# zbMATH — the first resource for mathematics

On small-sample confidence intervals for parameters in discrete distributions. (English) Zbl 1209.62041
Summary: The traditional definition of a confidence interval requires the coverage probability at any value of the parameter to be at least the nominal confidence level. In constructing such intervals for parameters in discrete distributions, less conservative behavior results from inverting a single two-sided test than inverting two separate one-sided tests of half the nominal level each. We illustrate for a variety of discrete problems, including interval estimation of a binomial parameter, the difference and the ratio of two binomial parameters for independent samples, and the odds ratio.

##### MSC:
 62F25 Parametric tolerance and confidence regions
SAS; StatXact
Full Text:
##### References:
 [1] Agresti, Simple and effective confidence intervals for proportions and difference of proportions result from adding two successes and two failures, The American Statistician 54 pp 280– (2000) · Zbl 1250.62016 · doi:10.2307/2685779 [2] Anscombe, The validity of comparative experiments (with discussion), Journal of the Royal Statistical Society, Series A 111 pp 181– (1948) · doi:10.2307/2984159 [3] Baptista, Exact two-sided confidence limits for the odds ratio in a 2 $$\times$$ 2 table, Journal of the Royal Statistical Society, Series C 26 pp 214– (1977) [4] Berger, P values maximized over a confidence set for the nuisance parameter, Journal of the American Statistical Association 89 pp 1012– (1994) · Zbl 0804.62018 · doi:10.2307/2290928 [5] Blaker, Confidence curves and improved exact confidence intervals for discrete distributions, Canadian Journal of Statistics 28 pp 783– (2000) · Zbl 0966.62016 · doi:10.2307/3315916 [6] Blyth, Binomial confidence intervals, Journal of the American Statistical Association 78 pp 108– (1983) · Zbl 0503.62028 · doi:10.2307/2287116 [7] Casella, Refining binomial confidence intervals, Canadian Journal of Statistics 14 pp 113– (1986) · Zbl 0592.62029 · doi:10.2307/3314658 [8] Casella, Statistical Inference (1990) [9] Casella, Refining Poisson confidence intervals, Canadian Journal of Statistics 17 pp 45– (1989) · Zbl 0672.62046 [10] Chan, Test-based exact confidence intervals for the difference of two binomial proportions, Biometrics 55 pp 1202– (1999) · Zbl 1059.62534 · doi:10.1111/j.0006-341X.1999.01202.x [11] Clopper, The use of confidence or fiducial limits illustrated in the case of the binomial, Biometrika 26 pp 404– (1934) · JFM 60.1175.02 · doi:10.1093/biomet/26.4.404 [12] Coe , P. R. 1998 A SAS macro to calculate exact confidence intervals for the difference of two proportions Proceedings of the 23rd Annual SAS Users Group International Conference 1400 1405 [13] Coe, Small sample confidence intervals for the difference, ratio and odds ratio of two success probabilities, Communications in Statistics-Simulation and Computation 22 pp 925– (1993) · Zbl 0791.62031 · doi:10.1080/03610919308813135 [14] Cornfield, A statistical problem arising from retrospective studies, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 4 pp 135– (1956) · Zbl 0070.14707 [15] Cox, Analysis of Binary Data (1970) · Zbl 0199.53301 [16] Crow, Confidence intervals for a proportion, Biometrika 43 pp 423– (1956) · Zbl 0074.14001 · doi:10.1093/biomet/43.3-4.423 [17] Crow, Confidence intervals for the expectation of a Poisson variable, Biometrika 46 pp 441– (1959) · Zbl 0093.32202 · doi:10.1093/biomet/46.3-4.441 [18] Cytel, StatXact 4 for Windows (1999) [19] Gart, Point and interval estimation of the common odds ratio in the combination of 2 $$\times$$ 2 tables with fixed marginals, Biometrika 57 pp 471– (1970) · Zbl 0203.51302 [20] Gart, Approximate interval estimation of the ratio of binomial parameters: A review and corrections for skewness, Biometrics 44 pp 323– (1988) · Zbl 0707.62073 · doi:10.2307/2531848 [21] Garwood, Fiducial limits for the Poisson distribution, Biometrika 28 pp 437– (1936) · Zbl 0015.26206 [22] Kabaila, Exact short Poisson confidence intervals, Canadian Journal of Statistics 29 (2001) · Zbl 1013.62027 · doi:10.2307/3316053 [23] Kim, Improved exact inference about conditional association in three-way contingency tables, Journal of the American Statistical Association 90 pp 632– (1995) · Zbl 0925.62207 · doi:10.2307/2291076 [24] Koopman, Confidence intervals for the ratio of two binomial proportions, Biometrics 40 pp 513– (1984) · doi:10.2307/2531405 [25] Lee, Likelihood-weighted confidence intervals for the difference of two binomial proportions, Biometrical Journal 39 pp 387– (1997) · Zbl 1063.62533 · doi:10.1002/bimj.4710390402 [26] Mee, Confidence bounds for the difference between two probabilities (letter), Biometrics 40 pp 1175– (1984) [27] Mehta, Comparison of exact, mid-p, and Mantel-Haenszel confidence intervals for the common odds ratio across several 2 $$\times$$ 2 contingency tables, The American Statistician 46 pp 146– (1992) · doi:10.2307/2684185 [28] Mehta, Computing an exact confidence interval for the common odds ratio in several 2 by 2 contingency tables, Journal of the American Statistical Association 80 pp 969– (1985) · Zbl 0581.65100 · doi:10.2307/2288562 [29] Miettinen, Comparative analysis of two rates, Statistics in Medicine 4 pp 213– (1985) · doi:10.1002/sim.4780040211 [30] Newcombe, Two-sided confidence intervals for the single proportion: Comparison of seven methods, Statistics in Medicine 17 pp 857– (1998a) · doi:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E [31] Newcombe, Interval estimation for the difference between independent proportions: Comparison of eleven methods, Statistics in Medicine 17 pp 873– (1998b) · doi:10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I [32] Neyman, On the problem of confidence limits, Annals of Mathematical Statistics 6 pp 111– (1935) · Zbl 0012.36303 · doi:10.1214/aoms/1177732585 [33] Santner, The Statistical Analysis of Discrete Data (1989) · Zbl 0702.62005 · doi:10.1007/978-1-4612-1017-7 [34] Santner, Small-sample confidence intervals for p1-p2 and p1/p2 in 2 $$\times$$ 2 contingency tables, Journal of the American Statistical Association 75 pp 386– (1980) · Zbl 0458.62025 · doi:10.2307/2287464 [35] Santner, Invariant small sample confidence-intervals for the difference of 2 success probabilities, Communications in Statistics-Simulation and Computation 22 pp 33– (1993) · Zbl 0825.62420 · doi:10.1080/03610919308813080 [36] Soms, Exact confidence intervals, based on the Z statistic, for the difference between two proportions, Communications in Statistics-Simulation and Computation 18 pp 1325– (1989a) · Zbl 0695.62097 · doi:10.1080/03610918908812824 [37] Soms, Some recent results for exact confidence intervals for the difference between two proportions, Communications in Statistics-Simulation and Computation 18 pp 1343– (1989b) · Zbl 0695.62096 [38] Sterne, Some remarks on confidence or fiducial limits, Biometrika 41 pp 275– (1954) · Zbl 0055.12807 [39] Stevens, Fiducial limits of the parameter of a discontinuous distribution, Biometrika 37 pp 117– (1950) · Zbl 0037.36701 · doi:10.1093/biomet/37.1-2.117 [40] Suissa, Exact unconditional sample sizes for the 2 by 2 binomial trial, Journal of the Royal Statistical Society, Series A 148 pp 317– (1985) · Zbl 0585.62101 · doi:10.2307/2981892 [41] Vollset, Fast computation of exact confidence limits for the common odds ratio in a series of 2 $$\times$$ 2 tables, Journal of the American Statistical Association 86 pp 404– (1991) · doi:10.2307/2290585 [42] Walton, A note on nonrandomized Neyman-shortest unbiased confidence intervals for the binomial and Poisson parameters, Biometrika 57 pp 223– (1970) · Zbl 0193.16705 · doi:10.1093/biomet/57.1.223
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.