# binom.bayes: Binomial confidence intervals using Bayesian inference In binom: Binomial Confidence Intervals For Several Parameterizations

## Description

Uses a beta prior on the probability of success for a binomial distribution, determines a two-sided confidence interval from a beta posterior. A plotting function is also provided to show the probability regions defined by each confidence interval.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```binom.bayes(x, n, conf.level = 0.95, type = c("highest", "central"), prior.shape1 = 0.5, prior.shape2 = 0.5, tol = .Machine\$double.eps^0.5, maxit = 1000, ...) binom.bayes.densityplot(bayes, npoints = 500, fill.central = "lightgray", fill.lower = "steelblue", fill.upper = fill.lower, alpha = 0.8, ...) ```

## Arguments

 `x` Vector of number of successes in the binomial experiment. `n` Vector of number of independent trials in the binomial experiment. `conf.level` The level of confidence to be used in the confidence interval. `type` The type of confidence interval (see Details). `prior.shape1` The value of the first shape parameter to be used in the prior beta. `prior.shape2` The value of the second shape parameter to be used in the prior beta. `tol` A tolerance to be used in determining the highest probability density interval. `maxit` Maximum number of iterations to be used in determining the highest probability interval. `bayes` The output `data.frame` from `binom.bayes`. `npoints` The number of points to use to draw the density curves. Higher numbers give smoother densities. `fill.central` The color for the central density. `fill.lower,fill.upper` The color(s) for the upper and lower density. `alpha` The alpha value for controlling transparency. `...` Ignored.

## Details

Using the conjugate beta prior on the distribution of p (the probability of success) in a binomial experiment, constructs a confidence interval from the beta posterior. From Bayes theorem the posterior distribution of p given the data x is:

`p|x ~ Beta(x + prior.shape1, n - x + prior.shape2)`

The default prior is Jeffrey's prior which is a Beta(0.5, 0.5) distribution. Thus the posterior mean is `(x + 0.5)/(n + 1)`.

The default type of interval constructed is "highest" which computes the highest probability density (hpd) interval which assures the shortest interval possible. The hpd intervals will achieve a probability that is within tol of the specified conf.level. Setting type to "central" constructs intervals that have equal tail probabilities.

If 0 or n successes are observed, a one-sided confidence interval is returned.

## Value

For `binom.bayes`, a `data.frame` containing the observed proportions and the lower and upper bounds of the confidence interval.

For `binom.bayes.densityplot`, a `ggplot` object that can printed to a graphics device, or have additional layers added.

## Author(s)

Sundar Dorai-Raj ([email protected])

## References

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1997) Bayesian Data Analysis, London, U.K.: Chapman and Hall.

`binom.confint`, `binom.cloglog`, `binom.logit`, `binom.probit`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```# Example using highest probability density. hpd <- binom.bayes( x = 0:10, n = 10, type = "highest", conf.level = 0.8, tol = 1e-9) print(hpd) binom.bayes.densityplot(hpd) # Remove the extremes from the plot since they make things hard # to see. binom.bayes.densityplot(hpd[hpd\$x != 0 & hpd\$x != 10, ]) # Example using central probability. central <- binom.bayes( x = 0:10, n = 10, type = "central", conf.level = 0.8, tol = 1e-9) print(central) binom.bayes.densityplot(central) # Remove the extremes from the plot since they make things hard # to see. binom.bayes.densityplot(central[central\$x != 0 & central\$x != 10, ]) ```