Compute confidence intervals for binomial proportions following the most popular methods. (Wald, Wilson, Agresti-Coull, Jeffreys, Clopper-Pearson etc.)
BinomCI(x, n, conf.level = 0.95, sides = c("two.sided", "left", "right"),
method = c("wilson", "wald", "agresti-coull", "jeffreys",
"modified wilson", "wilsoncc","modified jeffreys",
"clopper-pearson", "arcsine", "logit", "witting", "pratt"),
rand = 123)number of successes.
number of trials.
confidence level, defaults to 0.95.
a character string specifying the side of the confidence interval, must be one of "two.sided" (default),
"left" or "right". You can specify just the initial letter. "left" would be analogue to a hypothesis of
"greater" in a t.test.
character string specifing which method to use; this can be one out of:
"wald", "wilson", "wilsoncc", "agresti-coull", "jeffreys",
"modified wilson", "modified jeffreys", "clopper-pearson",
"arcsine", "logit", "witting" or "pratt". Defaults to "wilson".
Abbreviation of method are accepted. See details.
seed for random number generator; see details.
A vector with 3 elements for estimate, lower confidence intervall and upper for the upper one.
All arguments are being recycled.
The Wald interval is obtained by inverting the acceptance region of the Wald large-sample normal test.
The Wilson interval, which is the default, 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).
The Agresti-Coull interval was proposed by Agresti and Coull (1998) and is a slight modification of the Wilson interval. The Agresti-Coull intervals are never shorter than the Wilson intervals; cf. Brown et al (2001).
The Jeffreys interval is an implementation of the equal-tailed Jeffreys prior interval as given in Brown et al (2001).
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 Wilson cc interval is a modification of the Wilson interval adding a continuity correction term.
The modified Jeffreys interval is a modification of the Jeffreys interval for
x == 0 | x == 1 and x == n-1 | x == n as proposed by
Brown et al (2001).
The Clopper-Pearson 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.
For more details we refer to Brown et al (2001) as well as Witting (1985).
The Pratt interval is obtained by extremely accurate normal approximation. (Pratt 1968)
Some approaches for the confidence intervals can potentially yield negative results or values beyond 1. These would be reset such as not to exceed the range of [0, 1].
And 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 Clopper-Pearson 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 Agresti-Coull, Wilson, or Jeffreys, for larger n as providing more reliable coverage than the alternatives. Also note that the point estimate for the Agresti-Coull method is slightly larger than for other methods because of the way this interval is calculated.
Agresti A. and Coull B.A. (1998) Approximate is better than "exact" for interval estimation of binomial proportions. American Statistician, 52, pp. 119-126.
Brown L.D., Cai T.T. and Dasgupta A. (2001) Interval estimation for a binomial proportion Statistical Science, 16(2), pp. 101-133.
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
# NOT RUN {
BinomCI(x=37, n=43, method=c("wald", "wilson", "agresti-coull", "jeffreys",
"modified wilson", "modified jeffreys", "clopper-pearson", "arcsine", "logit",
"witting", "pratt")
)
# the confidence interval computed by binom.test
# corresponds to the Clopper-Pearson interval
BinomCI(x=42, n=43, method="clopper-pearson")
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")
# }
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