Predicting Confidence Interval for the Proportion at the Time of Study Planning in Small Clinical Trials

Abstract

Confidence intervals are commonly used to assess the precision of parameter estimations. Particularly in small clinical trials, such assessment may be used in place of a power calculation. We discuss `future' confidence interval prediction with binomial outcomes for small clinical trials and sample size calculation, where the term `future' confidence interval emphasizes the confidence interval as a function of a random sample that is not observed at the planning stage of a study. We propose and discuss three probabilistic approaches to future confidence interval prediction when the sample size is small. We demonstrate substantial differences among these approaches in terms of the interval width prediction and sample size calculation. We show that the approach based on the expectation of the boundaries has the most desirable properties and is easy to implement. In this chapter, we primarily discuss prediction of the Clopper-Pearson exact confidence interval, and then extend our discussion to other confidence interval methods. In particular, we discuss the arcsine transformation as a viable alternative to the exact confidence interval.

Appendix

The following R codes are to calculate the predicted width of confidence intervals proposed in Sects. 11.2 and 11.3. The parameters needed are n, p, and alpha for the sample size, true success rate, and 100(1-alpha)% confidence interval. The results of each function consist of predicted lower, upper bounds, and width.

R codes for Sect. 11.2

#####Approach 1###### approach1<-function(n,p,alpha){ x<-n*p lc<-1/(1+(n-x+1)/(x*qf(0.025,2*x,2*(n-x+1)))) rc<-1/(1+(n-x)/((x+1)*qf(0.975,2*(x+1),2*(n-x)))) cat(round(lc,4),round(rc,4),round((rc-lc),4), fill=T) } #Usage approach1(20,0.07,0.05) #####Approach 2-1#### approach2.1<-function(n,p,alpha){ x<-seq(0,n,1) probs<-pbinom(x,n,p) L<-max(x[probs<=(alpha/2)],-0.5)+0.5 U<-min(min(x[probs>=(1-alpha/2)])+1-0.5,n) cat(round(L/n,4),round(U/n,4), round((U/n-L/n),4), fill=T) } #Usage approach2.1(20,0.07,0.05) #####Approach 2-2#### #Functions to solve Eq. (11.6) incbetaL<-function(x,p,n,alpha){ duhaeyo<-0 ele1<-x+1 ele2<-n for(i in ele1:ele2){ duhaeyo<-duhaeyo+factorial(ele2)/(factorial(i)*factorial (ele2-i))* p^i*(1-p)^(ele2-i) } crit<-abs(duhaeyo-(1-alpha/2)) return(crit) } incbetaU<-function(x,p,n,alpha){ duhaeyo<-0 ele1<-x ele2<-n for(i in ele1:ele2){ duhaeyo<-duhaeyo+factorial(ele2)/(factorial(i)*factorial (ele2-i))* p^i*(1-p)^(ele2-i) } crit<-abs(duhaeyo-(alpha/2)) return(crit) } approach2.2<-function(nn,pp,alphaa){ nval<-nn pval<-pp alphaval<-alphaa x<-nval*pval clow<-1/(1+(nval-x+1)/(x*qf((alphaval/2),2*x,2*(nval-x+1)))) cupp<-1/(1+(nval-x)/((x+1)*qf((1-alphaval/2),2*(x+1), 2*(nval-x)))) lowval<-optimize(incbetaL, c(((clow-0.1)*nval), ((clow+0.1)*nval)),p=pval, n=nval, tol = 0.0001, alpha=alphaval) uppval<-optimize(incbetaU, c(((cupp-0.1)*nval), ((cupp+0.1)*nval)),p=pval, n=nval, tol = 0.0001, alpha=alphaval) lc<-max(lowval[[1]]/nval,0) rc<-min(uppval[[1]]/nval,1) cat(c(lc,rc,(rc-lc))) } #Usage approach2.2(20,0.07,0.05) ####Approach 3####### approach3<-function(n,p,alpha){ truep<-p x<-seq(0,n,1) probs<-dbinom(x,n,truep) counts<-0 clc<-c() crc<-c() for(i in 1:(n+1)){ lc<-binom.test(x[i], n, conf.level = (1-alpha))$conf.int[1] rc<-binom.test(x[i], n, conf.level = (1-alpha))$conf.int[2] clc<-c(clc,lc*probs[i]) crc<-c(crc,rc*probs[i]) } cat(round(sum(clc),4),round(sum(crc),4),round((sum(crc)-sum(clc)), 4), fill=T) } #Usage approach3(20,0.07,0.05)

R codes for Sect. 11.3

#Approach 1 approach1<-function(n,p,alpha){ c<-qnorm((1-alpha/2)) x<-n*p lc<-max(sin(asin(sqrt((3/8+x)/(n+3/4)))-c/(2*sqrt(n)))^2,0) rc<-min(sin(asin(sqrt((3/8+x)/(n+3/4)))+c/(2*sqrt(n)))^2,1) cat(round(lc,4),round(rc,4),round((rc-lc),4)) } #Approach 2 approach2<-function(n,p,alpha){ c<-qnorm((1-alpha/2)) lc<-max(((sin(asin(sqrt(p))-c/(2*sqrt(n)))^2)*(1+3/(4*n)) -3/(8*n)),0) rc<-min(((sin(asin(sqrt(p))+c/(2*sqrt(n)))^2)*(1+3/(4*n)) -3/(8*n)),1) cat(round(lc,4),round(rc,4),round((rc-lc),4)) } #Approach 3 approach3<-function(n,p,alpha){ c<-qnorm((1-alpha/2)) truep<-p x<-seq(0,n,1) probs<-dbinom(x,n,truep) counts<-0 clc<-c() crc<-c() for(i in 1:length(probs)){ lc<-max(sin(asin(sqrt((3/8+x[i])/(n+3/4)))-c/(2*sqrt(n)))^2,0) rc<-min(sin(asin(sqrt((3/8+x[i])/(n+3/4)))+c/(2*sqrt(n)))^2,1) clc<-c(clc,lc*probs[i]) crc<-c(crc,rc*probs[i]) } crc<-c(crc,1*probs[(n+1)]) cat(round(sum(clc),4),round(sum(crc),4),round((sum(crc) -sum(clc)),4)) }