Smoothed Empirical Likelihood Inference for ROC Curves with Missing Data

Author(s)
Yueheng An

ABSTRACT

The receiver operating characteristic (ROC) curve has been widely used in scientific research fields. After using the random hot deck imputation, we propose the smoothed empirical likelihood ratio statistic for the ROC curve with missing data. Its asymptotic distribution is a scaled chi-square distribution and empirical likelihood confidence intervals for ROC curves are constructed. The simulation study shows that the proposed interval estimates perform well based on the coverage probability for different sample sizes and response rates.

The receiver operating characteristic (ROC) curve has been widely used in scientific research fields. After using the random hot deck imputation, we propose the smoothed empirical likelihood ratio statistic for the ROC curve with missing data. Its asymptotic distribution is a scaled chi-square distribution and empirical likelihood confidence intervals for ROC curves are constructed. The simulation study shows that the proposed interval estimates perform well based on the coverage probability for different sample sizes and response rates.

Cite this paper

Y. An, "Smoothed Empirical Likelihood Inference for ROC Curves with Missing Data,"*Open Journal of Statistics*, Vol. 2 No. 1, 2012, pp. 21-27. doi: 10.4236/ojs.2012.21003.

Y. An, "Smoothed Empirical Likelihood Inference for ROC Curves with Missing Data,"

References

[1] X.-H. Zhou, D. K. McClish and N. A. Obuchowski, “Statistical Methods in Diagnostic Medicine,” Wiley, New York, 2002.

[2] M. S. Pepe, “The Statistical Evaluation of Medical Tests for Classification and Prediction,” Oxford University Press, Oxford, 2003.

[3] A. B. Owen, “Empirical Likelihood,” Chapman & Hall Ltd, London, 2001. doi:10.1201/9781420036152

[4] A. B. Owen, “Empirical Likelihood Ratio Confidence Intervals for a Single Functional,” Biometrika, Vol. 75, No. 2, 1988, pp. 237-249. doi:10.1093/biomet/75.2.237

[5] A. B. Owen, “Empirical Likelihood Ratio Confidence Regions. The Annals of Statistics,” Biometrika, Vol. 18, 1990, pp. 90-120.

[6] S. X. Chen and P. Hall, “Smoothed Empirical Likelihood Confidence Intervals for Quantiles,” The Annals of Statistics, Vol. 21, No. 3, 1993, pp. 1166-1181. doi:10.1214/aos/1176349256

[7] G. Claeskens, B.-Y. Jing, L. Peng and W. Zhou, “Empirical Likelihood Confidence Regions for Comparison Distributions and ROC Curves,” The Canadian Journal of Statistics, Vol. 31, 2003, pp. 173-190.

[8] H. Su, Y. Qin and H. Liang, “Empirical Likelihood-Based Confidence Interval of ROC Curves,” Statistics in Biopharmaceutical Research, Vol. 1, No. 4, 2009, pp. 407- 414. doi:10.1198/sbr.2009.0044

[9] H. Liang and Y. Zhou, “Semiparametirc Inference for ROC Curves with Censoring,” Scandinavian Journal of Statistics, Vol. 35, No. 2, 2008, pp. 212-227. doi:10.1111/j.1467-9469.2007.00580.x

[10] Y. S. Qin and Y. J. Qian, “Empirical Likelihood Confidence Intervals for the Differences of Quantiles with Missing Data,” Acta Mathematicae Applicatae Sinica (English Series), Vol. 25, No. 1, 2009, pp. 105-116. doi:10.1007/s10255-006-6116-0

[11] Q. Wang and J. N. K. Rao, “Empirical Likelihood-Based Inference under for Missing Response Data,” The Annals of Statistics, Vol. 30, No. 3, 2002, pp. 896-924. doi:10.1214/aos/1028674845

[12] R. J. A. Little and D. B. Rubin, “Statistical Analysis with Missing Data,” 2nd Edition, Wiley& John Sons, New York, 2002.

[1] X.-H. Zhou, D. K. McClish and N. A. Obuchowski, “Statistical Methods in Diagnostic Medicine,” Wiley, New York, 2002.

[2] M. S. Pepe, “The Statistical Evaluation of Medical Tests for Classification and Prediction,” Oxford University Press, Oxford, 2003.

[3] A. B. Owen, “Empirical Likelihood,” Chapman & Hall Ltd, London, 2001. doi:10.1201/9781420036152

[4] A. B. Owen, “Empirical Likelihood Ratio Confidence Intervals for a Single Functional,” Biometrika, Vol. 75, No. 2, 1988, pp. 237-249. doi:10.1093/biomet/75.2.237

[5] A. B. Owen, “Empirical Likelihood Ratio Confidence Regions. The Annals of Statistics,” Biometrika, Vol. 18, 1990, pp. 90-120.

[6] S. X. Chen and P. Hall, “Smoothed Empirical Likelihood Confidence Intervals for Quantiles,” The Annals of Statistics, Vol. 21, No. 3, 1993, pp. 1166-1181. doi:10.1214/aos/1176349256

[7] G. Claeskens, B.-Y. Jing, L. Peng and W. Zhou, “Empirical Likelihood Confidence Regions for Comparison Distributions and ROC Curves,” The Canadian Journal of Statistics, Vol. 31, 2003, pp. 173-190.

[8] H. Su, Y. Qin and H. Liang, “Empirical Likelihood-Based Confidence Interval of ROC Curves,” Statistics in Biopharmaceutical Research, Vol. 1, No. 4, 2009, pp. 407- 414. doi:10.1198/sbr.2009.0044

[9] H. Liang and Y. Zhou, “Semiparametirc Inference for ROC Curves with Censoring,” Scandinavian Journal of Statistics, Vol. 35, No. 2, 2008, pp. 212-227. doi:10.1111/j.1467-9469.2007.00580.x

[10] Y. S. Qin and Y. J. Qian, “Empirical Likelihood Confidence Intervals for the Differences of Quantiles with Missing Data,” Acta Mathematicae Applicatae Sinica (English Series), Vol. 25, No. 1, 2009, pp. 105-116. doi:10.1007/s10255-006-6116-0

[11] Q. Wang and J. N. K. Rao, “Empirical Likelihood-Based Inference under for Missing Response Data,” The Annals of Statistics, Vol. 30, No. 3, 2002, pp. 896-924. doi:10.1214/aos/1028674845

[12] R. J. A. Little and D. B. Rubin, “Statistical Analysis with Missing Data,” 2nd Edition, Wiley& John Sons, New York, 2002.