Multiply Robust Estimation of Quantile Treatment Effects with Missing Responses

Abstract

Causal inference and missing data have attracted significant research interests in recent years, while the current literature usually focuses on only one of these two issues. In this paper, we develop two multiply robust methods to estimate the quantile treatment effect (QTE), in the context of missing data. Compared to the commonly used average treatment effect, QTE provides a more complete picture of the difference between the treatment and control groups. The first one is based on inverse probability weighting, the resulting QTE estimator is root-n consistent and asymptotic normal, as long as the class of candidate models of propensity scores contains the correct model and so does that for the probability of being observed. The second one is based on augmented inverse probability weighting, which further relaxes the restriction on the probability of being observed. Simulation studies are conducted to investigate the performance of the proposed method, and the motivated CHARLS data are analyzed, exhibiting different treatment effects at various quantile levels.

Keywords

Augmented inverse probability weighting / CHARLS / Missing data / Multiply robust / Quantile treatment effect / 62G05 / 62G35

References

[1] Bang H, Robins JM. Doubly robust estimation in missing data and causal inference models. Biometrics. 2005, 61: 962-973.

[2] Cai, Z.W., Fang, Y., Lin, M., Tang, S.F.: Inferences for partially conditional quantile treatment effect model. Technical Report 202005, Department of Economics, University of Kansas (2020)

[3] Chen CX, Shen BY, Liu AY, Wu RL, Wang M. A multiple robust propensity score method for longitudinal analysis with intermittent missing data. Biometrics. 2021, 77(2): 519-532.

[4] Chen LJ, Li DY, Zhou C. Distributed inference for the extreme value index. Biometrika. 2021, 109(1): 257-264.

[5] Chen X, Liu WD, Zhang YC. Quantile regression under memory constraint. Ann. Stat.. 2019, 47(6): 3244-3273.

[6] Christelis D, Dobrescu LI. The causal effect of social activities on cognition: evidence from 20 European countries. Soc. Sci. Medi.. 2020, 247: 1-9

[7] Donald S, Hsu YC. Estimation and inference for distribution functions and quantile functions in treatment effect models. J. Econom.. 2014, 178: 383-397.

[8] Firpo S. Efficient semiparametric estimation of quantile treatment effects. Econometrica. 2007, 75(1): 259-276.

[9] Han, P.S., Kong, L.L., Zhao, J.W., Zhou, X.C.: A general framework for quantile estimation with incomplete data. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 81(2), 305–333 (2019)

[10] Han PS, Wang L. Estimation with missing data: beyond double robustness. Biometrika. 2013, 100: 417-430.

[11] Han PS. Multiply robust estimation in regression analysis with missing data. J. Am. Stat. Assoc.. 2014, 109(507): 1159-1173.

[12] Hernán MA, Robins JM. Causal Inference: What If. 2020, Boca Raton, Chapman & Hall/CRC

[13] Hu, Y.Q., Lei, X.Y., Smith, J.P., Zhao, Y.H.: Effects of social activities on cognitive functions: evidence from CHARLS. In: Aging in Asia: Findings From New and Emerging Data Initiatives, pp. 279–308, Chap. 12. National Academies Press (US), Washington (2012)

[14] Imbens GW, Rubin DB. Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. 2015, Cambridge, Cambridge University Press.

[15] Lin HZ, Zhou FY, Wang QX, Zhou L, Qin J. Robust and efficient estimation for the treatment effect in causal inference and missing data problems. J. Econom.. 2018, 205(2): 363-380.

[16] Liu HX, Fan XJ, Luo HY, Zhou ZL, Shen C, Hu NB, Zhai XM. Comparison of depressive symptoms and its influencing factors among the elderly in urban and rural areas: evidence from the china health and retirement longitudinal study (CHARLS). Int. J. Environ. Res. Public Health. 2021, 18(8): 3886.

[17] Lunceford JK, Davidian M. Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Stat. Med.. 2004, 23: 2937-2960.

[18] Melly B. Estimation of counterfactual distributions using quantile regression. 2006, Monograph (discussion paper), University of St.Gallen

[19] Rosenbaum PR, Rubin DB. Reducing bias in observational studies using subclassification on the propensity score. J. Am. Stat. Assoc.. 1984, 79387516-524.

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

[21] Rubin DB. Multiple Imputation for Nonresponse in Surveys. 2004, New York, Wiley

[22] Seaman SR, White IR. Review of inverse probability weighting for dealing with missing data. Stat. Methods Med. Res.. 2013, 22: 278-295.

[23] Tan ZQ. Bounded, efficient and doubly robust estimation with inverse weighting. Biometrika. 2010, 97(3): 661-682.

[24] Wei Y, Ma YY, Carroll RJ. Multiple imputation in quantile regression. Biometrika. 2012, 99: 423-438.

[25] Xie YY, Cotton C, Zhu YY. Multiply robust estimation of causal quantile treatment effects. Stat. Med.. 2020, 39: 4238-4251.

[26] Zhang YC. Extremal quantile treatment effects. Ann. Stat.. 2018, 46(6B): 3707-3740.

[27] Zhang ZW, Chen Z, Troendle JF, Zhang J. Causal inference on quantiles with an obstetric application. Biometrics. 2012, 68: 697-706.

Funding

National Natural Science Foundation of China(11871376)

Natural Science Foundation of Shanghai(21ZR1420700)

Open Research Fund of KLATASDS-MOE

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature