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Causal inference: a missing data perspective. (English) Zbl 1397.62125

Summary: Inferring causal effects of treatments is a central goal in many disciplines. The potential outcomes framework is a main statistical approach to causal inference, in which a causal effect is defined as a comparison of the potential outcomes of the same units under different treatment conditions. Because for each unit at most one of the potential outcomes is observed and the rest are missing, causal inference is inherently a missing data problem. Indeed, there is a close analogy in the terminology and the inferential framework between causal inference and missing data. Despite the intrinsic connection between the two subjects, statistical analyses of causal inference and missing data also have marked differences in aims, settings and methods. This article provides a systematic review of causal inference from the missing data perspective. Focusing on ignorable treatment assignment mechanisms, we discuss a wide range of causal inference methods that have analogues in missing data analysis, such as imputation, inverse probability weighting and doubly robust methods. Under each of the three modes of inference – Frequentist, Bayesian and Fisherian randomization – we present the general structure of inference for both finite-sample and super-population estimands, and illustrate via specific examples. We identify open questions to motivate more research to bridge the two fields.

MSC:

62G05 Nonparametric estimation
62H12 Estimation in multivariate analysis
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[1] Abadie, A. and Imbens, G. W. (2006). Large sample properties of matching estimators for average treatment effects. Econometrica74 235–267. · Zbl 1112.62042
[2] Abadie, A. and Imbens, G. (2011). Bias corrected matching estimators for average treatment effects. J. Bus. Econom. Statist.29 1–11. · Zbl 1214.62031
[3] Andrews, D. W. (2000). Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space. Econometrica68 399–405. · Zbl 1015.62044
[4] Angrist, J. D., Imbens, G. W. and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. J. Amer. Statist. Assoc.91 444–455. · Zbl 0897.62130
[5] Angrist, J. D. and Pischke, J.-S. (2008). Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton Univ. Press, Princeton, NJ. · Zbl 1159.62090
[6] Athey, S. and Imbens, G. W. (2015). Machine learning methods for estimating heterogeneous causal effects. Available at arXiv:1504.01132. · Zbl 1357.62190
[7] Athey, S., Imbens, G. W. and Wager, S. (2018). Approximate residual balancing: De-biased inference of average treatment effects in high dimensions. J. R. Stat. Soc. Ser. B. Stat. Methodol. To appear. Available at https://arxiv.org/abs/1604.07125.
[8] Athey, S., Imbens, G., Pham, T. and Wager, S. (2017). Estimating average treatment effects: Supplementary analyses and remaining challenges. Am. Econ. Rev.107 278–281.
[9] Bang, H. and Robins, J. M. (2005). Doubly robust estimation in missing data and causal inference models. Biometrics61 962–972. · Zbl 1087.62121
[10] Belloni, A., Chernozhukov, V. and Hansen, C. (2014). Inference on treatment effects after selection among high-dimensional controls. Rev. Econ. Stud.81 608–650.
[11] Belloni, A., Chernozhukov, V., Fernández-Val, I. and Hansen, C. (2017). Program evaluation and causal inference with high-dimensional data. Econometrica85 233–298.
[12] Bickel, P. J. and Doksum, K. A. (2015). Mathematical Statistics: Basic Ideas and Selected Topics, Volume I, 2nd ed. CRC Press, Boca Raton, FL. · Zbl 1380.62002
[13] Bloniarz, A., Liu, H., Zhang, C.-H., Sekhon, J. S. and Yu, B. (2016). Lasso adjustments of treatment effect estimates in randomized experiments. Proc. Natl. Acad. Sci. USA113 7383–7390. · Zbl 1357.62098
[14] Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017). Stan: A probabilistic programming language. J. Stat. Softw.76 1–32.
[15] Chapin, F. S. (1947). Experimental Designs in Sociological Research. Harper, New York.
[16] Chen, H., Geng, Z. and Zhou, X.-H. (2009). Identifiability and estimation of causal effects in randomized trials with noncompliance and completely nonignorable missing data. Biometrics65 675–682. · Zbl 1172.62041
[17] Cheng, J. and Small, D. S. (2006). Bounds on causal effects in three-arm trials with non-compliance. J. R. Stat. Soc. Ser. B. Stat. Methodol.68 815–836. · Zbl 1110.62152
[18] Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2016). Double/debiased machine learning for treatment and causal parameters. Preprint. Available at arXiv:1608.00060.
