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The cost of using exact confidence intervals for a binomial proportion. (English) Zbl 1348.62092
Summary: 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.

MSC:
62F25 Parametric tolerance and confidence regions
62F12 Asymptotic properties of parametric estimators
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[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. · Zbl 0966.62016 · doi:10.2307/3315916
[4] Blyth, C.R., Still, H.A. (1983). Binomial confidence intervals., Journal of the American Statistical Association , 78 108-116. · Zbl 0503.62028 · doi:10.2307/2287116
[5] Brown, L.D., Cai, T.T., DasGupta, A. (2001). Interval estimation for a binomial proportion., Statistical Science , 16 , 101-133. · Zbl 1059.62533 · doi:10.1214/ss/1009213286
[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. · Zbl 1012.62026 · doi:10.1214/aos/1015362189
[7] Cai, T.T. (2005). One-sided confidence intervals in discrete distributions., Journal of Statistical Planning and Inference , 131 , 63-88. · Zbl 1062.62053 · doi:10.1016/j.jspi.2004.01.005
[8] Casella, G. (1986). Refining binomial confidence intervals., The Canadian Journal of Statistics , 14 , 113-129. · Zbl 0592.62029 · doi:10.2307/3314658
[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. · JFM 60.1175.02
[10] Cramér, H. (1946)., Mathematical Methods of Statistics , Princeton University Press. · Zbl 0063.01014
[11] Crow, E.L. (1956). Confidence intervals for a proportion., Biometrika , 43 , 423-435. · Zbl 0074.14001 · doi:10.1093/biomet/43.3-4.423
[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. · doi:10.1080/02664763.2012.713919
[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. · Zbl 1009.62524 · doi:10.1081/STA-100002143
[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. · Zbl 1126.62011 · doi:10.1080/03610910701569218
[18] M’Lan, C.E., Joseph, L., Wolfson, D.B. (2008). Bayesian sample size determination for binomial proportions., Bayesian Analysis , 3 , 269-296. · Zbl 1330.62064 · doi:10.1214/08-BA310
[19] Newcombe, R.G. (2011). Measures of location for confidence intervals for proportions., Communications in Statistics - Theory and Methods , 40 , 1743-1767. · Zbl 1277.62089 · doi:10.1080/03610921003646406
[20] Newcombe, R.G. (2012)., Confidence Intervals for Proportions and Related Measures of Effect Size . Chapman & Hall. · Zbl 1263.62089
[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. · Zbl 1270.62063 · doi:10.1080/03610920903576580
[22] Piegorsch, W.W. (2004). Sample sizes for improved binomial confidence intervals., Computational Statistics & Data Analysis , 46 , 309-316. · Zbl 1429.62101
[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. · Zbl 1171.62021 · doi:10.1016/j.jspi.2009.03.021
[25] Sterne, T.H. (1954). Some remarks on confidence or fiducial limits., Biometrika , 41 , 275-278. · Zbl 0055.12807
[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. · Zbl 06298502
[28] Thulin, M. (2014). On split sample and randomized confidence intervals for binomial proportions., Statistics and Probability Letters , 92 , 65-71. · Zbl 1396.62048
[29] Vos, P.W., Hudson, S. (2005). Evaluation criteria for discrete confidence intervals., The American Statistician , 59 , 137-142. · Zbl 05680633 · doi:10.1198/000313005X42453
[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. · doi:10.1080/02664763.2012.740629
[33] Wilson, E.B. (1927). Probable inference, the law of succession and statistical inference., Journal of the American Statistical Association , 22 , 209-212.
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