Mann-Whitney Test with Adjustments to Pre-treatment Variables for Missing Values and Observational Study

Chen, Songxi (2013): Mann-Whitney Test with Adjustments to Pre-treatment Variables for Missing Values and Observational Study. Published in:

Abstract

The conventional Wilcoxon/Mann-Whitney test can be invalid for comparing treatment effects in the presence of missing values or in observational studies. This is because the missingness of the outcomes or the participation in the treatments may depend on certain pre-treatment variables. We propose an approach to adjust the Mann-Whitney test by correcting the potential bias via consistently estimating the conditional distributions of the outcomes given the pre-treatment variables. We also propose semiparametric extensions of the adjusted Mann-Whitney test which leads to dimension reduction for high dimensional covariate. A novel bootstrap procedure is devised to approximate the null distribution of the test statistics for practical implementations. Results from simulation studies and an economic observational study data analysis are presented to demonstrate the performance of the proposed approach.

• Item Type: MPRA Paper
• Language: English
• Keywords: Dimension reduction; Kernel smoothing; Mann-Whitney statistic; Missing outcomes;Observational studies;Selection bias.
• Subjects: C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory; C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs; C - Mathematical and Quantitative Methods > C9 - Design of Experiments; G - Financial Economics > G0 - General

References

Bilker, W. and Wang, M.-C. (1996), “A semiparametric extension of the Mann-Whitney test for randomly truncated data,” Biometrics, 52, 10–20.

Breslow, N. (2003), “Are statistical contributions to medicine undervalued?” Biometrics,59, 1–8.

Cheng, P. E. (1994), “Nonparametric-estimation of mean functionals with data missing at random,” Journal of the American Statistical Association, 89, 81–87.

Cheung, Y. K. (2005), “Exact two-sample inference with missing data,” Biometrics, 61,524–531.

Chu, C. K. and Chen, K. F. (1995), “Nonparametric regression estimates using misspecified binary responses,” Biometrika, 82, 315–325.

Dehejia, R. H. and Wahba, S. (1999), “Causal effects in nonexperimental studies: reevaluation the evaluation of training programs,” Journal of the American Statistical Association,94, 1053–1062.

Fan, J. and Gijbels, I. (1996), Local Polynomial Modeling and Its Applications, Chapman and Hall, London.

Hahn, J. (1998), “On the role of the propensity score in efficient semiparametric estimation of average treatment effects,” Econometrica, 66, 315–331.

Hall, P., Racine, J., and Li, Q. (2004), “Cross-validation and the estimation of conditional probability densities,” Journal of the American Statistical Association, 99, 1015–1026.

Hall, P. and Yao, Q. (2005), “Approximating conditional distribution functions using dimension reduction,” The Annals of Statistics, 33, 1404–1421.

Hansen, L. P. (1982), “Large sample properties of generalized method of moments estimators,”Econometrica, 50, 1029–1054.

H¨ardle, W. (1990), Applied Nonparametric Regression, Cambridge: Cambridge University Press.

Hirano, K., Imbens, G. W., and Ridder, G.(2003), “Efficient estimation of average treatment effects using the estimated propensity score,” Econometrica, 71, 1161–1189.

Hoeffding, W. (1948), “A class of statistics with asymptotically normal distribution,” The Annals of Mathematical Statistics, 19, 293–325.

Horvitz, D. G. and Thompson, D. J. (1952), “A generalization of sampling without replacement from a finite universe,” Journal of the American Statistical Association, 47, 663–685.

Hu, Z., Follmann, D. A., and Qin, J.(2010), “Semiparametric dimension reduction estimation for mean response with missing data,” Biometrika, 97, 305–319.

Hu, Z., Follmann, D. A., and Qin, J.(2011), “Dimension reducted kernel estimation for distribution function with incomplete data,” Journal of Stat, 141, 3084–3093.

Imbens, G. (2004), “Nonparametric estimation of average treatment effects under exogeneity:a review,” Review of Economics and Statistics, 86, 4–30.

Korn, E. L. and Baumrind, S. (1998), “Clinician preferences and the estimation of causal treatment differences,” Statistical Science, 13, 209–235.

Koroljuk, V. S. and Borovskich, Y. V. (1994), Theory of U-Statistics, Kluwer, Dordrecht.

Kuk, A. Y. C. (1993), “A kernel method for estimating finite population functions using auxiliary information,” Biometrika, 80, 385–392.

Lalonde, R. J. (1986), “Evaluating the econometric evaluations of training programs with experimental data,” American Economic Review, 76, 604–620.

Little, R. and Rubin, D. (2002), Statistical Analysis With Missing Data, Wiley, 2nd ed.

Matloff, N. S. (1981), “Use of regression functions for improved estimation of means,”Biometrika, 68, 685–689.

Newey, W. K. and McFadden, D. (1994), “Large sample estimation and hypothesis testing,”Handbook of Econometrics, Vol 4, ed. by R. Engle and D. McFadden. New York: North Holland.

Qin, J., Shao, J., and Zhang, B. (2008), “Efficient and doubly robust imputation for covariate-dependent missing responses,” Journal of the American Statistical Association, 103, 797–810.

Rosenbaum, P. R. (2002), Observational Studies, Springer-Verlag: New York.

Rosenbaum, P. R. and Rubin, D. B. (1983), “The central role of the propensity score in observational studies for causal effects,” Biometrika, 70, 41–55.

Rubin, D. B. (1976), “Inference and missing values (with discussion),” Biometrika, 63, 581–592.

Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, John Wiley.

Tsiatis, A. A. (2006), Semiparametric Theory and Missing Data, Springer-Verlag: New York.

Wang, C. Y., Wang, S., Gutierrez, R. G., and Carroll, R. J. (1998), “Local linear regression for generalized linear models with missing data,” The Annals of Statistics, 26, 1028–1050.