R/OneSampleBernoulli.R

In BayesDIP: Bayesian Decreasingly Informative Priors for Early Termination Phase II Trials

# ------------------------------------------------------------------------------
#     prior --> list containing the information for prior
#          [[1]] - the prior distribution type:
#                1 - DIP
#                2 - Beta(a,b)
#          [[2]] - a: first parameter of the Beta distribution
#          [[3]] - b: second parameter of the Beta distribution
# ------------------------------------------------------------------------------

#' One sample Bernoulli model
#'
#' For a given planned sample size, the efficacy and futility boundaries,
#' return the power, the type I error, the expected sample size and its
#' standard deviation, the probability of reaching the efficacy and futility boundaries.
#'
#'
#' @param prior A list of length 3 containing the distributional information of the prior.
#' The first element is a number specifying the type of prior. Options are
#' \enumerate{
#' \item DIP ;
#' \item Beta(a,b), where a = shape, b = scale}
#' The second and third elements of the list are the parameters a and b, respectively.
#' @param N The planned sample size.
#' @param p0 The null response rate, which could be taken as the standard or historical rate.
#' @param p1 The response rate of the new treatment.
#' @param d The target improvement (minimal clinically meaningful difference).
#' @param ps The efficacy boundary (upper boundary).
#' @param pf The futility boundary (lower boundary).
#' @param alternative less (lower values imply greater efficacy) or greater (larger
#' values imply greater efficacy).
#' @param seed The seed for simulations.
#' @param sim The number of simulations.
#' @return A list of the arguments with method and computed elements
#' @examples
#' # with traditional Bayesian prior Beta(1,1)
#' OneSampleBernoulli(list(2,1,1), N = 100, p0 = 0.3, p1 = 0.5, d = 0.05,
#'                    ps = 0.98, pf = 0.05, alternative = "greater",
#'                    seed = 202210, sim = 10)
#' # with DIP
#' OneSampleBernoulli(list(1,0,0), N = 100, p0 = 0.3, p1 = 0.5, d = 0.05,
#'                    ps = 0.98, pf = 0.05, alternative = "greater",
#'                    seed = 202210, sim = 10)
#' @importFrom stats rbeta rbinom rgamma rnorm rpois
#' @export OneSampleBernoulli

OneSampleBernoulli <- function(prior, N = 100, p0, p1, d = 0,
                               ps = 0.95, pf = 0.05,
                               alternative = c("less", "greater"), seed = 202209, sim = 5000) {

  alternative <- match.arg(alternative)
  # Define the inputs
  if(prior[[1]] == 1){
     prior[[2]] <- NA
     prior[[3]] <- NA
  }
  ## N limit
  if(!is.null(N) && (!is.numeric(N) || N <= 0 ))
    stop("N must be positive number and greater than 10")
  ## p0 limit
  if(!is.null(p0) && (!is.numeric(p0) || (p0 < 0 | p0 > 1)))
    stop("p0 must be numeric in [0,1]")
  ## p1 limit
  if(!is.null(p1) && (!is.numeric(p1) || (p1 < 0 | p1 > 1)))
    stop("p1 must be numeric in [0,1]")
  ## d limit
  if(!is.null(d) && (!is.numeric(d) || (d < 0 | d > abs(p1-p0))))
    stop("d must be numeric in [0, |p1-p0|]")
  ## efficacy boundary limit
  if(!is.null(ps) && (!is.numeric(ps) || (ps < 0.8 | ps > 1)))
    stop("ps (efficacy boundary) must be numeric in [0.8,1]")
  ## futility boundary limit
  if(!is.null(pf) && (!is.numeric(pf) || (pf < 0 | pf > 0.2)))
    stop("pf (futility boundary) must be numeric in [0,0.2]")
  ## set.seed
  if(!is.numeric(seed))
    stop("seed must be numeric")
  ## number of simulation
  if(!is.numeric(sim))
    stop("simulation number  must be numeric")

  set.seed(seed)

  # Functions to calculate the posterior
  Bernoulli <- function(a,b,y){posterior<-rbeta(1000, a+sum(y), b+(length(y)-sum(y)))}
  Bernoulli.DIP <- function(p0, y, N){
    j<-length(y)
    posterior<-rbeta(1000,1+sum(y)+p0*(N-j),1+(j-sum(y))+(1-p0)*(N-j))
  }

  # Simulated Data
  # calculate power
  n.enrolled <- NULL
  cat1s <- 0
  cat1f <- 0
  for (k in 1:sim) {
    y.data<-NULL
    j<-0
    cat<-0
    cats<-0
    catf<-0
    pp_stop<-0.5
    while(cat == 0)
    {
      j<-j+1
      y.data<-append(y.data,rbinom(1,1,p1))
      if(j>=10)
      {
        if (prior[[1]] == 2){
            p1_s<-Bernoulli(a = prior[[2]], b = prior[[3]], y = y.data)
        }else if (prior[[1]] == 1){
            p1_s <- Bernoulli.DIP(p0, y = y.data, N = N)
        }

        if (alternative == "greater"){
          pp_stop<-sum(p1_s>p0+d)/length(p1_s)
        }else if (alternative == "less"){
          pp_stop<-sum(p1_s<p0-d)/length(p1_s)
        }

      }
      if(pp_stop>=ps){cats<-1}
      if(pp_stop<pf){catf<-1}
      cat<-cats+catf
      if(j==N){cat<-1}
    }
    if(cats==1){cat1s<-cat1s+1}
    if(cats==0){cat1s<-cat1s}
    if(catf==1){cat1f<-cat1f+1}
    if(catf==0){cat1f<-cat1f}

    # Recruited Sample Size
    n.enrolled <- append(n.enrolled, j)
  }
  ss <- round(mean(n.enrolled), digits = 1)
  sd <- round(sd(n.enrolled), digits = 2)
  fut.rate <- cat1f/sim
  power <- cat1s/sim


  # calculate type I error
  cat1s <- 0
  cat1f <- 0
  for (k in 1:sim) {
    y.data<-NULL
    j<-0
    cat<-0
    cats<-0
    catf<-0
    pp_stop<-0.5
    while(cat == 0)
    {
      j<-j+1
      y.data<-append(y.data,rbinom(1,1,p0)) # under the null hypothesis p1 = p0
      if(j>=10)
      {
        if (prior[[1]] == 2){
          p1_s<-Bernoulli(a = prior[[2]], b = prior[[3]], y = y.data)
        }else if (prior[[1]] == 1){
          p1_s <- Bernoulli.DIP(p0, y = y.data, N = N)
        }

        if (alternative == "greater"){
          pp_stop<-sum(p1_s>p0+d)/length(p1_s)
        }else if (alternative == "less"){
          pp_stop<-sum(p1_s<p0-d)/length(p1_s)
        }

      }
      if(pp_stop>=ps){cats<-1}
      if(pp_stop<pf){catf<-1}
      cat<-cats+catf
      if(j==N){cat<-1}
    }
    if(cats==1){cat1s<-cat1s+1}
    if(cats==0){cat1s<-cat1s}
    if(catf==1){cat1f<-cat1f+1}
    if(catf==0){cat1f<-cat1f}
  }
  t1error <- cat1s/sim

  # Outputs
  if (prior[[1]] == 1) {method = "DIP"
  } else if (prior[[1]] == 2) {method = paste("Beta(",prior[[2]], ",", prior[[3]], ")", sep="")}

  z <- list(method = method, power = power, type_I_error = t1error,
            expected_sample_size = ss, expected_sample_size_std = sd,
            the_prob_efficacy = power, the_prob_futility = fut.rate)
  z
}