The Annals of Statistics

On construction of the smallest one-sided confidence interval for the difference of two proportions

Weizhen Wang

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For any class of one-sided 1−α confidence intervals with a certain monotonicity ordering on the random confidence limit, the smallest interval, in the sense of the set inclusion for the difference of two proportions of two independent binomial random variables, is constructed based on a direct analysis of coverage probability function. A special ordering on the confidence limit is developed and the corresponding smallest confidence interval is derived. This interval is then applied to identify the minimum effective dose (MED) for binary data in dose-response studies, and a multiple test procedure that controls the familywise error rate at level α is obtained. A generalization of constructing the smallest one-sided confidence interval to other discrete sample spaces is discussed in the presence of nuisance parameters.

Article information

Ann. Statist., Volume 38, Number 2 (2010), 1227-1243.

First available in Project Euclid: 19 February 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions
Secondary: 62J15: Paired and multiple comparisons 62P10: Applications to biology and medical sciences

Binomial distribution coverage probability minimum effective dose multiple tests Poisson distribution set inclusion


Wang, Weizhen. On construction of the smallest one-sided confidence interval for the difference of two proportions. Ann. Statist. 38 (2010), no. 2, 1227--1243. doi:10.1214/09-AOS744.

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  • Barnard, G. A. (1947). Significance tests for 2 × 2 tables. Biometrika 34 123–138.
  • Bol’shev, L. N. (1965). On the construction of confidence limits. Theory Probab. Appl. 10 173–177 (English translation).
  • Bol’shev, L. N. and Loginov, E. A. (1966). Interval estimates in the presence of nuisance parameters. Theory Probab. Appl. 11 82–94 (English translation).
  • Bretz, F., Pinheiro, J. C. and Branson, M. (2005). Combining multiple comparisons and modeling techniques in dose-response studies. Biometrics 61 738–748.
  • Casella, G. and Berger, R. L. (1990). Statistical Inference. Duxbury Press, Belmont, CA.
  • Hsu, J. C. and Berger, R. L. (1999). Stepwise confidence intervals without multiplicity adjustment for dose-response and toxicity studies. J. Amer. Statist. Assoc. 94 468–482.
  • Marcus, R., Peritz, E. and Gabriel, K. R. (1976). On closed testing procedures with special reference to ordered analysis of variance. Biometrika 63 655–660.
  • Martin, A. A. and Silva, M. A. (1994). Choosing the optimal unconditional test for comparing two independent proportions. Comput. Statist. Data Anal. 17 555–574.
  • Tamhane, A. C. and Dunnett, C. W. (1999). Stepwise multiple test procedures with biometric applications. J. Statist. Plann. Inferences 82 55–68.
  • Tamhane, A. C., Hochberg, Y. and Dunnett, C. W. (1996). Multiple test procedures for dose finding. Biometrics 52 21–37.
  • Wang, W. (2006). Smallest confidence intervals for one binomial proportion. J. Statist. Plann. Inference 136 4293–4306.
  • Wang, W. and Peng, J. (2008). A step-up test procedure to identify the minimum effective dose. Technical report, Dept. Mathematics and Statistics, Wright State Univ.