R/pi_pp.R

In PPQplan: Process Performance Qualification (PPQ) Plans in Chemistry, Manufacturing and Controls (CMC) Statistical Analysis

#' Probability of Passing PPQ Test using Prediction Interval
#'
#' The function for calculating the probability of passing critical quality attributes (CQA) PPQ test .
#'
#' @usage pi_pp(Llim, Ulim, mu, sigma, n, n.batch, alpha)
#' @param Llim lower specification limit
#' @param Ulim upper specification limit
#' @param mu hypothetical mean of the attribute
#' @param sigma hypothetical standard deviation of the attribute
#' @param n sample size (number of locations) per batch
#' @param n.batch number of batches for passing PPQ during validation
#' @param alpha significant level for constructing the prediction interval.
#' @return
#' A numeric value of the passing/acceptance probability
#' @seealso \code{rl_pp}.
#' @references
#' Burdick, R. K., LeBlond, D. J., Pfahler, L. B., Quiroz, J., Sidor, L., Vukovinsky, K., & Zhang, L. (2017).
#' Statistical Applications for Chemistry, Manufacturing and Controls (CMC) in the Pharmaceutical Industry.
#' \emph{Springer}.
#' @author Yalin Zhu
#' @examples
#' \dontrun{
#' pi_pp(sigma=0.5, mu=2.5, n=10, n.batch=1, Llim=1.5, Ulim=3.5, alpha=0.05)
#'
#' sapply(X=c(0.1,0.5, 1,2,3,4,5,10), FUN = pi_pp, mu=97, n=10, Llim=95, Ulim=105,
#' n.batch=1, alpha=0.05)
#' sapply(X=c(0.1,0.5, 1,2,3,4,5,10), FUN = pi_pp, mu=100, n=10, Llim=95, Ulim=105,
#' n.batch=1, alpha=0.05)
#' }
#' @export
pi_pp <- function(Llim=1.5, Ulim=3.5, mu, sigma, n=10, n.batch=1, alpha=0.05){
  t.cv <- qt((1-alpha/2), df=n-1);
  k <- t.cv*sqrt(1+1/n)
  Func <- function(V){
    (pnorm(q = Ulim-k*sqrt(V), mean = mu, sd = sigma/sqrt(n))-pnorm(q = Llim+k*sqrt(V), mean = mu, sd = sigma/sqrt(n)))*dchisq(x = (n-1)*V/sigma^2, df = n-1)
  }
  (min(1, integrate(Func, lower = 0, upper = ((Ulim-Llim)/(2*k))^2, rel.tol = 1e-10)$value*(n-1)/sigma^2))^n.batch
}