##########################################################################
## R code for sample size calculation written by Youlan Rao
##
## Reference: 
## Determination of Sample Size for Validation Study in Pharmacogenomics 
## - Rao, Y., Lee, Y., and Hsu, J. C., 
## January 2010, Technical Report No. 834, 
## Department of Statistics, The Ohio State University. 
## Section 2.2 
##########################################################################

# gamma: minimum level of sensitivity
# 1-beta: the minimal probability of successful validation
# Sen_v: true sensitivity 

mstar<-fn.samplesize(gamma=0.80, beta=0.05, Sen_v=0.90)

# function to find m*: the smallest number of subjects in the subgroup
# such that for any m>=m*, the sensitivity rquirement is satisfied.
# continuous approximation(CA) + Backward grid search

fn.samplesize<-function(gamma, beta=0.05, Sen_v)
{
  if(gamma>=Sen_v)
  {return(0)}
  else
  # find m0: the sharper upper bound than by Hoeffding's inequality   
  { m0<-ceiling(fn.samplesize.CA(gamma, beta, Sen_v))
  # Backward grid search
  N_m=1:m0
  rhs<-NULL
  for (i in 1:length(N_m))
      {
        n_m=N_m[i]      
        # use regularized incomplete beta function. 
        rhs[i]<-pbeta(1-Sen_v, n_m-(ceiling(n_m*gamma)-1), ceiling(n_m*gamma))
      }
 
  jc<-length(N_m)
  while((rhs[jc]<=beta) & (jc>1)){jc<-jc-1}
  if (jc>1) {ind<-jc+1}else {ind<-jc}
  mstar<-N_m[ind]
  return(mstar) 
  }
}


# continuous approximation(CA)
# function to find m0: sharper upper bound than by Hoeffding's inequality
fn.samplesize.CA<-function(gamma, beta=0.05, Sen_v)
{
  if(gamma>=Sen_v)
  {return(0)}
  else
  # mH: the upper bound given by Hoeffding's inequality
  { mH<--log(beta)/(2*(Sen_v-gamma)^2)
  # m0: sharper upper bound than by Hoeffding's inequality
    m0<-uniroot(function(m) pbeta(1-Sen_v, m-m*gamma, m*gamma+1)-beta, c(1,mH), tol=1e-10)$root
  return(m0)
  }
}