The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

The Weighted Polya Posterior Confidence Interval For the Difference Between Two Independent Proportions

독립표본에서 두 모비율의 차이에 대한 가중 POLYA 사후분포 신뢰구간

  • 이승천 (한신대학교 정보통계학과)
  • Published : 2006.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

The Wald confidence interval has been considered as a standard method for the difference of proportions. However, the erratic behavior of the coverage probability of the Wald confidence interval is recognized in various literatures. Various alternatives have been proposed. Among them, Agresti-Caffo confidence interval has gained the reputation because of its simplicity and fairly good performance in terms of coverage probability. It is known however, that the Agresti-Caffo confidence interval is conservative. In this note, a confidence interval is developed using the weighted Polya posterior which was employed to obtain a confidence interval for the binomial proportion in Lee(2005). The resulting confidence interval is simple and effective in various respects such as the closeness of the average coverage probability to the nominal confidence level, the average expected length and the mean absolute error of the coverage probability. Practically it can be used for the interval estimation of the difference of proportions for any sample sizes and parameter values.

모비율 차이의 구간 추정에서 표준으로 인식되고 있는 Wald 신뢰구간은 모비율 구간 추정과 마찬가지로 포함확률의 근사성에서 문제가 있다는 것이 알려져 있다. 이에 대한 대안으로 모비율 차이의 신뢰구간에 대한 많은 연구가 있어 왔으나 대부분의 신뢰구간은 매우 복잡한 과정을 통해 얻어지게 되어 있어 실용성에 대한 문제가 제기될 수 있다. 이와 비교하여 Agresti와 Caffo(2000)에 의해 제시된 신뢰구간은 매우 간편한 식에 의해 구할 수 있어 이해하기 쉽고 포함확률과 포함확률의 평균절대오차에 있어 다른 복잡한 신뢰 구간과 필적할 수 있다. 그러나 Agresti-Caffo 신뢰 구간은 포함확률이 명목 신뢰수준을 상회하는 보수적인 구간으로 알려져 있다. 본 논문에서는 이승천(2005)에서 이항비율의 신뢰구간을 구하기 위해 사용된 가중 Polya 사후분포를 이용하여 두 모비율 차이의 신뢰구간을 구하였다. 이렇게 구하여진 신뢰구간은 간편성은 물론 Agresti-Caffo 신뢰구간의 보수성을 개선하였다.

Keywords

References

  1. 이승천 (2005). 이항비율의 가중 Polya posterior 구간추정, <응용통계연구> 18, 607-615 https://doi.org/10.5351/KJAS.2005.18.3.607
  2. 정형철, 전명식, 김대학 (2003). 모비율 차이의 신뢰구간들에 대한 비교연구, <응용통계연구> 16, 377-393
  3. Agresti, A. and Coull, B. A. (1998). Approximation is better than 'exact' for interval estimation of binomial proportions, The American Statistician, 52, 119-126 https://doi.org/10.2307/2685469
  4. 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 https://doi.org/10.2307/2685779
  5. Anbar, D. (1983). On estimating the difference between two probabilities with special reference to clinical trials, Biometrics, 39, 257-262 https://doi.org/10.2307/2530826
  6. Beal, S. L. (1987). Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples, Biometrics, 43, 941-950 https://doi.org/10.2307/2531547
  7. Blyth, C. R. and Still, H. A. (1983) Binomial confidence intervals, Journal of the American Statistical Association, 78, 108-116 https://doi.org/10.2307/2287116
  8. Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion, Statistical Science, 16, 101-133
  9. Brown, L. D., Cai, T. T. and DasGupta, A. (2002). Confidence intervals for a binomial proportion and asymptotic expansions, The Annals of Statistics, 30. 160-201 https://doi.org/10.1214/aos/1015362189
  10. Chan, I. S. F. and Zhang, Z. (1999). Test-based exact confidence intervals for the difference of two binomial proportions, Biometrics, 55. 1202-1209 https://doi.org/10.1111/j.0006-341X.1999.01202.x
  11. Feller, W. (1968). 'An introduction of probability theory and its applications, volumn I, Wiley, New York
  12. Ghosh, B. K. (1979). A comparison of some approximate confidence intervals for the binomial parameter, Journal of the American Statistical Association, 74, 894-900 https://doi.org/10.2307/2286420
  13. Ghosh, M. and Meeden, G. D. (1998) 'Bayesian methods for finite population sampling, Chapman & Hall, London
  14. Mee, R. (1984). Confidence bounds for the difference between two probabilities, Biometrics, 40, 1175-1176
  15. Meeden, G. D. (1999). Interval estimators for the population mean for skewed distributions with a small sample size, Journal of Applied Statistics, 26, 81-96 https://doi.org/10.1080/02664769922674
  16. Newcombe, R. (1998a) Two-sided confidence intervals for the single proportion: Comparison of seven methods, Statistics in Medicine, 17, 857-872 https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E
  17. Newcombe, R. (1998a) Interval estimation for the difference between independent proportions: Comparison of eleven methods, Statistics in Medicine, 17, 873-890 https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I
  18. Santner, T. J. and Snell, M. K. (1980). Small-sample confidence intervals for $P_{l}\;-\;P_{2}\;and\;P_{1}/P_{2}\;in\;2\;{\time}\;2$ contingency tables, Statistics in Medicine, 17, 873-890 https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I
  19. Vollet, S. E. (1993). Confidence intervals for a binomial proportion, Statistics in Medicine, 12, 809-824 https://doi.org/10.1002/sim.4780120902
  20. Wilson, E. B. (1927). Probable inference, the law of succession and statistical inference, Journal of the American Statistical Association, 22. 209-212 https://doi.org/10.2307/2276774

Cited by

  1. A Bayesian approach to obtain confidence intervals for binomial proportion in a double sampling scheme subject to false-positive misclassification vol.37, pp.4, 2008, https://doi.org/10.1016/j.jkss.2008.05.001
  2. The Role of Artificial Observations in Misclassified Binary Data with Common False-Positive Error vol.25, pp.4, 2012, https://doi.org/10.5351/KJAS.2012.25.4.697