# Strategies for Bias Reduction in Estimation of Marginal Means with Data Missing at Random

## Abstract

Incomplete data are common in many fields of research, and interest often lies in estimating a marginal mean based on available information. This paper is concerned with the comparison of different strategies for estimating the marginal mean of a response when data are missing at random. We evaluate these methods based on the asymptotic bias, empirical bias and efficiency. We show that complete case analysis gives biased results when data are missing at random, but inverse probability weighted estimating equations (IPWEE) and a method based on the expected conditional mean (ECM) yield consistent estimators.. While these methods give estimators which behave similarly in the contexts studied they are based on quite different assumptions. The IPWEE approach requires analysts to specify a model for the missing data mechanism whereas the ECM approach requires a model for the distribution of auxiliary variables driving the missing data mechanism. The latter can be a challenge in practice, particularly when the covariates are of high dimension or are a mixture of continuous and categorical variables. The IPWEE approach therefore has considerable appeal in many practical settings.

## Keywords

Consistent Estimator Covariate Vector Covariate Distribution Inverse Probability Weight Miss Data Mechanism## References

- 1.R.E. Bellman,
*Adaptive Control Processes*(Princeton University Press, Princeton, 1961)zbMATHGoogle Scholar - 2.R. Cameron, K.S. Brown, J.A. Best, C.L. Pelkman, C.L. Madill, S.R. Manske, M.E. Payne, Effectiveness of a social influences smoking prevention program as a function of provider type, training method, and social risk. Am. J. Public Health
**89**, 1827–1831 (1999)CrossRefGoogle Scholar - 3.P.J. Diggle, P. Heagerty, K.Y. Liang, S.L. Zeger,
*Analysis of Longitudinal Data*, 2nd edn. (Oxford University Press, London, 2002)Google Scholar - 4.J.H. Friedman,
*An Overview of Predictive Learning and Function Approximation*, ed. by V. Cherkassky, J.H. Friedman, H. Wechsler. From Statistics to Neural Networks. Proc. NATO/ASI Workshop (Springer, Berlin, 1994), pp. 1–61Google Scholar - 5.G.N. Hortobagyi, R.L. Theriault, A. Lipton, L. Porter, D. Blayney, C. Sinoff, H. Wheeler, J.F. Simeone, J.J. Seaman, R.D. Knight, M. Heffernan, K. Mellars, D.J. Reitsma, Long-term prevention of skeletal complications of metastatic breast cancer with Pamidronate. J. Clin. Oncol.
**16**, 2038–2044 (1998)Google Scholar - 6.D.G. Horvitz, D.J. Thompson, A generalization of sampling without replacement from a finite universe, J. Am. Stat. Assoc.
**47**663–685 (1952)MathSciNetzbMATHCrossRefGoogle Scholar - 7.J.D.Y. Kang, J.L. Schafer, Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Stat. Sci.
**22**, 523–539 (2007)MathSciNetzbMATHCrossRefGoogle Scholar - 8.K.Y. Liang, S.L. Zeger, Longitudinal data analysis using generalized linear models. Biometrika
**73**, 13–22 (1986)MathSciNetzbMATHCrossRefGoogle Scholar - 9.R.J.A. Little, D.B. Rubin,
*Statistical Analysis with Missing Data*(Wiley, 2nd edn. 2002)Google Scholar - 10.C.R. Loader, Local likelihood density estimation. Ann. Stat.
**24**, 1602–1618 (1996)MathSciNetzbMATHGoogle Scholar - 11.P. McCullagh, J.A. Nelder,
*Generalized Linear Models*(Chapman and Hall, London, 1989)zbMATHGoogle Scholar - 12.J. Qin, B. Zhang, Empirical-likelihood-based inference in missing response problems and its application in observational studies. J. Roy. Stat. Soc. B
**69**, 101–122 (2007)MathSciNetCrossRefGoogle Scholar - 13.J.M. Robins, A. Rotnitzky, L.P. Zhao, Estimation of regression coefficients when some regressor are not always observed. J. Am. Stat. Assoc.
**89**, 846–866 (1994)MathSciNetzbMATHCrossRefGoogle Scholar - 14.J.M. Robins, A. Rotnitzky, L.P. Zhao, Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. J. Am. Stat. Assoc.
**90**, 106–121 (1995)MathSciNetzbMATHCrossRefGoogle Scholar - 15.D.B. Rubin, Inference and Missing data. Biometrika
**63**, 581–592 (1976)zbMATHGoogle Scholar - 16.D.B. Rubin,
*Multiple Imputation for Nonresponse in Surveys*(Wiley, New York, 1987)CrossRefGoogle Scholar - 17.J.L. Schafer,
*Analysis of Incomplete Multivariate Data*(Chapman and Hall, New York, 1997)zbMATHCrossRefGoogle Scholar