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ExactCIone

Introduction

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).

library(ExactCIone)

Admissible exact CI for binomial proportion \(p\)

Suppose \(X\sim bino(n,p)\), 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 \(1-\alpha\) 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 0.8107447

Use “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] 0.95

The one-sided intervals are the one-sided \(1-\alpha\) 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 1.0000000

Admissible exact CI for the poisson mean \(\lambda\)

Suppose \(X\sim poi(\lambda)\), the sample space of \(X\) is \(\{0,1,...\}\). Wang (2014) proposed an admissible interval for \(\lambda\) which is obtained by uniformly shrinking the initial \(1-\alpha\) 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] 0.95

The one-sided intervals are the one-sided \(1-\alpha\) 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 7.753657

Admissible exact confidence intervals for N, the number of balls in an urn.

Suppose \(X\sim Hyper(M,N,n)\). The sample space is \(\{0,\ldots,\min(M,n)\}\). When \(M\) and \(n\) are known, Wang (2015) construct an admissible confidence interval for \(N\) by uniformly shrinking the initial \(1-\alpha\) 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] 0.9500001

The one-sided \(1-\alpha\) 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   1077

Admissible exact CI for \(M\), the number of white balls in an urn

Suppose \(X\sim Hyper(M,N,n)\). When N and n are known, Wang (2015) construct an admissible confidence interval for N by uniformly shrinking the initial \(1-\alpha\) 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] 0.9500005

The one-sided \(1-\alpha\) 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  2000

Reference

Clopper, 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.

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