gap (version 1.5-1)

ccsize: Power and sample size for case-cohort design

Description

The power of the test is according to $$\Phi\left(Z_\alpha+m^{1/2}\theta\sqrt{\frac{p_1p_2p_D}{q+(1-q)p_D}}\right)$$ where \(\alpha\) is the significance level, \(\theta\) is the log-hazard ratio for two groups, \(p_j\), j=1, 2, are the proportion of the two groups in the population. \(m\) is the total number of subjects in the subcohort, \(p_D\) is the proportion of the failures in the full cohort, and \(q\) is the sampling fraction of the subcohort.

Usage

ccsize(n, q, pD, p1, theta, alpha, beta = 0.2, power = FALSE, verbose = FALSE)

Value

The returned value is a value indicating the power or required sample size.

Arguments

n

the total number of subjects in the cohort.

q

the sampling fraction of the subcohort.

pD

the proportion of the failures in the full cohort.

p1

proportions of the two groups (p2=1-p1).

theta

log-hazard ratio for two groups.

alpha

type I error -- significant level.

beta

type II error.

power

if specified, the power for which sample size is calculated.

verbose

error messages are explicitly printed out.

Author

Jing Hua Zhao

Details

Alternatively, the sample size required for the subcohort is $$m=nBp_D/(n-B(1-p_D))$$ where \(B=(Z_{1-\alpha}+Z_\beta)^2/(\theta^2p_1p_2p_D)\), and \(n\) is the size of cohort.

When infeaisble configurations are specified, a sample size of -999 is returned.

References

Cai J, Zeng D. Sample size/power calculation for case-cohort studies. Biometrics 2004, 60:1015-1024

See Also

pbsize

Examples

if (FALSE) {
# Table 1 of Cai & Zeng (2004).
outfile <- "table1.txt"
cat("n","pD","p1","theta","q","power\n",file=outfile,sep="\t")
alpha <- 0.05
n <- 1000
for(pD in c(0.10,0.05))
{
   for(p1 in c(0.3,0.5))
   {
      for(theta in c(0.5,1.0))
      {
         for(q in c(0.1,0.2))
         {
            power <- ccsize(n,q,pD,p1,alpha,theta)
            cat(n,"\t",pD,"\t",p1,"\t",theta,"\t",q,"\t",signif(power,3),"\n",
                file=outfile,append=TRUE)
         }
      }
   }
}
n <- 5000
for(pD in c(0.05,0.01))
{
   for(p1 in c(0.3,0.5))
   {
      for(theta in c(0.5,1.0))
      {
         for(q in c(0.01,0.02))
         {
            power <- ccsize(n,q,pD,p1,alpha,theta)
            cat(n,"\t",pD,"\t",p1,"\t",theta,"\t",q,"\t",signif(power,3),"\n",
                file=outfile,append=TRUE)
         }
      }
   }
}
table1<-read.table(outfile,header=TRUE,sep="\t")
unlink(outfile)
# ARIC study
outfile <- "aric.txt"
n <- 15792
pD <- 0.03
p1 <- 0.25
alpha <- 0.05
theta <- c(1.35,1.40,1.45)
beta1 <- 0.8
s_nb <- c(1463,722,468)
cat("n","pD","p1","hr","q","power","ssize\n",file=outfile,sep="\t")
for(i in 1:3)
{
  q <- s_nb[i]/n
  power <- ccsize(n,q,pD,p1,alpha,log(theta[i]))
  ssize <- ccsize(n,q,pD,p1,alpha,log(theta[i]),beta1)
  cat(n,"\t",pD,"\t",p1,"\t",theta[i],"\t",q,"\t",signif(power,3),"\t",ssize,"\n",
      file=outfile,append=TRUE)
}
aric<-read.table(outfile,header=TRUE,sep="\t")
unlink(outfile)
# EPIC study
outfile <- "epic.txt"
n <- 25000
alpha <- 0.00000005
power <- 0.8
s_pD <- c(0.3,0.2,0.1,0.05)
s_p1 <- seq(0.1,0.5,by=0.1)
s_hr <- seq(1.1,1.4,by=0.1)
cat("n","pD","p1","hr","alpha","ssize\n",file=outfile,sep="\t")
# direct calculation
for(pD in s_pD)
{
   for(p1 in s_p1)
   {
      for(hr in s_hr)
      {
         ssize <- ccsize(n,q,pD,p1,alpha,log(hr),power)
         if (ssize>0) cat(n,"\t",pD,"\t",p1,"\t",hr,"\t",alpha,"\t",ssize,"\n",
                          file=outfile,append=TRUE)
      }
   }
}
epic<-read.table(outfile,header=TRUE,sep="\t")
unlink(outfile)
# exhaustive search
outfile <- "search.txt"
s_q <- seq(0.01,0.5,by=0.01)
cat("n","pD","p1","hr","nq","alpha","power\n",file=outfile,sep="\t")
for(pD in s_pD)
{
   for(p1 in s_p1)
   {
      for(hr in s_hr)
      {
         for(q in s_q)
         {
            power <- ccsize(n,q,pD,p1,alpha,log(hr))
            cat(n,"\t",pD,"\t",p1,"\t",hr,"\t",q*n,"\t",alpha,"\t",power,"\n",
                file=outfile,append=TRUE)
         }
      }
   }
}
search<-read.table(outfile,header=TRUE,sep="\t")
unlink(outfile)
}