Confidence we seek…

November 18, 2009
By

(This article was first published on Stats raving mad » R, and kindly contributed to R-bloggers)


Estimating a proportion at first looks elementary. Hail to aymptotics, right? Well, initially it might seem efficient to iuse the fact that frac{hat{p}-p}{sqrt{frac{hat{p}(1-hat{p})}{n}}} tilde {} N(0,1). In other words the classical confidence interval relies on the inversion of Wald’s test.

A function to ease the computation is the following (not really needed!).

waldci<- function(x,n,level){
phat<-sum(x)/n
results<-phat + c(-1,1)*qnorm(1-level/2)*sqrt(phat*(1-phat)/n)
print(results)
}

An exact confidence interval is the so-called Blake’s confidence interval

blakerci <- function(x,n,level,tolerance=1e-05){
  lower = 0
  upper = 1
  if (x!=0){lower = qbeta((1-level)/2, x, n-x+1)
    while (acceptbin(x, n, lower + tolerance) < (1 - level))
      lower = lower+tolerance
   }
  if (x!=n){upper = qbeta(1 - (1-level)/2, x+1, n-x)
    while (acceptbin(x, n, upper - tolerance) < (1 - level))
      upper = upper-tolerance
   }
 c(lower,upper)
}
# Computes the Blaker exact ci (Canadian J. Stat 2000)
# for a binomial success probability
# for x successes out of n trials with
# confidence coefficient = level; uses acceptbin function

acceptbin = function(x, n, p){
 #computes the Blaker acceptability of p when x is observed
 # and X is bin(n, p)
   p1 = 1 - pbinom(x - 1, n, p)
   p2 = pbinom(x, n, p)
   a1 = p1 + pbinom(qbinom(p1, n, p) - 1, n, p)
   a2 = p2 + 1 - pbinom(qbinom(1 - p2, n, p), n, p)
   return(min(a1,a2))
}

A comparison is easily made along the following lines of code

list.counts.bl=NA
list.counts.wl=NA
prob=c(0.05,0.1,.9,.97)

for (j in 1:4){
p=prob[j]
n=9
size=1
level=0.05
N=30000
counts.bl=0
counts.wl=0

for (i in 1:N)
{
tmp.sample=rbinom(n,size,p)
x=sum(tmp.sample)
if (blakerci(x,n,level,tolerance=1e-05)[1]<p && blakerci(x,n,level,tolerance=1e-05)[2]>p) counts.bl=counts.bl+1
if (waldci(x,n,level)[1]<p && waldci(x,n,level)[2]>p) counts.wl=counts.wl+1
}

list.counts.bl[j]=counts.bl
list.counts.wl[j]=counts.wl
}

list.counts.bl/N
list.counts.wl/N

You can see the difference at extremes!

tt=matrix(data=c(list.counts.bl,list.counts.wl),nrow=2,ncol=4,byrow=TRUE)/N
colnames(tt)=prob
rownames(tt)=list("Blaker","Wald")
> round(tt,3)
        0.05   0.1   0.9  0.97
Blaker 0.632 0.391 0.384 0.762
Wald   0.368 0.605 0.605 0.238

The blacerci function is adapted form A. Agresti’s site [link].

Look it up…

Lawrence D. Brown, T. Tony Cai and Anirban DasGupta, “Interval Estimation for a Binomial Proportion”, Statistical Science 2001, Vol. 16, No. 2, pp. 101–133 [pdf]

Laura A. Thompson, An R (and S-PLUS) Manual to Accompany Agresti’s Categorical Data Analysis, 2009 [pdf | R code]

To leave a comment for the author, please follow the link and comment on his blog: Stats raving mad » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , , , , , ,

Comments are closed.