# 독립표본에서 두 모비율의 차이에 대한 가중 POLYA 사후분포 신뢰구간

• 이승천 (한신대학교 정보통계학과)
• Published : 2006.03.01

#### 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.

#### References

1. 이승천 (2005). 이항비율의 가중 Polya posterior 구간추정, <응용통계연구> 18, 607-615 https://doi.org/10.5351/KJAS.2005.18.3.607
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 https://doi.org/10.2307/2685469
3. 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
4. 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
5. 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
6. 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
7. 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
8. Feller, W. (1968). 'An introduction of probability theory and its applications, volumn I, Wiley, New York
9. 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
10. Ghosh, M. and Meeden, G. D. (1998) 'Bayesian methods for finite population sampling, Chapman & Hall, London
11. Mee, R. (1984). Confidence bounds for the difference between two probabilities, Biometrics, 40, 1175-1176
12. 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
13. 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
14. 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
15. 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
16. Vollet, S. E. (1993). Confidence intervals for a binomial proportion, Statistics in Medicine, 12, 809-824 https://doi.org/10.1002/sim.4780120902
17. 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
18. 정형철, 전명식, 김대학 (2003). 모비율 차이의 신뢰구간들에 대한 비교연구, <응용통계연구> 16, 377-393
19. Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion, Statistical Science, 16, 101-133
20. 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

#### 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