Ŷ2 ) (0, 1) Ŷ2(0, 0) = (1,0) opt (0,1) opt DTR: Estimated gain per student in T = 0 from heterogeneous optimal W 0 over homogeneous optimal treatment W 0 = 0: ] W W 0 =1 optimal [Ŷ 0 =1 2 Ŷ W 0=0 2 #obs used in T = 0 s.t. W 0 = 1 opt = mean(y 2 ) = 0; min(y 2 ) = 4.06; max(y 2 ) = Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 26 / 27

27 Conclusion Conclusion We developed a nonparametric framework using supervised learning to estimate heterogeneous causal regime effects. Our model addresses the dynamic treatment setting, the population heterogeneity, and the difficulty in setting up sequential randomized experiments in reality. We introduced machine learning approach, in particular deep learning, which demonstrates its estimation power and which is robust to model misspecification. We also introduced matching based testing method for the estimation of heterogeneous treatment regime effects. A matching based kernel estimator for variance of these effects is introduced in Appendix. Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 27 / 27

28 Appendix Variance Estimation - Matching Kernel Approach Matching M matching pairs [ ] (x d,m 0, y d,m 2 ); (x d,m 0, y d,m 2 ), m = 1,..., M. Fix x new 0. To estimate σ 2 (x new Let x mean,m 0 = x d,m 0 +x d,m 0 0 ) = Var(Y2 d Y d 2 ɛ m = τ m τ m. 2. An estimator for σ 2 (x0 new ) is x new 0 ), we define M σ 2 (x0 new m=1 ) = K (H 1 [x mean,m 0 x0 new ]) ɛ 2 m M m=1 K (H 1 [x mean,m. 0 x0 new ]) Thai T. Pham Estimation of Heterogeneous Causal Regime Effects 27 / 27

The Supervised Learning Approach To Estimating Heterogeneous Causal Regime Effects

The Supervised Learning Approach To Estimating Heterogeneous Causal Regime Effects Thai T. Pham August, 2016 Abstract We develop a nonparametric framework using supervised learning to estimate heterogeneous