[19] Chib, S. and Jacobi, L. (2016). Bayesian fuzzy regression discontinuity analysis and returns to compulsory schooling. J. Appl. Econometrics31 1026–1047.
[20] Chung, E. and Romano, J. P. (2013). Exact and asymptotically robust permutation tests. Ann. Statist.41 484–507. · Zbl 1267.62064
[21] Cochran, W. G. (1953). Sampling Techniques, 1st ed. Wiley, New York. · Zbl 0051.10707
[22] Cochran, W. G. (1957). Analysis of covariance: Its nature and uses. Biometrics13 261–281.
[23] Cochran, W. G. (2007). Sampling Techniques, 3rd ed. Wiley, New York. · Zbl 0051.10707
[24] Cornfield, J., Haenszel, W., Hammond, E. et al. (1959). Smoking and lung cancer: Recent evidence and a discussion of some questions. J. Natl. Cancer Inst.22 173–203.
[25] Crump, R. K., Hotz, V. J., Imbens, G. W. and Mitnik, O. A. (2009). Dealing with limited overlap in estimation of average treatment effects. Biometrika96 187–199. · Zbl 1163.62083
[26] Dawid, A. P. (2000). Causal inference without counterfactuals. J. Amer. Statist. Assoc.95 407–424. · Zbl 0999.62003
[27] Dawid, A. P. Musio, M. and Murtas, R. (2017). The probability of causation. Law, Probability and Risk16 163–179. · Zbl 1357.62036
[28] Dempster, A., Laird, N. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B. Stat. Methodol.39 1–38. · Zbl 0364.62022
[29] Ding, P. (2014). Three occurrences of the hyperbolic-secant distribution. Amer. Statist.68 32–35.
[30] Ding, P. and Dasgupta, T. (2016). A potential tale of two-by-two tables from completely randomized experiments. J. Amer. Statist. Assoc.111 157–168.
[31] Ding, P., Feller, A. and Miratrix, L. (2016). Randomization inference for treatment effect variation. J. R. Stat. Soc. Ser. B. Stat. Methodol.78 655–671.
[32] Ding, P. and Geng, Z. (2014). Identifiability of subgroup causal effects in randomized experiments with nonignorable missing covariates. Stat. Med.33 1121–1133.
[33] Ding, P. and Lu, J. (2017). Principal stratification analysis using principal scores. J. R. Stat. Soc. Ser. B. Stat. Methodol.79 757–777.
[34] Ding, P. and VanderWeele, T. J. (2016). Sensitivity analysis without assumptions. Epidemiology27 368–377.
[35] Ding, P., Geng, Z., Yan, W. and Zhou, X.-H. (2011). Identifiability and estimation of causal effects by principal stratification with outcomes truncated by death. J. Amer. Statist. Assoc.106 1578–1591. · Zbl 1234.62142
[36] Ding, W. and Song, P. X.-K. (2016). EM algorithm in Gaussian copula with missing data. Comput. Statist. Data Anal.101 1–11. · Zbl 1466.62056
[37] Elliott, M., Raghunathan, T. and Li, Y. (2010). Bayesian inference for causal mediation effects using principal stratification with dichotomous mediators and outcomes. Biostatistics11 353–372.
[38] Fan, Y., Guerre, E. and Zhu, D. (2017). Partial identification of functionals of the joint distribution of “potential outcomes”. J. Econometrics197 42–59. · Zbl 1443.62444
[39] Fan, Y. and Park, S. S. (2010). Sharp bounds on the distribution of treatment effects and their statistical inference. Econometric Theory26 931–951. · Zbl 1191.62061
[40] Feller, A., Greif, E., Miratrix, L. and Pillai, N. (2016). Principal stratification in the twilight zone: Weakly separated components in finite mixture models. Preprint. Available at arXiv:1602.06595.
