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Continuous treatment effect estimation via generative adversarial de-confounding

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Abstract

One fundamental problem in causal inference is the treatment effect estimation in observational studies, and its key challenge is to handle the confounding bias induced by the associations between covariates and treatment variable. In this paper, we study the problem of effect estimation on continuous treatment from observational data, going beyond previous work on binary treatments. Previous work on binary treatment focuses on de-confounding by balancing the distribution of covariates between the treated and control groups with either propensity score or confounder balancing techniques. In the continuous setting, those methods would fail as we can hardly evaluate the distribution of covariates under each treatment status. To tackle the case of continuous treatments, we propose a novel Generative Adversarial De-confounding (GAD) algorithm to eliminate the associations between covariates and treatment variable with two main steps: (1) generating an “calibration” distribution without associations between covariates and treatment by randomly perturbation on treatment variable; (2) learning sample weights that transfer the distribution of observed data to the “calibration” distribution for de-confounding with a Generative Adversarial Network. We show, both theoretically and with empirical experiments, that our GAD algorithm can remove the associations between covariates and treatment, hence, precisely estimating the causal effect of continuous treatment. Extensive experiments on both synthetic and real-world datasets demonstrate that our algorithm outperforms the state-of-the-art methods for effect estimation of continuous treatment with observational data.

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Notes

  1. Units represent the objects of treatment. For example, in medical experiments, the units refer to the patients who take a particular medication.

  2. \({\mathbf {X}}'\) should have the identical marginal distribution with the observed covariates, that is \(P({\mathbf {X}}') = P({\mathbf {X}})\).

  3. https://en.wikipedia.org/wiki/Moment_(mathematics).

References

  • Arjovsky M, Chintala S, Bottou L (2017) Wasserstein generative adversarial networks. In: Proceedings of the 34th international conference on machine learning, PMLR, proceedings of machine learning research, vol 70, pp 214–223

  • Athey S, Imbens G (2016) Recursive partitioning for heterogeneous causal effects. Proc Natl Acad Sci 113:7353–7360

    Article  MathSciNet  MATH  Google Scholar 

  • Athey S, Imbens GW, Wager S (2018) Approximate residual balancing: debiased inference of average treatment effects in high dimensions. J R Stat Soc: Ser B (Stat Methodol) 80(4):597–623

    Article  MathSciNet  MATH  Google Scholar 

  • Austin PC (2011) An introduction to propensity score methods for reducing the effects of confounding in observational studies. Multivar Behav Res 46(3):399–424

    Article  Google Scholar 

  • Bang H, Robins JM (2005) Doubly robust estimation in missing data and causal inference models. Biometrics 61(4):962–973

    Article  MathSciNet  MATH  Google Scholar 

  • Chan D, Ge R, Gershony O, Hesterberg T, Lambert D (2010) Evaluating online ad campaigns in a pipeline: causal models at scale. In: Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, pp 7–16

  • Chan KG, Yam SC, Zhang Z (2016) Globally efficient non-parametric inference of average treatment effects by empirical balancing calibration weighting. J R Stat Soc Ser B Stat Methodol 78(3):673–700

    Article  MathSciNet  MATH  Google Scholar 

  • Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C et al (2016) Double machine learning for treatment and causal parameters. arXiv preprint arXiv:1608.00060

  • Duchi J, Namkoong H (2018) Learning models with uniform performance via distributionally robust optimization. arXiv preprint arXiv:1810.08750

  • Egel D, Graham BS, de Xavier Pinto CC (2008) Inverse probability tilting for moment condition models with missing data. Single equation models eJournal, Econometrics

    MATH  Google Scholar 

  • Fan J, Imai K, Liu H, Ning Y, Yang X (2016) Improving covariate balancing propensity score: a doubly robust and efficient approach. Technical report

  • Flores CA, Flores-Lagunes A (2009) Identification and estimation of causal mechanisms and net effects of a treatment under unconfoundedness. IZA Institute of Labor Economics Discussion Paper Series

  • Fong C, Hazlett C, Imai K et al (2018) Covariate balancing propensity score for a continuous treatment: application to the efficacy of political advertisements. Ann Appl Stat 12(1):156–177

    Article  MathSciNet  MATH  Google Scholar 

  • Galagate D (2016) Causal inference with a continuous treatment and outcome: alternative estimators for parametric dose-response functions with applications. Ph.D. thesis

  • Galvao AF, Wang L (2015) Uniformly semiparametric efficient estimation of treatment effects with a continuous treatment. J Am Stat Assoc 110(512):1528–1542

    Article  MathSciNet  MATH  Google Scholar 

  • Goodfellow I, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A, Bengio Y (2014) Generative adversarial nets. In: Advances in neural information processing systems, pp 2672–2680

  • Hainmueller J (2012) Entropy balancing for causal effects: a multivariate reweighting method to produce balanced samples in observational studies. Polit Anal 20(1):25–46

    Article  Google Scholar 

  • Hill JL (2011) Bayesian nonparametric modeling for causal inference. J Comput Graph Stat 20(1):217–240

