Blaker's binomial confidence limits adjusted so that logical inconsistencies criticized by Vos and Hudson (2008) are avoided.
number of successes.
number of trials.
additional arguments to be passed to
Length 2 vector – the lower and upper (adjusted) confidence limits.
The stopping rule used is not fully justified:
1 - alpha confidence bounds
x successes in
may be expressed as
and can be generalized this way to real
(i. e. not only integer) values of
The stopping rule used in
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
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
replaces the lower (upper) Blaker's confidence limit
x successes in
with the infimum (supremum) of the Blaker's lower (upper)
confidence limits over such pairs
m is not less that
y/m is not less (greater) than
Note that Lecoutre & Poitevineau (2014), refering to the
Vos & Hudson, proposed a modification of the Blaker's
Their adjustment, however, eliminates only a subset of
“discrepancies” treated by
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 email@example.com
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, http://dx.doi.org/10.1080/03610918.2014.911900.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
binom.blaker.VHadj.limits(6,13) #  0.2150187 0.7395922 ## Note that the lower limit differs from the ## unadjusted version: binom.blaker.limits(6,13) #  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) #  0.2150187 0.7096627 binom.blaker.VHadj.limits(7,15) #  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.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.