This vignette serves an introduction to the R package ‘ExactCIone’, which is aimed at constructing the admissible exact confidence intervals (CI) for the binomial proportion, the poisson mean and the total number of subjects with a certain attribute or the total number of the subjects for the hypergeometric distribution. Both one-sided and two-sided CI are of interest. This package can be used to calculate the intervals constructed methods developed by Wang (2014) and Wang (2015).
Suppose X ∼ bino(n, p), the sample space of X is {0, 1, ..., n}. Wang (2014) proposed an admissible interval for p which is obtained by uniformly shrinking the initial 1 − α Clopper-Pearson interval from the middle to both sides of the sample space iteratively. This interval is admissible so that any proper sub-interval of it cannot assure the confidence coefficient. So the interval cannot be shortened anymore.
# Compute the 95% confidence interval when x=2, n=5.
WbinoCI(x=2,n=5,conf.level=0.95)
#> $CI
#> x lower upper
#> [1,] 2 0.0764403 0.8107447Use “details=TRUE” to show the CIs of the whole sample space.
WbinoCI(x=2,n=5,conf.level=0.95,details=TRUE)
#> $CI
#> x lower upper
#> [1,] 2 0.0764403 0.8107447
#>
#> $CIM
#> x lower upper
#> [1,] 0 0.00000000 0.5000000
#> [2,] 1 0.01020614 0.6574084
#> [3,] 2 0.07644030 0.8107447
#> [4,] 3 0.18925530 0.9235597
#> [5,] 4 0.34259163 0.9897939
#> [6,] 5 0.49999997 1.0000000
#>
#> $icp
#> [1] 0.95The one-sided intervals are the one-sided 1 − α Clopper-Pearson intervals (Clopper and Pearson, 1934). Also show all the CIs when “details=TRUE”.
WbinoCI_lower(x=2,n=5,conf.level=0.95)
#> $CI
#> sample lower upper
#> [1,] 2 0.07644039 1
WbinoCI_lower(x=2,n=5,conf.level=0.95,details=TRUE)
#> $CI
#> sample lower upper
#> [1,] 0 0.00000000 1
#> [2,] 1 0.01020622 1
#> [3,] 2 0.07644039 1
#> [4,] 3 0.18925538 1
#> [5,] 4 0.34259168 1
#> [6,] 5 0.54928027 1
WbinoCI_upper(x=2,n=5,conf.level=0.95)
#> $CI
#> sample lower upper
#> [1,] 2 0 0.8107446
WbinoCI_upper(x=2,n=5,conf.level=0.95,details=TRUE)
#> $CI
#> sample lower upper
#> [1,] 0 0 0.4507197
#> [2,] 1 0 0.6574083
#> [3,] 2 0 0.8107446
#> [4,] 3 0 0.9235596
#> [5,] 4 0 0.9897938
#> [6,] 5 0 1.0000000Suppose X ∼ poi(λ), the sample space of X is {0, 1, ...}. Wang (2014) proposed an admissible interval for λ which is obtained by uniformly shrinking the initial 1 − α Clopper-Pearson interval one by one from 0 to the sample point of interest. This interval is admissible so that any proper sub-interval of it cannot assure the confidence coefficient, which means the interval cannot be shortened anymore.
# The admissible CI for poisson mean when the observed sample is x=3.
WpoisCI(x=3,conf.level = 0.95)
#> $CI
#> x lower upper
#> [1,] 3 0.8176914 8.395386
#We show the intervals from 0 to the sample of interest when "details=TRUE".
WpoisCI(x=3,conf.level = 0.95,details = TRUE)
#> $CI
#> x lower upper
#> [1,] 3 0.8176914 8.395386
#>
#> $CIM
#> x lower upper
#> [1,] 0 0.00000000 3.453832
#> [2,] 1 0.05129329 5.491160
#> [3,] 2 0.35536150 6.921952
#> [4,] 3 0.81769144 8.395386
#>
#> $icp
#> [1] 0.95The one-sided intervals are the one-sided 1 − α Clopper-Pearson intervals which is givend by Garwood (1936). Also shows all the CIs when “details=TRUE”.