[41] Firth, D. and Bennett, K. E. (1998). Robust models in probability sampling. J. R. Stat. Soc. Ser. B. Stat. Methodol.60 3–21. · Zbl 0910.62009
[42] Fisher, R. A. (1935). The Design of Experiments, 1st ed. Oliver and Boyd, Edinburgh.
[43] Frangakis, C. and Rubin, D. B. (1999). Addressing complications of intention-to-treat analysis in the combined presence of all-or-none treatment-noncompliance and subsequent missing outcomes. Biometrika86 365–378. · Zbl 0934.62110
[44] Frangakis, C. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics58 21–29. · Zbl 1209.62288
[45] Frumento, P., Mealli, F., Pacini, B. and Rubin, D. B. (2012). Evaluating the effect of training on wages in the presence of noncompliance, nonemployment, and missing outcome data. J. Amer. Statist. Assoc.107 450–466. · Zbl 1328.62609
[46] Frumento, P., Mealli, F., Pacini, B. and Rubin, D. B. (2016). The fragility of standard inferential approaches in principal stratification models relative to direct likelihood approaches. Stat. Anal. Data Min.9 58–70.
[47] Gallop, R., Small, D., Lin, J., Elliot, M., Joffe, M. and Have, T. T. (2009). Mediation analysis with principal stratification. Stat. Med.28 1108–1130.
[48] Gelfand, A. and Smith, A. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc.85 398–409. · Zbl 0702.62020
[49] Gelman, A., Meng, X.-L. and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statist. Sinica6 733–807. · Zbl 0859.62028
[50] Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014). Bayesian Data Analysis, 3nd ed. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1279.62004
[51] Gilbert, P. and Hudgens, M. (2008). Evaluating candidate principal surrogate endpoints. Biometrics64 1146–1154. · Zbl 1152.62389
[52] Graham, B. S., de Xavier Pinto, C. C. and Egel, D. (2012). Inverse probability tilting for moment condition models with missing data. Rev. Econ. Stud.79 1053–1079.
[53] Grilli, L. and Mealli, F. (2008). Nonparametric bounds on the causal effect of university studies on job opportunities using principal stratification. J. Educ. Behav. Stat.33 111–130.
[54] Gustafson, P. (2009). What are the limits of posterior distributions arising from nonidentified models, and why should we care? J. Amer. Statist. Assoc.104 1682–1695. · Zbl 1205.62169
[55] Gustafson, P. (2015). Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data. CRC Press, Boca Raton, FL. · Zbl 1329.62009
[56] Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica66 315–331. · Zbl 1055.62572
[57] Hainmueller, J. (2012). Entropy balancing for causal effects: A multivariate reweighting method to produce balanced samples in observational studies. Polit. Anal.20 25–46.
[58] Hájek, J. (1971). Comment on a paper by D. Basu. In Foundations of Statistical Inference (V. P. Godambe and D. A. Sprott, eds.) 236. Holt, Rinehart and Winston, Toronto.
[59] Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica47 153–161. · Zbl 0392.62093
[60] Heckman, J., Lopes, H. and Piatek, R. (2014). Treatment effects: A Bayesian perspective. Econometric Rev.33 36–67.
[61] Hirano, K. and Imbens, G. W. (2004). The propensity score with continuous treatments. In Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives. 73–84. Wiley, Chichester. · Zbl 05274806
[62] Hirano, K., Imbens, G. W. and Ridder, G. (2003). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica71 1161–1189. · Zbl 1152.62328
[63] Ho, D. E., Imai, K., King, G. and Stuart, E. A. (2011). MatchIt: Nonparametric preprocessing for parametric causal inference. J. Stat. Softw.42 1–28.
[64] Hoeffding, W. (1952). The large-sample power of tests based on permutations of observations. Ann. Math. Stat.23 169–192. · Zbl 0046.36403
[65] Holland, P. (1986). Statistics and causal inference (with discussion). J. Amer. Statist. Assoc.81 945–970. · Zbl 0607.62001
[66] Horvitz, D. and Thompson, D. (1952). A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc.47 663–685. · Zbl 0047.38301
[67] Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability1 221–233. Univ. California Press, Berkeley, CA. · Zbl 0212.21504
[68] Ichino, A., Mealli, F. and Nannicini, T. (2008). From temporary help jobs to permanent employment: What can we learn from matching estimators and their sensitivity? J. Appl. Econometrics23 305–327.
