Confidence Intervals for the Difference of Binomial Proportions in Two Doubly Sampled Data

Lee, Seung-Chun

  • Received : 20100300
  • Accepted : 20100400
  • Published : 2010.05.31


The construction of asymptotic confidence intervals is considered for the difference of binomial proportions in two doubly sampled data subject to false-positive error. The coverage behaviors of several likelihood based confidence intervals and a Bayesian confidence interval are examined. It is shown that a hierarchical Bayesian approach gives a confidence interval with good frequentist properties. Confidence interval based on the Rao score is also shown to have good performance in terms of coverage probability. However, the Wald confidence interval covers true value less often than nominal level.


Profile likelihood;Rao score;hierarchical Bayesian approach;coverage probability;expected width;double sampling


  1. Agresti, A. and Caffo, B. (2000). Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures, The American Statistician, 54, 280-288.
  2. Agresti, A. and Coull, B. A. (1998). Approximation is better than “exact” for interval estimation of binomial proportions, The American Statistician, 52, 119-126.
  3. Agresti, A. and Min, Y. (2005). Simple improved confidence intervals for comparing matched proportions, Statistics in Medicine, 24, 729-740.
  4. Barndorff-Nielsen, O. E. and Cox, D. R. (1994). Inference and asymptotics, Chapman & Hall, London.
  5. Barnett, V., Haworth, J. and Smith, T. M. F. (2001). A two-phase sampling scheme with applications to auditing or sed quis custodiet ipsos custodes? Journal of Royal Statistical Society, Serie A, 164, 407-422.
  6. Blyth, C. R. and Still, H. A. (1983). Binomial confidence intervals, Journal of the American Statistical Association, 78, 108-116.
  7. Boese, D. H., Young, D. M. and Stamey, J. D. (2006). Confidence intervals for a binomial parameter based on binary data subject to false-positive misclassification, Computational Statistics and Data Analysis, 50, 3369-3385.
  8. Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion, Statistical Science, 16, 101-133.
  9. Efron, B. and Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information, Biometrika, 65, 457-482.
  10. Geng, Z. and Asano, C. (1989). Bayesian estimation methods for categorical data with misclassifications, Communications in Statistics, Theory and Methods, 18, 2935-2954.
  11. Hildesheim, A., Mann, V., Brinton, L. A., Szklo, M., Reeves, W. C. and Rawls, W. E. (1991). Herpes simplex virus type 2: A possible interaction with human papillomavirus types 16/18 in the development of invasion cervical cancer, International Journal of Cancer, 49, 335-340.
  12. Lee, S.-C. (2007). An improved confidence interval for the population proportion in a double sampling scheme subject to false-positive misclassification, Journal of the Korean Statistical Society, 36, 275-284.
  13. Lee, S.-C. and Byun, J.-S. (2008). A Bayesian approach to obtain confidence intervals for binomial proportion in a double sampling scheme subject to false-positive misclassification, Journal of the Korean Statistical Society, 37, 393-403.
  14. Lie, R. T., Heuch, I. and Irgens, L. M. (1994). Maximum likelihood estimation of proportion of congenital malformations using double registration systems, Biometrics, c, 433-444.
  15. Moors, J. J. A., van der Genugten, B. B. and Strijbosch, L. W. G. (2000). Repeated audit controls, Statistica Neerlandica, 54, 3-13.
  16. Price, R. M. and Bonett, D. G. (2004). An improved confidence interval for a linear function of binomial proportions, Computational Statistics and Data Analysis, 45, 449-456.
  17. Raats, V. M. and Moors, J. J. A. (2003). Double-checking auditors: A Bayesian approach, The Statistician. 52, 351-365.
  18. Tenenbein, A. (1970). A double sampling scheme for estimating from binomial data with misclassifications, Journal of the American Statistical Association, 65, 1350-1361.
  19. York, J., Madigan, D., Heuch, I. and Lie, R. T. (1995). Birth defects registered by double sampling: A Bayesian approach incorporating covariates and model uncertainty, Applied Statistics, 44, 227-242.

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