## Statistical Science

### Introduction to Double Robust Methods for Incomplete Data

#### Abstract

Most methods for handling incomplete data can be broadly classified as inverse probability weighting (IPW) strategies or imputation strategies. The former model the occurrence of incomplete data; the latter, the distribution of the missing variables given observed variables in each missingness pattern. Imputation strategies are typically more efficient, but they can involve extrapolation, which is difficult to diagnose and can lead to large bias. Double robust (DR) methods combine the two approaches. They are typically more efficient than IPW and more robust to model misspecification than imputation. We give a formal introduction to DR estimation of the mean of a partially observed variable, before moving to more general incomplete-data scenarios. We review strategies to improve the performance of DR estimators under model misspecification, reveal connections between DR estimators for incomplete data and “design-consistent” estimators used in sample surveys, and explain the value of double robustness when using flexible data-adaptive methods for IPW or imputation.

#### Article information

Source
Statist. Sci., Volume 33, Number 2 (2018), 184-197.

Dates
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.ss/1525313141

Digital Object Identifier
doi:10.1214/18-STS647

Mathematical Reviews number (MathSciNet)
MR3797709

Zentralblatt MATH identifier
1397.62176

#### Citation

Seaman, Shaun R.; Vansteelandt, Stijn. Introduction to Double Robust Methods for Incomplete Data. Statist. Sci. 33 (2018), no. 2, 184--197. doi:10.1214/18-STS647. https://projecteuclid.org/euclid.ss/1525313141