WpoisCI_lower(x=3,conf.level = 0.95)
#> $CI
#> sample
#> [1,] 3 0.8176914 Inf
WpoisCI_lower(x=3,conf.level = 0.95,details = TRUE)
#> $CI
#> x lower upper
#> [1,] 3 0.8176914 Inf
#>
#> $CIM
#> sample lower upper
#> [1,] 0 0.00000000 Inf
#> [2,] 1 0.05129329 Inf
#> [3,] 2 0.35536151 Inf
#> [4,] 3 0.81769145 Inf
WpoisCI_upper(x=3,conf.level = 0.95)
#> $CI
#> sample
#> [1,] 3 0 7.753657
WpoisCI_upper(x=3,conf.level = 0.95,details = TRUE)
#> $CI
#> x lower upper
#> [1,] 3 0 7.753657
#>
#> $CIM
#> sample lower upper
#> [1,] 0 0 2.995732
#> [2,] 1 0 4.743865
#> [3,] 2 0 6.295794
#> [4,] 3 0 7.753657Suppose X ∼ Hyper(M, N, n). The sample space is {0, …, min (M, n)}. When M and n are known, Wang (2015) construct an admissible confidence interval for N by uniformly shrinking the initial 1 − α Clopper-Pearson type interval from 0 to min (M, n). Also this interval cannot be shortened more.
# For hyper(M,N,n), construct 95% CI for N on the observed sample x when n,M are known.
WhyperCI_N(x=5,n=10,M=800,conf.level = 0.95)
#> $CI
#> x lower upper
#> [1,] 5 1031 3591
# It shows CIs for all the sample point When "details=TRUE".
WhyperCI_N(x=5,n=10,M=800,conf.level = 0.95,details=TRUE)
#> $CI
#> x lower upper
#> [1,] 5 1031 3591
#>
#> $CIM
#> x lower upper
#> [1,] 0 3003 Inf
#> [2,] 1 1837 156370
#> [3,] 2 1459 21746
#> [4,] 3 1295 9160
#> [5,] 4 1151 5326
#> [6,] 5 1031 3591
#> [7,] 6 943 3002
#> [8,] 7 878 2096
#> [9,] 8 831 1779
#> [10,] 9 805 1411
#> [11,] 10 800 1150
#>
#> $icp
#> [1] 0.9500001The one-sided 1 − α CI for N is the one-sided Clopper-Pearson type interval (Konijn, 1973).
WhyperCI_N_lower(x=0,n=10,M=800,conf.level = 0.95)
#> $CI
#> x
#> [1,] 0 3095 Inf
WhyperCI_N_lower(x=0,n=10,M=800,conf.level = 0.95,details=TRUE)
#> $CI
#> sample lower upper
#> [1,] 0 3095 Inf
#>
#> $CIM
#> sample lower upper
#> [1,] 0 3095 Inf
#> [2,] 1 2033 Inf
#> [3,] 2 1581 Inf
#> [4,] 3 1321 Inf
#> [5,] 4 1151 Inf
#> [6,] 5 1031 Inf
#> [7,] 6 943 Inf
#> [8,] 7 878 Inf
#> [9,] 8 831 Inf
#> [10,] 9 805 Inf
#> [11,] 10 800 Inf
WhyperCI_N_upper(x=0,n=10,M=800,conf.level = 0.95)
#> $CI
#> x
#> [1,] 0 0 Inf
WhyperCI_N_upper(x=0,n=10,M=800,conf.level = 0.95,details=TRUE)
#> $CI
#> sample lower upper
#> [1,] 0 0 Inf
#>
#> $CIM
#> sample lower upper
#> [1,] 0 0 Inf
#> [2,] 1 0 156370
#> [3,] 2 0 21746
#> [4,] 3 0 9160
#> [5,] 4 0 5326
#> [6,] 5 0 3591
#> [7,] 6 0 2631
#> [8,] 7 0 2030
#> [9,] 8 0 1619
#> [10,] 9 0 1318
#> [11,] 10 0 1077Suppose X ∼ Hyper(M, N, n). When N and n are known, Wang (2015) construct an admissible confidence interval for N by uniformly shrinking the initial 1 − α Clopper-Pearson type interval from the mid-point of the sample space to 0. This interval is admissible so that any proper sub-interval of it cannot assure the confidence coefficient. This means the interval cannot be shortened anymore.
