Electronic Journal of Statistics

The cost of using exact confidence intervals for a binomial proportion

Måns Thulin

Full-text: Open access

Enhanced PDF (332 KB)

Abstract

When computing a confidence interval for a binomial proportion $p$ one must choose between using an exact interval, which has a coverage probability of at least $1-\alpha$ for all values of $p$, and a shorter approximate interval, which may have lower coverage for some $p$ but that on average has coverage equal to $1-\alpha$. We investigate the cost of using the exact one and two-sided Clopper–Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper–Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper–Pearson interval and Bayesian intervals based on noninformative priors.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 817-840.

Dates
First available in Project Euclid: 16 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1402927499

Digital Object Identifier
doi:10.1214/14-EJS909

Mathematical Reviews number (MathSciNet)
MR3217790

Zentralblatt MATH identifier
1348.62092

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Asymptotic expansion binomial distribution confidence interval expected length sample size determination proportion

Citation

Thulin, Måns. The cost of using exact confidence intervals for a binomial proportion. Electron. J. Statist. 8 (2014), no. 1, 817--840. doi:10.1214/14-EJS909. https://projecteuclid.org/euclid.ejs/1402927499


Export citation

References

  • [1] Abramson, J.S., Takvorian, R.W., Fisher, D.C., Feng, Y., Jacobsen, E.D., et al. (2013). Oral clofarabine for relapsed/refractory non-Hodgkin lymphomas: results of a phase 1 study. Leukemia & Lymphoma, 54, 1915–1920.
  • [2] Agresti, A., Coull, B.A. (1998). Approximate is better than “exact” for interval estimation of a binomial proportion. The American Statistician, 52, 119–126.
  • [3] Blaker, H. (2000). Confidence curves and improved exact confidence intervals for discrete distributions. The Canadian Journal of Statistics, 28, 783–798.
  • [4] Blyth, C.R., Still, H.A. (1983). Binomial confidence intervals. Journal of the American Statistical Association, 78 108–116.
  • [5] Brown, L.D., Cai, T.T., DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16, 101–133.
  • [6] Brown, L.D., Cai, T.T., DasGupta, A. (2002). Confidence intervals for a binomial proportion and asymptotic expansions. The Annals of Statistics, 30, 160–201.
  • [7] Cai, T.T. (2005). One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference, 131, 63–88.
  • [8] Casella, G. (1986). Refining binomial confidence intervals. The Canadian Journal of Statistics, 14, 113–129.
  • [9] Clopper, C.J., Pearson, E.S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26, 404–413.
  • [10] Cramér, H. (1946). Mathematical Methods of Statistics, Princeton University Press.
  • [11] Crow, E.L. (1956). Confidence intervals for a proportion. Biometrika, 43, 423–435.
  • [12] Gonçalves, L., de Oliviera, M.R., Pascoal, C., Pires, A. (2012). Sample size for estimating a binomial proportion: comparison of different methods. Journal of Applied Statistics, 39, 2453–2473.
  • [13] Ibrahim, T., Farolfi, A., Scarpi, E., Mercatali, L., Medri, L., et al. (2013). Hormonal receptor, human epidermal growth factor receptor-2, and Ki67 discordance between primary breast cancer and paired metastases: clinical impact. Oncology, 84, 150–157.
  • [14] Johnson, N.L., Kemp, A.W., Kotz, S. (2005). Univariate Discrete Distributions, 3rd edition, Wiley.
  • [15] Katsis, A. (2001). Calculating the optimal sample size for binomial populations. Communications in Statistics – Theory and Methods, 30, 665–678.
  • [16] Klaschka, J. (2010). On calculation of Blaker’s binomial confidence limits. COMPSTAT’10.
  • [17] Krishnamoorthy, K., Peng, J. (2007). Some properties of the exact and score methods for binomial proportion and sample size calculation. Communications in Statistics – Simulation and Computation, 36, 1171–1186.
  • [18] M’Lan, C.E., Joseph, L., Wolfson, D.B. (2008). Bayesian sample size determination for binomial proportions. Bayesian Analysis, 3, 269–296.
  • [19] Newcombe, R.G. (2011). Measures of location for confidence intervals for proportions. Communications in Statistics – Theory and Methods, 40, 1743–1767.
  • [20] Newcombe, R.G. (2012). Confidence Intervals for Proportions and Related Measures of Effect Size. Chapman & Hall.
  • [21] Newcombe, R.G., Nurminen, M.M. (2011). In defence of score intervals for proportions and their differences. Communications in Statistics – Theory and Methods, 40, 1271–1282.
  • [22] Piegorsch, W.W. (2004). Sample sizes for improved binomial confidence intervals. Computational Statistics & Data Analysis, 46, 309–316.
  • [23] Reiczigel, J. (2003). Confidence intervals for the binomial parameter: some new considerations. Statistics in Medicine, 22, 611–621.
  • [24] Staicu, A.-M. (2009). Higher-order approximations for interval estimation in binomial settings. Journal of Statistical Planning and Inference, 139, 3393–3404.
  • [25] Sterne, T.H. (1954). Some remarks on confidence or fiducial limits. Biometrika, 41, 275–278.
  • [26] Sullivan, A.K., Raben, D., Reekie, J., Rayment M., Mocroft, A., et al. (2013). Feasibility and effectiveness of indicator condition-guided testing for HIV: results from HIDES I (HIV Indicator Diseases across Europe Study). PLoS ONE, 8, e52845.
  • [27] Thulin, M. (2014). Coverage-adjusted confidence intervals for a binomial proportion. Scandinavian Journal of Statistics, 41, 291–300.
  • [28] Thulin, M. (2014). On split sample and randomized confidence intervals for binomial proportions. Statistics and Probability Letters, 92, 65–71.
  • [29] Vos, P.W., Hudson, S. (2005). Evaluation criteria for discrete confidence intervals. The American Statistician, 59, 137–142.
  • [30] Vos, P.W., Hudson, S. (2008). Problems with binomial two-sided tests and the associated confidence intervals. Australian & New Zealand Journal of Statistics, 50, 81–89.
  • [31] Ward, L.G., Heckman, M.G., Warren, A.I., Tran, K. (2013). Dosing accuracy of insulin aspart FlexPens after transport through the pneumatic tube system. Hospital Pharmacy, 48, 33–38.
  • [32] Wei, L., Hutson, A.D. (2013). A comment on sample size calculations for binomial confidence intervals. Journal of Applied Statistics, 40, 311–319.
  • [33] Wilson, E.B. (1927). Probable inference, the law of succession and statistical inference. Journal of the American Statistical Association, 22, 209–212.

The Institute of Mathematical Statistics and the Bernoulli Society

Electronic Journal of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more