binom.blaker.acc {BlakerCI}  R Documentation 
Blaker's binomial acceptability function, optionally unimodalized.
Description
Calculates values of the acceptability function for the binomial
distribution (see function acceptbin
in Blaker (2000))
in a sequence of points (for, e.g., plotting purposes).
The acceptability function may optionally be
“unimodalized”, i.e. replaced with the smallest
greater or equal unimodal function.
Usage
binom.blaker.acc(x, n, p, type = c("orig", "unimod"),
acc.tol = 1e10, ...)
Arguments
x 
number of successes. 
n 
number of trials. 
p 
vector (length 1 allowed) of hypothesized binomial parameters (between 0 and 1). In case of more than one point, an increasing sequence required. 
type 
for 
acc.tol 
numerical tolerance (relevant only for 
... 
additional arguments to be passed to

Details
For type = "orig"
, essentially the same is calculated
as – for single points – by acceptbin
function
from Blaker (2000).
Single values of the “unimodalized” acceptability function
(for type = "unimod"
) are computed by an iterative
numerical algorithm implemented in internal function
binom.blaker.acc.single.p
.
The function cited is called just once in each of the intervals
where the acceptability function is continuous
(namely in the leftmost one of those points of p
that fall into the interval when dealing with points
below x/n
, and the rightmost one when above
x/n
). The rest is done by function
cummax
.
This is considerably faster than calling
binom.blaker.acc.single.p
for every point of p
.
Note that applying cummax
directly to
a vector of unmodified acceptability values
is even faster and provides a unimodal output;
it may, nevertheless, lack accuracy (see Examples).
Value
Vector of acceptability values (with or without unimodalization)
in points of p
.
Note
Inspired by M.P. Fay (2010), mentioning “unavoidable inconsistencies” between tests with nonunimodal acceptability functions and confidence intervals derived from them. When the acceptability functions are unimodalized and the test modified accordingly (i.e. pvalues slightly increased in some cases), a perfectly matching testCI pair is obtained.
Author(s)
Jan Klaschka klaschka@cs.cas.cz
References
Blaker, H. (2000) Confidence curves and improved exact confidence
intervals for discrete distributions.
Canadian Journal of Statistics 28: 783798.
(Corrigenda: Canadian Journal of Statistics 29: 681.)
Fay, M.P. (2010). Twosided Exact Tests and Matching Confidence Intervals for Discrete Data. R Journal 2(1): 5358.
Examples
p < seq(0,1,length=1001)
acc < binom.blaker.acc(3,10,p)
acc1 < binom.blaker.acc(3,10,p,type="unimod")
## The two functions look the same at first glance.
plot(p,acc,type="l")
lines(p,acc1,col="red")
legend(x=.7,y=.8,c("orig","unimod"),col=c("black","red"),lwd=1)
## There is, nevertheless, a difference.
plot(p,acc1acc,type="l")
## Focussing on the difference about p=0.4:
p < seq(.395,.405,length=1001)
acc < binom.blaker.acc(3,10,p)
acc1 < binom.blaker.acc(3,10,p,type="unimod")
plot(p,acc,type="l",ylim=c(.749,.7495))
lines(p,acc1,col="red")
legend(x=.402,y=.7494,c("orig","unimod"),col=c("black","red"),lwd=1)
## Difference between type="unimod" and mere applying
## cummax to values obtained via type="orig":
p < seq(0,1,length=1001)
x < 59
n < 355
## Upper confidence limit (at 0.95 level) is slightly above 0.209:
binom.blaker.limits(x,n) ## [1] 0.1300807 0.2090809
## Unmodified acceptability value fall below 0.05 at p = .209
## left to the limit (so that the null hypothesis p = .209
## would be rejected despite the fact that p lies within
## the confidence interval):
acc < binom.blaker.acc(59,355,p)
rbind(p,acc)[,207:211]
## [,1] [,2] [,3] [,4] [,5]
## p 0.20600000 0.20700000 0.20800000 0.20900000 0.21000000
## acc 0.06606867 0.05759836 0.05014189 0.04999082 0.04330283
##
## Modified acceptability is above 0.05 at p = 0.05 (so that
## hypothesis p = 0.05 is not rejected by the modified test):
acc1 < binom.blaker.acc(59,355,p,type="unimod")
rbind(p,acc1)[,207:211]
## [,1] [,2] [,3] [,4] [,5]
## p 0.20600000 0.20700000 0.20800000 0.20900000 0.21000000
## acc1 0.06608755 0.05759836 0.05014189 0.05000009 0.04331409
##
## Applying cummax to unmodified acceptabilities guarantees unimodality
## but lacks accuracy, leaving the value at p = 0.209 below 0.05:
m < max(which(p <= 59/355))
acc2 < acc
acc2[1:m] < cummax(acc2[1:m])
acc2[1001:(m+1)] < cummax(acc2[1001:(m+1)])
rbind(p,acc2)[,207:211]
## [,1] [,2] [,3] [,4] [,5]
## p 0.20600000 0.20700000 0.20800000 0.20900000 0.21000000
## acc2 0.06606867 0.05759836 0.05014189 0.04999082 0.04330283