    Article  MathSciNet  Google Scholar 

  • Hirano K, Imbens GW (2004) The propensity score with continuous treatments. Applied Bayesian modeling and causal inference from incomplete-data perspectives 226164:73–84

    MathSciNet  MATH  Google Scholar 

  • Holland PW (1986) Statistics and causal inference. J Am Stat Assoc 81(396):945–960

    Article  MathSciNet  MATH  Google Scholar 

  • Imai K, Ratkovic M (2014) Covariate balancing propensity score. J R Stat Soc: Ser B (Stat Methodol) 76(1):243–263

    Article  MathSciNet  MATH  Google Scholar 

  • Imai K, Van Dyk DA (2004) Causal inference with general treatment regimes: generalizing the propensity score. J Am Stat Assoc 99(467):854–866

    Article  MathSciNet  MATH  Google Scholar 

  • Imbens GW (2004) Nonparametric estimation of average treatment effects under exogeneity: a review. Rev Econ Stat 86(1):4–29

    Article  MathSciNet  Google Scholar 

  • Imbens GW, Rubin DB (2015) Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Kallus N (2019) Generalized optimal matching methods for causal inference. J Mach Learn Res (forthcoming)

  • Kallus N, Santacatterina M (2019) Kernel optimal orthogonality weighting: a balancing approach to estimating effects of continuous treatments. arXiv, Methodology

  • Kallus N, Zhou A (2018) Policy evaluation and optimization with continuous treatments. In: International conference on artificial intelligence and statistics, pp 1243–1251

  • Kennedy EH, Ma Z, McHugh MD, Small DS (2017) Non-parametric methods for doubly robust estimation of continuous treatment effects. J R Stat Soc: Ser B (Stat Methodol) 79(4):1229–1245

    Article  MathSciNet  MATH  Google Scholar 

  • Kohavi R, Longbotham R (2011) Unexpected results in online controlled experiments. ACM SIGKDD Explor Newsl 12(2):31–35

    Article  Google Scholar 

  • Kreif N, Grieve R, Díaz I, Harrison D (2015) Evaluation of the effect of a continuous treatment: a machine learning approach with an application to treatment for traumatic brain injury. Health Econ 24(9):1213–1228

    Article  Google Scholar 

  • Kuang K, Cui P, Li B, Jiang M, Yang S (2017) Estimating treatment effect in the wild via differentiated confounder balancing. In: Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining, pp 265–274. ACM

  • Kuang K, Cui P, Athey S, Xiong R, Li B (2018) Stable prediction across unknown environments. In: Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, pp 1617–1626

  • Kuang K, Cui P, Li B, Jiang M, Wang Y, Wu F, Yang S (2019) Treatment effect estimation via differentiated confounder balancing and regression. ACM Trans Knowl Discov Data (TKDD) 14(1):1–25

    Google Scholar 

  • Kuang K, Cui P, Zou H, Li B, Tao J, Wu F, Yang S (2020) Data-driven variable decomposition for treatment effect estimation. IEEE Trans Knowl Data Eng. https://doi.org/10.1109/TKDE.2020.3006898

    Article  Google Scholar 

  • Kuang K, Li L, Geng Z, Xu L, Zhang K, Liao B, Huang H, Ding P, Miao W, Jiang Z (2020b) Causal inference. Engineering 6(3):253–263

    Article  Google Scholar 

  • Künzel SR, Sekhon JS, Bickel PJ, Yu B (2019) Metalearners for estimating heterogeneous treatment effects using machine learning. Proc Natl Acad Sci 116(10):4156–4165

    Article  Google Scholar 

  • Li F, Li L, Yin J, Zhang Y, Zhou Q, Kuang K (2020a) How to interpret machine knowledge. Engineering 6(3):218–220

    Article  Google Scholar 

  • Li M, Kuang K, Zhu Q, Chen X, Guo Q, Wu F (2020b) IB-M: a flexible framework to align an interpretable model and a black-box model. In: 2020 IEEE international conference on bioinformatics and biomedicine (BIBM), pp 643–649. IEEE

  • Liu J, Ma Y, Wang L (2018) An alternative robust estimator of average treatment effect in causal inference. Biometrics 74(3):910–923

    Article  MathSciNet  MATH  Google Scholar 

  • Liu Y, Dieng A, Roy S, Rudin C, Volfovsky A (2019) Interpretable almost matching exactly for causal inference. AISTATS

  • Louizos C, Shalit U, Mooij J, Sontag D, Zemel R, Welling M (2017) Causal effect inference with deep latent-variable models. In: Proceedings of the 31st annual conference on neural information processing systems

  • Lu C, Wang S (2020) The general-purpose intelligent agent. Engineering 6(3):221–226

    Article  Google Scholar 

  • McCaffrey DF, Ridgeway G, Morral AR (2004) Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychol Methods 9(4):403

    Article  Google Scholar 

  • Neugebauer R, van der Laan M (2007) Nonparametric causal effects based on marginal structural models. J Stat Plan Inference 137(2):419–434