[69] Imai, K. (2008). Sharp bounds on the causal effects in randomized experiments with “truncation-by-death”. Statist. Probab. Lett.78 144–149. · Zbl 1131.62099
[70] Imai, K. and Ratkovic, M. (2014). Covariate balancing propensity score. J. R. Stat. Soc. Ser. B. Stat. Methodol.76 243–263.
[71] Imai, K. and van Dyk, D. (2004). Causal treatment with general treatment regimes: Generalizing the propensity score. J. Amer. Statist. Assoc.99 854–866. · Zbl 1117.62361
[72] Imbens, G. W. (2000). The role of the propensity score in estimating dose-response functions. Biometrika87 706–710. · Zbl 1120.62334
[73] Imbens, G. W. (2003). Sensitivity to exogeneity assumptions in program evaluation. Am. Econ. Rev.93 126–132.
[74] Imbens, G. W. (2004). Nonparametric estimation of average treatment effects under exogeneity: A review. Rev. Econ. Stat.86 4–29.
[75] Imbens, G. W. and Angrist, J. (1994). Identification and estimation of local average treatment effects. Econometrica62 467–476. · Zbl 0800.90648
[76] Imbens, G. W. and Rubin, D. B. (1997). Bayesian inference for causal effects in randomized experiments with noncompliance. Ann. Statist.25 305–327. · Zbl 0877.62005
[77] Imbens, G. W. and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge Univ. Press, New York. · Zbl 1355.62002
[78] Kang, J. D. and Schafer, J. L. (2007). Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statist. Sci.22 523–539. · Zbl 1246.62073
[79] Li, X. and Ding, P. (2017). General forms of finite population central limit theorems with applications to causal inference. J. Amer. Statist. Assoc.112 1759–1769.
[80] Li, F., Mattei, A. and Mealli, F. (2015). Evaluating the causal effect of university grants on student dropout: Evidence from a regression discontinuity design using principal stratification. Ann. Appl. Stat.9 1906–1931. · Zbl 1397.62585
[81] Li, F., Morgan, K. and Zaslavsky, A. (2018). Balancing covariates via propensity score weighting. J. Amer. Statist. Assoc. To appear. Available at https://doi.org/10.1080/01621459.2016.1260466.
[82] Li, F., Baccini, M., Mealli, F., Zell, E. R., Frangakis, C. E. and Rubin, D. B. (2014). Multiple imputation by ordered monotone blocks with application to the anthrax vaccine research program. J. Comput. Graph. Statist.23 877–892.
[83] Lin, W. (2013). Agnostic notes on regression adjustments to experimental data: Reexamining freedman’s critique. Ann. Appl. Stat.7 295–318. · Zbl 1454.62217
[84] Lindley, D. V. (1972). Bayesian Statistics: A Review. SIAM, Philadelphia, PA.
[85] Little, R. J. (1988). Missing-data adjustments in large surveys. J. Bus. Econom. Statist.6 287–296.
[86] Little, R. and An, H. (2004). Robust likelihood-based analysis of multivariate data with missing values. Statist. Sinica14 949–968. · Zbl 1073.62050
[87] Little, R. J. and Rubin, D. B. (2002). Statistical Analysis with Missing Data, 2nd ed. Wiley-Interscience, Hoboken, NJ. · Zbl 1011.62004
[88] Liublinska, V. and Rubin, D. B. (2014). Sensitivity analysis for a partially missing binary outcome in a two-arm randomized clinical trial. Stat. Med.33 4170–4185.
[89] Lu, J., Ding, P. and Dasgupta, T. (2015). Treatment effects on ordinal outcomes: Causal estimands and sharp bounds. Preprint. Available at arXiv:1507.01542.
[90] Lunceford, J. K. and Davidian, M. (2004). Stratification and weighting via the propensity score in estimation of causal treatment effects: A comparative study. Stat. Med.23 2937–2960.
[91] Manski, C. F. (1990). Nonparametric bounds on treatment effects. Am. Econ. Rev.80 319–323.
