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.