# Blaker's binomial acceptability function, optionally unimodalized. 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. ```r binom.blaker.acc(x, n, p, type = c("orig", "unimod"), acc.tol = 1e-10, ...) ``` ## 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 `type = "orig"`, original acceptability function calculated. For `type = "unimod"`, smallest unimodal function greater or equal to the acceptability function calculated instead. - `acc.tol`: numerical tolerance (relevant only for `type = "unimod"`). - `...`: additional arguments to be passed to `binom.blaker.acc.single.p`; in fact, just `maxiter` (see `BlakerCI-internal`). ## 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). ## Returns Vector of acceptability values (with or without unimodalization) in points of `p`. ## References Blaker, H. (2000) Confidence curves and improved exact confidence intervals for discrete distributions. Canadian Journal of Statistics 28: 783-798. (Corrigenda: Canadian Journal of Statistics 29: 681.) Fay, M.P. (2010). Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data. R Journal 2(1): 53-58. ## Author(s) Jan Klaschka klaschka@cs.cas.cz ## Note Inspired by M.P. Fay (2010), mentioning unavoidable inconsistencies between tests with non-unimodal acceptability functions and confidence intervals derived from them. When the acceptability functions are unimodalized and the test modified accordingly (i.e. p-values slightly increased in some cases), a perfectly matching test-CI pair is obtained. ## Examples ```r 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,acc1-acc,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 ```

binom.blaker.acc function