[92] Mattei, A. and Mealli, F. (2011). Augmented designs to assess principal strata direct effects. J. R. Stat. Soc. Ser. B. Stat. Methodol.73 729–752. · Zbl 1228.62059
[93] Mattei, A., Mealli, F. and Pacini, B. (2014). Identification of causal effects in the presence of nonignorable missing outcome values. Biometrics70 278–288. · Zbl 1419.62405
[94] Mealli, F. and Rubin, D. B. (2015). Clarifying missing at random and related definitions, and implications when coupled with exchangeability. Biometrika102 995–1000. · Zbl 1390.62042
[95] Mealli, F., Imbens, G. W., Ferro, S. and Biggeri, A. (2004). Analyzing a randomized trial on breast self-examination with noncompliance and missing outcomes. Biostatistics5 207–222. · Zbl 1096.62120
[96] Mebane, W. R. Jr and Poast, P. (2013). Causal inference without ignorability: Identification with nonrandom assignment and missing treatment data. Polit. Anal.21 233–251.
[97] Meng, X.-L. (1994). Posterior predictive \(p\)-values. Ann. Statist.22 1142–1160. · Zbl 0820.62027
[98] Mercatanti, A. (2004). Analyzing a randomized experiment with imperfect compliance and ignorable conditions for missing data: Theoretical and computational issues. Comput. Statist. Data Anal.46 493–509. · Zbl 1429.62055
[99] Mercatanti, A. and Li, F. (2014). Do debit cards increase household spending? Evidence from a semiparametric causal analysis of a survey. Ann. Appl. Stat.8 2405–2508. · Zbl 1454.62488
[100] Mercatanti, A. and Li, F. (2017). Do debit cards decrease cash demand? Causal inference and sensitivity analysis using principal stratification. J. R. Stat. Soc. Ser. C. Appl. Stat.66 759–776.
[101] Miratrix, L. W., Sekhon, J. S. and Yu, B. (2013). Adjusting treatment effect estimates by post-stratification in randomized experiments. J. R. Stat. Soc. Ser. B. Stat. Methodol.75 369–396.
[102] Mitra, R. and Reiter, J. P. (2011). Estimating propensity scores with missing covariate data using general location mixture models. Stat. Med.30 627–641.
[103] Mitra, R. and Reiter, J. P. (2016). A comparison of two methods of estimating propensity scores after multiple imputation. Stat. Methods Med. Res.25 188–204.
[104] Molinari, F. (2010). Missing treatments. J. Bus. Econom. Statist.28 82–95. · Zbl 1198.62162
[105] Murray, J. S. and Reiter, J. P. (2016). Multiple imputation of missing categorical and continuous values via Bayesian mixture models with local dependence. J. Amer. Statist. Assoc.111 1466–1479.
[106] Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics79 147–168. · Zbl 0873.62049
[107] Neyman, J. (1935). Statistical problems in agricultural experimentation. Suppl. J. R. Stat. Soc.2 107–180. · JFM 63.1103.02
[108] Neyman, J. (1990). On the application of probability theory to agricultural experiments. Essay on principles. Section 9. Statist. Sci.5 465–472. · Zbl 0955.01560
[109] Nolen, T. L. and Hudgens, M. G. (2011). Randomization-based inference within principal strata. J. Amer. Statist. Assoc.106 581–593. · Zbl 1232.62152
[110] Qin, J. (2017). Biased Sampling, Over-Identified Parameter Problems and Beyond. Springer, Singapore. · Zbl 1441.62008
[111] Richardson, T. S., Evans, R. J. and Robins, J. M. (2010). Transparent parameterizations of models for potential outcomes. In Bayesian Statistics9 569–610. Oxford Univ. Press, Oxford.
[112] Robins, J. M. (1986). A new approach to causal inference in mortality studies with sustained exposure periods—Application to control of the healthy worker survivor effect. Math. Modelling7 1393–1512. · Zbl 0614.62136
[113] Robins, J. M. and Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. Stat. Med.16 285–319.