# For Hyper(M,N,n), construct the CI for M on the observed sample x when n, N are known.
# Also output CI for p=M/N.
WhyperCI_M(x=0,n=10,N=2000,conf.level = 0.95)
#> $CI
#> x lower upper
#> [1,] 0 0 608
#>
#> $CI_p
#> p lower upper
#> [1,] 0 0 0.304
WhyperCI_M(x=0,n=10,N=2000,conf.level = 0.95,details = TRUE)
#> $CI
#> X lower upper
#> [1,] 0 0 608
#>
#> $CIM
#> x lower upper
#> [1,] 0 0 608
#> [2,] 1 11 873
#> [3,] 2 74 1102
#> [4,] 3 176 1236
#> [5,] 4 301 1391
#> [6,] 5 446 1554
#> [7,] 6 609 1699
#> [8,] 7 764 1824
#> [9,] 8 898 1926
#> [10,] 9 1127 1989
#> [11,] 10 1392 2000
#>
#> $CIM_p
#> p lower_p upper_p
#> [1,] 0.0 0.0000 0.3040
#> [2,] 0.1 0.0055 0.4365
#> [3,] 0.2 0.0370 0.5510
#> [4,] 0.3 0.0880 0.6180
#> [5,] 0.4 0.1505 0.6955
#> [6,] 0.5 0.2230 0.7770
#> [7,] 0.6 0.3045 0.8495
#> [8,] 0.7 0.3820 0.9120
#> [9,] 0.8 0.4490 0.9630
#> [10,] 0.9 0.5635 0.9945
#> [11,] 1.0 0.6960 1.0000
#>
#> $icp
#> [1] 0.9500005The one-sided 1 − α CI for M is the one-sided Clopper-Pearson type interval (Konijn, 1973).
WhyperCI_M_lower(X=0,n=10,N=2000,conf.level = 0.95)
#> $CI
#> X N
#> [1,] 0 0 2000
WhyperCI_M_lower(X=0,n=10,N=2000,conf.level = 0.95,details = TRUE)
#> $CI
#> X N
#> [1,] 0 0 2000
#>
#> $CIM
#> sample lower upper
#> [1,] 0 0 2000
#> [2,] 1 11 2000
#> [3,] 2 74 2000
#> [4,] 3 176 2000
#> [5,] 4 301 2000
#> [6,] 5 446 2000
#> [7,] 6 609 2000
#> [8,] 7 788 2000
#> [9,] 8 988 2000
#> [10,] 9 1213 2000
#> [11,] 10 1484 2000
WhyperCI_M_upper(X=0,n=10,N=2000,conf.level = 0.95)
#> $CI
#> X
#> [1,] 0 0 516
WhyperCI_M_upper(X=0,n=10,N=2000,conf.level = 0.95,details = TRUE)
#> $CI
#> X
#> [1,] 0 0 516
#>
#> $CIM
#> sample lower upper
#> [1,] 0 0 516
#> [2,] 1 0 787
#> [3,] 2 0 1012
#> [4,] 3 0 1212
#> [5,] 4 0 1391
#> [6,] 5 0 1554
#> [7,] 6 0 1699
#> [8,] 7 0 1824
#> [9,] 8 0 1926
#> [10,] 9 0 1989
#> [11,] 10 0 2000Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial limits in the case of the binomial. “Biometrika” 26: 404-413.
Garwood, F. (1936). Fiducial Limits for the Poisson Distribution. “Biometrika” 28: 437-442.
Konijn, H. S. (1973). Statistical Theory of Sample Survey Design and Analysis, Amsterdam: North-Holland.
Wang, W. (2014). An iterative construction of confidence intervals for a proportion. “Statistica Sinica” 24: 1389-1410.
Wang, W. (2015). Exact Optimal Confidence Intervals for Hypergeometric Parameters. “Journal of the American Statistical Association” 110 (512): 1491-1499.