#### References

• [1] Bang, H. and Robins, J. M. (2005). Doubly robust estimation in missing data and causal inference models. Biometrics 61 962–972.
• [2] Belloni, A. and Chernozhukov, V. (2011). $l_{1}$-Penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82–130.
• [3] Belloni, A., Chernozhukov, V. and Hansen, C. (2016). Lasso methods for Gaussian instrumental variables models. Preprint. Available at arXiv:1012.1297.
• [4] Brookhart, M. A. and Van der Laan, M. J. (2006). A semiparametric model selection criterion with applications to the marginal structural model. Comput. Statist. Data Anal. 50 475–498.
• [5] Cao, W., Tsiatis, A. A. and Davidian, M. (2009). Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika 96 723–734.
• [6] Cassel, C. M., Sarndal, C. E. and Wretman, J. H. (1976). Some results on generalized difference estimation and generalized regression estimation for finite populations. Biometrika 63 615–620.
• [7] Cheng, G., Yu, Z. and Huang, J. Z. (2013). The cluster bootstrap consistency in generalized estimating equations. J. Multivariate Anal. 115 33–47.
• [8] Chernozhukov, V., Escanciano, J. C., Ichimura, H. and Newey, W. K. (2016). Locally robust semiparametric estimation. Preprint. Available at arXiv:1608.00033.
• [9] Farrell, M. H. (2015). Robust inference on average treatment effects with possibly more covariates than observations. J. Econometrics 189 1–23.
• [10] Gruber, S. and van der Laan, M. J. (2010). A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome. Int. J. Biostat. 6 Article 26.
• [11] Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47 663–685.
• [12] Kang, J. D. Y. 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.
• [13] Leeb, H. and Pötscher, B. M. (2005). Model selection and inference: Facts and fiction. Econometric Theory 21 21–59.
• [14] Leeb, H. and Pötscher, B. M. (2006). Performance limits for estimators of the risk or distribution of shrinkage-type estimators, and some general lower risk-bound results. Econometric Theory 22 69–97.
• [15] Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalised linear models. Biometrika 73 13–22.
• [16] Little, R. and An, H. (2004). Robust likelihood-based analysis of multivariate data with missing values. Statist. Sinica 14 949–968.
• [17] Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data. Wiley, New York.
• [18] Long, Q., Zhang, X. and Johnson, B. A. (2011). Robust estimation of area under ROC curve using auxiliary variables in the presence of missing biomarker values. Biometrics 67 559–567.
• [19] Meng, X.-L. (1994). Multiple-imputation inferences with uncongenial sources of input. Statist. Sci. 9 538–573.
• [20] Newey, W. K., Hsieh, F. and Robins, J. M. (2004). Twicing kernels and a small bias property of semiparametric estimators. Econometrica 72 947–962.
• [21] Paik, M. C. (1997). The generalized estimating equations approach when data are not missing completely at random. J. Amer. Statist. Assoc. 92 1320–1329.
• [22] Porter, K. E., Gruber, S., van der Laan, M. J. and Sekhon, J. S. (2011). The relative performance of targeted maximum likelihood estimators. Int. J. Biostat. 7 Article 31.
• [23] Qi, L., Wang, C. Y. and Prentice, R. L. (2005). Weighted estimators for proportional hazards regression with missing covariates. J. Amer. Statist. Assoc. 100 1250–1263.
• [24] Robins, J. and Rotnitzky, A. (1998). Discussion on the paper by Firth and Bennett. J. Roy. Statist. Soc. Ser. B 60 51–52.
• [25] Robins, J., Sued, M., Lei-Gomez, Q. and Rotnitzky, A. (2007). Comment: Performance of double-robust estimators when “inverse probability” weights are highly variable [MR2420458]. Statist. Sci. 22 544–559.
• [26] Robins, J. M. (2000). Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association Section on Bayesian Statistical Science 1999 6–10. Amer. Statist. Assoc., Alexandria, VA.
• [27] Robins, J. M. and Gill, R. D. (1997). Non-response models for the analysis of non-monotone ignorable missing data. Stat. Med. 16 39–56.
• [28] Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89 846–866.
• [29] Rotnitzky, A., Faraggi, D. and Schisterman, E. (2006). Doubly robust estimation of the area under the receiver-operating characteristic curve in the presence of verification bias. J. Amer. Statist. Assoc. 101 1276–1288.
• [30] Rotnitzky, A., Lei, Q. H., Sued, M. and Robins, J. M. (2012). Improved double-robust estimation in missing data and causal inference models. Biometrika 99 439–456.
• [31] Rotnitzky, A. and Vansteelandt, S. (2014). Double-robust methods. In Handbook of Missing Data Methodology (G. Molenberghs, G. Fitzmaurice, M. G. Kenward, A. Tsiatis and G. Verbeke, eds.) 185–212. CRC Press, Boca Raton, FL.
• [32] Scharfstein, D. O., Rotnitzky, A. and Robins, J. M. (1999). Adjusting for nonignorable drop-out using semiparametric nonresponse models: Rejoinder. J. Amer. Statist. Assoc. 94 1135–1146.
• [33] Schnitzer, M. E., Lok, J. J. and Bosch, R. J. (2016). Double robust and efficient estimation of a prognostic model for events in the presence of dependent censoring. Biostatistics 17 165–177.
• [34] Seaman, S. and Copas, A. (2009). Doubly robust generalized estimating equations for longitudinal data. Stat. Med. 28 937–955.
• [35] Seaman, S. R., Galati, J., Jackson, D. and Carlin, J. (2013). What is meant by “missing at random”? Statist. Sci. 28 257–268.
• [36] Seaman, S. R. and Vansteelandt, S. (2018). Supplement to “Introduction to double robust methods for incomplete data.” DOI:10.1214/18-STS647SUPP.
• [37] Tan, Z. (2006). A distributional approach for causal inference using propensity scores. J. Amer. Statist. Assoc. 101 1619–1637.
• [38] Tan, Z. (2008). Comment: Improved local efficiency and double robustness. Int. J. Biostat. 4 Article 10.
• [39] Tan, Z. (2010). Bounded, efficient and doubly robust estimation with inverse weighting. Biometrika 97 661–682.
• [40] Tsiatis, A. A. (2006). Semiparametric Theory and Missing Data. Springer, New York.
• [41] Tsiatis, A. A. and Davidian, M. (2014). Missing data methods: A semi-parametric perspective. In Handbook of Missing Data Methodology (G. Molenberghs, G. Fitzmaurice, M. G. Kenward, A. Tsiatis and G. Verbeke, eds.) Chapter 8. CRC Press, Boca Raton, FL.
• [42] Tsiatis, A. A., Davidian, M. and Cao, W. (2011). Improved doubly robust estimation when data are monotonely coarsened, with application to longitudinal studies with dropout. Biometrics 67 536–545.
• [43] van der Laan, M. J. and Rubin, D. B. (2006). Targeted maximum likelihood learning. Int. J. Biostat. 2 Art. 11.
• [44] Vansteelandt, S., Carpenter, J. and Kenward, M. G. (2015). Analysis of incomplete data using inverse probability weighting and doubly robust estimators. Methodology 6 37–48.
• [45] van der Laan, M. J. and Gruber, S. (2010). Collaborative double robust targeted maximum likelihood estimation. Int. J. Biostat. 6 Article 17.
• [46] Vermeulen, K. and Vansteelandt, S. (2015). Bias-reduced doubly robust estimation. J. Amer. Statist. Assoc. 110 1024–1036.
• [47] Wilson, A. and Reich, B. J. (2014). Confounder selection via penalized credible regions. Biometrics 70 852–861.
• [48] Wirth, K. E., Tchetgen Tchetgen, E. J. and Murray, M. (2010). Adjustment for missing data in complex surveys using doubly robust estimation. Epidemiology 21 863–871.

#### Supplemental materials

• Supplement to “Introduction to double robust methods for incomplete data”. Additional semi-parametric theory, proofs, connections between methods, empirical likelihood DR estimators and software.