[114] Robins, J. M., Rotnitzky, A. and Scharfstein, D. O. (2000). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In Statistical Models in Epidemiology, the Environment, and Clinical Trials 1–94. Springer, New York. · Zbl 0998.62091
[115] Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. J. Amer. Statist. Assoc.90 106–121. · Zbl 0818.62042
[116] Robins, J. M., van der Vaart, A. and Ventura, V. (2000). Asymptotic distribution of \(p\) values in composite null models. J. Amer. Statist. Assoc.95 1143–1156. · Zbl 1072.62522
[117] Rosenbaum, P. R. (1984a). Conditional permutation tests and the propensity score in observational studies. J. Amer. Statist. Assoc.79 565–574.
[118] Rosenbaum, P. R. (1984b). The consquences of adjustment for a concomitant variable that has been affected by the treatment. J. R. Stat. Soc. Ser. B. Stat. Methodol.147 656–666.
[119] Rosenbaum, P. R. (1987). Sensitivity analysis for certain permutation inferences in matched observational studies. Biometrika74 13–26. · Zbl 0605.62130
[120] Rosenbaum, P. R. (2002a). Covariance adjustment in randomized experiments and observational studies. Statist. Sci.17 286–327. · Zbl 1013.62117
[121] Rosenbaum, P. R. (2002b). Observational Studies, 2nd ed. Springer, New York. · Zbl 0985.62091
[122] Rosenbaum, P. R. (2010). Design of Observational Studies. Springer, New York. · Zbl 1308.62005
[123] Rosenbaum, P. R. and Rubin, D. B. (1983a). Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J. R. Stat. Soc. Ser. B. Stat. Methodol.45 212–218.
[124] Rosenbaum, P. R. and Rubin, D. B. (1983b). The central role of the propensity score in observational studies for causal effects. Biometrika70 41–55. · Zbl 0522.62091
[125] Rosenbaum, P. R. and Rubin, D. B. (1984). Reducing bias in observational studies using subclassification on the propensity score. J. Amer. Statist. Assoc.79 516–524.
[126] Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. J. Educ. Psychol.66 688–701.
[127] Rubin, D. B. (1975). Bayesian inference for causality: The role of randomization. In Proceedings of the Social Statistics Section of the American Statistical Association 233–239.
[128] Rubin, D. B. (1976). Inference and missing data. Biometrika63 581–592. · Zbl 0344.62034
[129] Rubin, D. B. (1977). Assignment to a treatment group on the basis of a covariate. Journal of Educational Statistics2 1–26.
[130] Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist.6 34–58. · Zbl 0383.62021
[131] Rubin, D. B. (1979). Using multivariate matched sampling and regression adjustment to control bias in observational studies. J. Amer. Statist. Assoc.74 318–324. · Zbl 0413.62047
[132] Rubin, D. B. (1980). Comment on “Randomization analysis of experimental data: The Fisher randomization test” by D. Basu. J. Amer. Statist. Assoc.75 591–593.
[133] Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applies statistician. Ann. Statist.12 1151–1172. · Zbl 0555.62010
[134] Rubin, D. B. (1986). Statistical matching using file concatenation with adjusted weights and multiple imputations. J. Bus. Econom. Statist.4 87–94.
[135] Rubin, D. B. (1998). More powerful randomization-based \(p\)-values in double-blind trials with non-compliance. Stat. Med.17 371–385.
[136] Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions. J. Amer. Statist. Assoc.100 322–331. · Zbl 1117.62418
[137] Rubin, D. B. (2006a). Causal inference through potential outcomes and principal stratification: Application to studies with “censoring” due to death. Statist. Sci.91 299–321. · Zbl 1246.62198
[138] Rubin, D. B. (2006b). Matched Sampling for Causal Effects. Cambridge Univ. Press, Cambridge. · Zbl 1118.62113
[139] Rubin, D. B. (2007). The design versus the analysis of observational studies for causal effects: Parallels with the design of randomized trials. Stat. Med.26 20–36.
[140] Rubin, D. B. (2008). For objective causal inference, design trumps analysis. Ann. Appl. Stat.2 808–840. · Zbl 1149.62089
[141] Scharfstein, D., Rotnitzky, A. and Robins, J. M. (1999). Adjusting for nonignorable drop-out using semiparametric nonresponse models (with discussion). J. Amer. Statist. Assoc.94 1096–1146. · Zbl 1072.62644
[142] Schwartz, S., Li, F. and Reiter, J. P. (2012). Sensitivity analysis for unmeasured confounding in principal stratification settings with binary variables. Stat. Med.31 949–962.
