# ciEX: Exact method of CI estimation

### Description

Exact method of CI estimation

### Usage

 `1` ```ciEX(n, alp, e) ```

### Arguments

 `n` - Number of trials `alp` - Alpha value (significance level required) `e` - Exact method indicator in [0, 1] 1: Clopper Pearson, 0.5: Mid P, The input can also be a range of values between 0 and 1.

### Details

Confidence interval for `p` (for all `x` = 0, 1, 2 ..`n`), based on inverting equal-tailed binomial tests with null hypothesis

H0: p = p0

and calculated from the cumulative binomial distribution. Exact two sided P-value is usually calculated as

P= 2[e*Pr(X = x) + min{(Pr(X < x), Pr(X > x))}]

where probabilities are found at null value of p and 0 <= e <= 1. The Confidence Interval of `n` given `alp` along with lower and upper abberation.

### Value

A dataframe with

 `x ` - Number of successes (positive samples) `LEX ` - Exact Lower limit `UEX ` - Exact Upper Limit `LABB ` - Likelihood Ratio Lower Abberation `UABB ` - Likelihood Ratio Upper Abberation `ZWI ` - Zero Width Interval `e ` - Exact method input

### References

[1] 1993 Vollset SE. Confidence intervals for a binomial proportion. Statistics in Medicine: 12; 809 - 824.

[2] 1998 Agresti A and Coull BA. Approximate is better than "Exact" for interval estimation of binomial proportions. The American Statistician: 52; 119 - 126.

[3] 1998 Newcombe RG. Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine: 17; 857 - 872.

[4] 2001 Brown LD, Cai TT and DasGupta A. Interval estimation for a binomial proportion. Statistical Science: 16; 101 - 133.

[5] 2002 Pan W. Approximate confidence intervals for one proportion and difference of two proportions Computational Statistics and Data Analysis 40, 128, 143-157.

[6] 2008 Pires, A.M., Amado, C. Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods. REVSTAT - Statistical Journal, 6, 165-197.

[7] 2014 Martin Andres, A. and Alvarez Hernandez, M. Two-tailed asymptotic inferences for a proportion. Journal of Applied Statistics, 41, 7, 1516-1529

`prop.test and binom.test` for equivalent base Stats R functionality, `binom.confint` provides similar functionality for 11 methods, `wald2ci` which provides multiple functions for CI calculation , `binom.blaker.limits` which calculates Blaker CI which is not covered here and `propCI` which provides similar functionality.

Other Basic methods of CI estimation: `PlotciAS`, `PlotciAllg`, `PlotciAll`, `PlotciBA`, `PlotciEX`, `PlotciLR`, `PlotciLT`, `PlotciSC`, `PlotciTW`, `PlotciWD`, `ciAS`, `ciAll`, `ciBA`, `ciLR`, `ciLT`, `ciSC`, `ciTW`, `ciWD`

### Examples

 ```1 2 3 4 5 6``` ```n=5; alp=0.05;e=0.5 ciEX(n,alp,e) #Mid-p n=5; alp=0.05;e=1 #Clopper-Pearson ciEX(n,alp,e) n=5; alp=0.05;e=c(0.1,0.5,0.95,1) #Range including Mid-p and Clopper-Pearson ciEX(n,alp,e) ```

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