BinomCI {DescTools}  R Documentation 
Confidence Intervals for Binomial Proportions
Description
Compute confidence intervals for binomial proportions according to a number of the most common proposed methods.
Usage
BinomCI(x, n, conf.level = 0.95, sides = c("two.sided", "left", "right"),
method = c("wilson", "wald", "waldcc", "agresticoull", "jeffreys",
"modified wilson", "wilsoncc","modified jeffreys",
"clopperpearson", "arcsine", "logit", "witting", "pratt",
"midp", "lik", "blaker"),
rand = 123, tol = 1e05, std_est = TRUE)
Arguments
x 
number of successes. 
n 
number of trials. 
conf.level 
confidence level, defaults to 0.95. 
sides 
a character string specifying the side of the confidence interval, must be one of 
method 
character string specifing which method to use; this can be one out of:

rand 
seed for random number generator; see details. 
tol 
tolerance for method 
std_est 
logical, specifying if the standard point estimator for the proportion value 
Details
All arguments are being recycled.
The Wald interval is obtained by inverting the acceptance region of the Wald largesample normal test.
The Wald with continuity correction interval is obtained by adding the term 1/(2*n) to the Wald interval.
The Wilson interval, which here is the default method, was introduced by Wilson (1927) and is the inversion of the CLT approximation to the family of equal tail tests of p = p0.
The Wilson interval is recommended by Agresti and Coull (1998) as well as by
Brown et al (2001). It is also returned as conf.int
from the function prop.test
with the correct
option set to FALSE
.
The Wilson cc interval is a modification of the Wilson interval adding a continuity correction term. This is returned as conf.int
from the function prop.test
with the correct
option set to TRUE
.
The modified Wilson interval is a modification of the Wilson interval for x close to 0 or n as proposed by Brown et al (2001).
The AgrestiCoull interval was proposed by Agresti and Coull (1998) and is a slight modification of the Wilson interval. The AgrestiCoull intervals are never shorter than the Wilson intervals; cf. Brown et al (2001). The internally used point estimator ptilde is returned as attribute.
The Jeffreys interval is an implementation of the equaltailed Jeffreys prior interval as given in Brown et al (2001).
The modified Jeffreys interval is a modification of the Jeffreys interval for
x == 0  x == 1
and x == n1  x == n
as proposed by
Brown et al (2001).
The ClopperPearson interval is based on quantiles of corresponding beta distributions. This is sometimes also called exact interval.
The arcsine interval is based on the variance stabilizing distribution for the binomial distribution.
The logit interval is obtained by inverting the Wald type interval for the log odds.
The Witting interval (cf. Beispiel 2.106 in Witting (1985)) uses randomization to obtain uniformly optimal lower and upper confidence bounds (cf. Satz 2.105 in Witting (1985)) for binomial proportions.
The Pratt interval is obtained by extremely accurate normal approximation. (Pratt 1968)
The Midp approach is used to reduce the conservatism of the ClopperPearson, which is known to be very pronounced. The method midp accumulates the tail areas.
The lower bound p_l
is found as the solution to the equation
\frac{1}{2} f(x;n,p_l) + (1F(x;m,p_l)) = \frac{\alpha}{2}
where f(x;n,p)
denotes the probability mass function (pmf) and
F(x;n,p)
the (cumulative) distribution function of the binomial
distribution with size n
and proportion p
evaluated at
x
.
The upper bound p_u
is found as the solution to the equation
\frac{1}{2} f(x;n,p_u) + F(x1;m,p_u) = \frac{\alpha}{2}
In case x=0 then the lower bound is zero and in case x=n then the upper bound is 1.
The Likelihoodbased approach is said to be theoretically appealing. Confidence intervals are based on profiling the binomial deviance in the neighbourhood of the MLE.
For the Blaker method refer to Blaker (2000).
For more details we refer to Brown et al (2001) as well as Witting (1985).
Some approaches for the confidence intervals are capable of violating the [0, 1] boundaries and potentially yield negative results or values beyond 1. These would be truncated such as not to exceed the valid range of [0, 1].
So now, which interval should we use? The Wald interval often has inadequate coverage, particularly for small n and values of p close to 0 or 1. Conversely, the ClopperPearson Exact method is very conservative and tends to produce wider intervals than necessary. Brown et al. recommends the Wilson or Jeffreys methods for small n and AgrestiCoull, Wilson, or Jeffreys, for larger n as providing more reliable coverage than the alternatives.
For the methods "wilson"
, "wilsoncc"
, "modified wilson"
, "agresticoull"
and "arcsine"
the internally used alternative point estimator for the proportion value can be returned (by setting std_est = FALSE
). The point estimate typically is slightly shifted towards 0.5 compared to the standard estimator. See the literature for the more details.
Value
A vector with 3 elements for estimate, lower confidence intervall and upper for the upper one.
For more than one argument each, a 3column matrix is returned.
Note
The base of this function once was binomCI()
from the SLmisc package. In the meantime, the code has been updated on several occasions and it has undergone numerous extensions and bug fixes.
Author(s)
Matthias Kohl <Matthias.Kohl@stamats.de>, Rand R. Wilcox (Pratt's method), Michael Hoehle <hoehle@math.su.se> (Midp), Ralph Scherer <shearer.ra76@gmail.com> (Blaker), Andri Signorell <andri@signorell.net> (interface issues and all the rest)
References
Agresti A. and Coull B.A. (1998) Approximate is better than "exact" for interval estimation of binomial proportions. American Statistician, 52, pp. 119126.
Brown L.D., Cai T.T. and Dasgupta A. (2001) Interval estimation for a binomial proportion Statistical Science, 16(2), pp. 101133.
Witting H. (1985) Mathematische Statistik I. Stuttgart: Teubner.
Pratt J. W. (1968) A normal approximation for binomial, F, Beta, and other common, related tail probabilities Journal of the American Statistical Association, 63, 1457 1483.
Wilcox, R. R. (2005) Introduction to robust estimation and hypothesis testing. Elsevier Academic Press
Newcombe, R. G. (1998) Twosided confidence intervals for the single proportion: comparison of seven methods, Statistics in Medicine, 17:857872 https://pubmed.ncbi.nlm.nih.gov/16206245/
Blaker, H. (2000) Confidence curves and improved exact confidence intervals for discrete distributions, Canadian Journal of Statistics 28 (4), 783798
See Also
binom.test
, binconf
, MultinomCI
, BinomDiffCI
, BinomRatioCI
Examples
BinomCI(x=37, n=43,
method=eval(formals(BinomCI)$method)) # return all methods
prop.test(x=37, n=43, correct=FALSE) # same as method wilson
prop.test(x=37, n=43, correct=TRUE) # same as method wilsoncc
# the confidence interval computed by binom.test
# corresponds to the ClopperPearson interval
BinomCI(x=42, n=43, method="clopperpearson")
binom.test(x=42, n=43)$conf.int
# all arguments are being recycled:
BinomCI(x=c(42, 35, 23, 22), n=43, method="wilson")
BinomCI(x=c(42, 35, 23, 22), n=c(50, 60, 70, 80), method="jeffreys")
# example Table I in Newcombe (1998)
meths < c("wald", "waldcc", "wilson", "wilsoncc",
"clopperpearson","midp", "lik")
round(cbind(
BinomCI(81, 263, m=meths)[, 1],
BinomCI(15, 148, m=meths)[, 1],
BinomCI(0, 20, m=meths)[, 1],
BinomCI(1, 29, m=meths)[, 1]), 4)
# returning p.tilde for agresticoull ci
BinomCI(x=81, n=263, meth="agresticoull", std_est = c(TRUE, FALSE))