[143] Seaman, S., Galati, J., Jackson, D. and Carlin, J. (2013). What is meant by “missing at random”? Statist. Sci.28 257–268. · Zbl 1331.62036
[144] Sekhon, J. S. (2011). Multivariate and propensity score matching software with automated balance optimization: The matching package for R. J. Stat. Softw.42 1–52.
[145] Stuart, E. (2010). Matching methods for causal inference: A review and a look forward. Statist. Sci.25 1–21. · Zbl 1328.62007
[146] Tanner, M. and Wong, W. (1987). The calculation of posterior distributions by data augmentation. J. Amer. Statist. Assoc.82 528–540. · Zbl 0619.62029
[147] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B. Stat. Methodol.58 267–288. · Zbl 0850.62538
[148] Tsiatis, A. A., Davidian, M., Zhang, M. and Lu, X. (2008). Covariate adjustment for two-sample treatment comparisons in randomized clinical trials: A principled yet flexible approach. Stat. Med.27 4658–4677.
[149] Tukey, J. W. (1993). Tightening the clinical trial. Controlled Clinical Trials14 266–285.
[150] van Buuren, S. (2012). Flexible Imputation of Missing Data. CRC press, Boca Raton, FL. · Zbl 1256.62005
[151] van der Laan, M. J. and Rose, S. (2011). Targeted Learning: Causal Inference for Observational and Experimental Data. Springer, New York.
[152] VanderWeele, T. (2008). Simple relations between principal stratification and direct and indirect effects. Statist. Probab. Lett.78 2957–2962. · Zbl 1317.62007
[153] Wager, S., Du, W., Taylor, J. and Tibshirani, R. J. (2016). High-dimensional regression adjustments in randomized experiments. Proc. Natl. Acad. Sci. USA113 12673–12678.
[154] White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica48 817–838. · Zbl 0459.62051
[155] Yang, S. and Ding, P. (2018). Asymptotic causal inference with observational studies trimmed by the estimated propensity scores. Biometrika. To appear. Available at https://arxiv.org/abs/1604.07125.
[156] Yang, F. and Small, D. S. (2016). Using post-outcome measurement information in censoring-by-death problems. J. R. Stat. Soc. Ser. B. Stat. Methodol.78 299–318.
[157] Yang, S., Wang, L. and Ding, P. (2017). Nonparametric identification of causal effects with confounders subject to instrumental missingness. Preprint. Available at arXiv:1702.03951.
[158] Zhang, G. and Little, R. J. (2009). Extensions of the penalized spline of propensity prediction method of imputation. Biometrics65 911–918. · Zbl 1172.62071
[159] Zhang, J. and Rubin, D. B. (2003). Estimation of causal effects via principal stratification when some outcomes are truncated by “death”. J. Educ. Behav. Stat.28 353–358.
[160] Zhang, J., Rubin, D. B. and Mealli, F. (2009). Likelihood-based analysis of the causal effects of job-training programs using principal stratification. J. Amer. Statist. Assoc.104 166–176. · Zbl 1388.62074
[161] Zhang, Z., Liu, W., Zhang, B., Tang, L. and Zhang, J. (2016). Causal inference with missing exposure information: Methods and applications to an obstetric study. Stat. Methods Med. Res.25 2053–2066.
[162] Zhou, J., Zhang, Z., Li, Z. and Zhang, J. (2015). Coarsened propensity scores and hybrid estimators for missing data and causal inference. Int. Stat. Rev.83 449–471.
[163] Zigler, C. and Belin, T. (2012). A Bayesian approach to improved estimation of causal effect predictiveness for a principal surrogate endpoint. Biometrics68 922–932. · Zbl 1272.62106
[164] Zubizarreta, J. R. (2015). Stable weights that balance covariates for estimation with incomplete outcome data. J. Amer. Statist. Assoc.110 910–922. · Zbl 1373.62051
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