    Article  MathSciNet  MATH  Google Scholar 

  • Olaya D, Coussement K, Verbeke W (2020) A survey and benchmarking study of multitreatment uplift modeling. Data Min Knowl Disc 34:273–308

    Article  MathSciNet  Google Scholar 

  • Pearl J (2009) Causality. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Ren K, Zheng T, Qin Z, Liu X (2020) Adversarial attacks and defenses in deep learning. Engineering 6(3):346–360

    Article  Google Scholar 

  • Robins J, Rotnitzky A (2001) Comment on inference for semiparametric models: some questions and an answer, by P.J. Bickel and J. Kwon. Stat Sin 11:920–936

    Google Scholar 

  • Robins JM, Hernan MA, Brumback B (2000) Marginal structural models and causal inference in epidemiology

  • Rojas-Carulla M, Schölkopf B, Turner R, Peters J (2018) Invariant models for causal transfer learning. J Mach Learn Res 19(1):1309–1342

    MathSciNet  MATH  Google Scholar 

  • Rong G, Mendez A, Assi EB, Zhao B, Sawan M (2020) Artificial intelligence in healthcare: review and prediction case studies. Engineering 6(3):291–301

    Article  Google Scholar 

  • Rosenbaum PR, Rubin DB (1983) The central role of the propensity score in observational studies for causal effects. Biometrika 70(1):41–55

    Article  MathSciNet  MATH  Google Scholar 

  • Rubin DB (1974) Estimating causal effects of treatments in randomized and nonrandomized studies. J Educ Psychol 66(5):688

    Article  Google Scholar 

  • Rudas K, Jaroszewicz S (2018) Linear regression for uplift modeling. Data Min Knowl Disc 32:1275–1305

    Article  MathSciNet  MATH  Google Scholar 

  • Rudin C (2019) Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat Mach Intell 1(5):206–215

    Article  Google Scholar 

  • Schölkopf B, Locatello F, Bauer S, Ke NR, Kalchbrenner N, Goyal A, Bengio Y (2021) Toward causal representation learning. Proc IEEE 109(5):612–634

    Google Scholar 

  • Soltys M, Jaroszewicz S, Rzepakowski P (2014) Ensemble methods for uplift modeling. Data Min Knowl Disc 29:1531–1559

    Article  MathSciNet  Google Scholar 

  • Tan Z (2010) Bounded, efficient and doubly robust estimation with inverse weighting. Biometrika 97:661–682

    Article  MathSciNet  MATH  Google Scholar 

  • Tian Q, Kuang K, Jiang K, Wu F, Wang Y (2021) Analysis and applications of class-wise robustness in adversarial training. arXiv preprint arXiv:2105.14240

  • Wager S, Athey S (2015) Estimation and inference of heterogeneous treatment effects using random forests. J Am Stat Assoc 113:1228–1242

    Article  MathSciNet  MATH  Google Scholar 

  • Westreich D, Lessler J, Funk MJ (2010) Propensity score estimation: neural networks, support vector machines, decision trees (cart), and meta-classifiers as alternatives to logistic regression. J Clin Epidemiol 63(8):826–833

    Article  Google Scholar 

  • Zhao Q (2016) Covariate balancing propensity score by tailored loss functions. arXiv, Methodology

  • Zhu Y, Coffman D, Ghosh D (2015) A boosting algorithm for estimating generalized propensity scores with continuous treatments. J Causal Inference 3:25–40

    Article  MathSciNet  Google Scholar 

  • Zou WY, Shyam S, Mui M, Wang M, Pedersen J, Ghahramani Z (2020) Learning continuous treatment policy and bipartite embeddings for matching with heterogeneous causal effects. arXiv:2004.09703

  • Zubizarreta J (2015) Stable weights that balance covariates for estimation with incomplete outcome data. J Am Stat Assoc 110:910–922

    Article  MathSciNet  MATH  Google Scholar 

  • Žliobaitė I (2017) Measuring discrimination in algorithmic decision making. Data Min Knowl Disc 31:1060–1089

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported in part by National Natural Science Foundation of China (Nos. 61625107, 62006207), National Key Research and Development Program of China (Nos. 2018AAA0101900, 2020YFC0832500), the Fundamental Research Funds for the Central Universities and Zhejiang Province Natural Science Foundation (No. LQ21F020020).

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Authors and Affiliations

  1. Department of Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang Province, China

    Kun Kuang, Yunzhe Li & Fei Wu

  2. Tsinghua University, Beijing, China

    Bo Li & Peng Cui

  3. Alibaba Group, Hangzhou, Zhejiang Province, China

    Hongxia Yang

  4. NetEase Fuxi AI Lab, Hangzhou, Zhejiang Province, China

    Jianrong Tao

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  1. Kun Kuang
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  2. Yunzhe Li
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Correspondence to Kun Kuang.

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Kuang, K., Li, Y., Li, B. et al. Continuous treatment effect estimation via generative adversarial de-confounding. Data Min Knowl Disc 35, 2467–2497 (2021). https://doi.org/10.1007/s10618-021-00797-x

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