binom.blaker.VHadj.limits {BlakerCI}R Documentation

Vos-Hudson adjustment of Blaker's binomial confidence limits


Blaker's binomial confidence limits adjusted so that logical inconsistencies criticized by Vos and Hudson (2008) are avoided.


binom.blaker.VHadj.limits(x, n, level = 0.95, tol = 1e-10, ...)



number of successes.


number of trials.


confidence level.


numerical tolerance.


additional arguments to be passed to binom.blaker.VHadj.lower.limit: maxiter, nmax, int.eps (see BlakerCI-internal).


Length 2 vector – the lower and upper (adjusted) confidence limits.


The stopping rule used is not fully justified:

The Clopper-Pearson 1 - alpha confidence bounds for x successes in n trials may be expressed as qbeta(alpha/2,x,n-x+1) and qbeta(1-alpha/2,x+1,n-x), and can be generalized this way to real (i. e. not only integer) values of x.

The stopping rule used in binom.blaker.VHadj.limits relies on the hypothesis that the generalized lower (upper) Clopper-Pearson confidence bounds grow (decrease) whenever the number of trials grows, and the proportion of successes grows (decreases) or remains unchanged (with obvious exceptions in extremes).

Though I firmly trust the hypothesis, I can prove it, so far, just for integer numbers of successes (i. e. for “ordinary” Clopper-Pearson confidence bounds, not the generalized ones), and lack a general proof. Should the hypothesis be invalid, the stopping rule implemented in binom.blaker.VHadj.limits would be incorrect, and the process of modifying the Blaker's confidence bounds could be incomplete in some cases.


Vos & Hudson (2008) gave examples of mutually contradictory inferences yielded by some binomial tests and confidence intervals, including the Blaker's confidence interval. Their objections may be interpreted as follows: When the number of trials is increased so that the success proportion increases (decreases) or remains the same, the lower (upper) confidence limit at the same confidence level should not decrease (increase).
The adjustment implemented in binom.blaker.VHadj.limits replaces the lower (upper) Blaker's confidence limit for x successes in n trials with the infimum (supremum) of the Blaker's lower (upper) confidence limits over such pairs y, m that m is not less that n, and y/m is not less (greater) than x/n.

Note that Lecoutre & Poitevineau (2014), refering to the criticism by Vos & Hudson, proposed a modification of the Blaker's confidence limits. Their adjustment, however, eliminates only a subset of “discrepancies” treated by binom.blaker.VHadj.limits, namely nonmonotonicities of upper (lower) Blaker's confidence bounds in the number of trials when the number of successes (failures) remains the same.


Jan Klaschka


Vos, P. W. & Hudson, S. (2008). Problems with binomial two-sided tests and the associated confidence intervals. Australian & New Zealand Journal of Statistics 50(1): 81-89.

Lecoutre, B. & Poitevineau, J. (2014). New results for computing Blaker's exact confidence interval limits for usual one-parameter discrete distributions. Communications in Statistics - Simulation and Computation,


binom.blaker.VHadj.limits(6,13) # [1] 0.2150187 0.7395922

## Note that the lower limit differs from the 
## unadjusted version:

binom.blaker.limits(6,13)       # [1] 0.2158050 0.7395922

## The (unadjusted) lower limit was replaced with the
## Blaker's lower limit (both unadjusted and adjusted) 
## assigned to 7 successes in 15 trials:

binom.blaker.limits(7,15)       # [1] 0.2150187 0.7096627
binom.blaker.VHadj.limits(7,15) # [1] 0.2150187 0.7096627

## The adjustment avoids a contradiction between
## inferences corresponding to
## 6 successes in 13 trials, and 7 successess in 15 trials: 
## Though the latter situation means a higher succes proportion
## in a higher number of trials, it is assigned a smaller
## (unadjusted) Blaker's 95% lower confidence limit. 

[Package BlakerCI version 1.0-6 Index]