# binCI: Confidence Intervals for One Binomial Proportion In binGroup: Evaluation and Experimental Design for Binomial Group Testing

## Description

Calculates the exact Clopper-Pearson and Blaker, the asymptotic second-order corrected, Wilson, Agresti-Coull and Wald confidence interval for a single binomial proportion

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```binCI(n, y, conf.level = 0.95, alternative = "two.sided", method = "CP") binCP(n, y, conf.level=0.95, alternative="two.sided") binBlaker(n,y,conf.level=0.95, tolerance=1e-04, alternative="two.sided") binAC(n, y, conf.level=0.95, alternative="two.sided") binSOC(n, y,conf.level=0.95,alternative="two.sided") binWald(n, y, conf.level=0.95, alternative="two.sided") binWilson(n, y,conf.level=0.95,alternative="two.sided") ```

## Arguments

 `n` number of trials (number of individuals under observation) `y` number of successes (number of individuals showing the trait of interest) `conf.level` nominal confidence level `alternative` character string defining the alternative hypothesis, either 'two.sided', 'less' or 'greater' where 'less' gives the only an upper bound with confidence level=conf.level 'greater' gives the only a lower bound with confidence level=conf.level and 'two.sided' gives a two-sided confidence interval with confidence level=conf.level `method` character string defining the method for CI calculation: where "CP" is Clopper-Pearson, an exact tail interval showing symmetric coverage probability (inversion of two one-sided tests), "Blaker" is the Blaker interval, an exact interval, inversion of one two.sided test, therefore defined only two.sided, but shorter than the two-sided Clopper-Pearson CI. Both guarantee to contain the true parameter with at least conf.level*100 percent probability, "AC" is Agresti-Coull, generalized Agresti-Coull interval, asymptotic method, "Score" is Wilson Score, asymptotic method derived from inversion of the Score test, "SOC" is the second order corrected interval, asymptotic method for one-sided problems (for details see Cai, 2005), and "Wald" the Wald interval, which cannot be recommended. `tolerance` precision of computation for the bounds of the Blaker interval

## Details

This function allows computation of confidence intervals for a binomial proportion from a standard binomial experiment. If an actual confidence level greater or equal to that specified in the conf.level argument shall always be guaranteed, the exact method of Clopper-Pearson (method="CP") can be recommended for one-sided and the improved method of Blaker (method="Blaker") can be recommended for two-sided hypotheses. If a mean confidence level close to that specified in the argument conf.level is required, but moderate violation of this level is acceptable, the Second-Order corrected (method="SOC"), Wilson Score (method="Wilson") or Agresti-Coull (method="AC") might be used, where SOC has the most symmetric coverage and Wilson and Agresti-Coull are in tendency conservative for the upper bound and proportions close to 0 and for the lower bound and proportions close to 1. The Wald CI might be used for large number of observations n>10000 or intermediate proportions.

For discussion of CI for a single binomial proportion see Brown et al. (2001) for two-sided and Cai (2005) for one-sided intervals.

## Value

A list containing:

 `conf.int ` the estimated confidence interval `estimate ` the point estimator

And the method, conf.level and alternative specified in the function call.

## Author(s)

Frank Schaarschmidt

## References

Blaker H (2000) Confidence curves and improved exact confidence intervals for discrete distributions. The Canadian Journal of Statistics 28 (4), 783-798.

Brown LD, Cai TT, DasGupta A (2001) Interval estimation for a binomial proportion. Statistical Science 16 (2), 101-133.

Cai TT(2005) One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference 131, 63-88.

`binom.test` for the exact confidence interval and test, `binTest` to calculate p.values of the exact, Score and Wald test.
 ```1 2 3 4 5 6 7 8 9``` ```# Default method is the two-sided 95% Clopper-Pearson CI: binCI(n=200, y=10) # other methods might result in # shorter intervals (but asymetric coverage): binCI(n=200,y=10, method="Blaker") binCI(n=200,y=10, method="Score") ```