Alternative CI for BE decision [Power / Sample Size]

posted by d_labes  – Berlin, Germany, 2017-02-09 11:18  – Posting: # 17041
Views: 19,281

(edited by d_labes on 2017-02-09 11:43)

Dear zizou,

all you have written about CI(s) is more or less correct.
But it's a misunderstanding.

The basic test to use for the BE decision is TOST with alpha = 0.05 by convention. And this is operationally identical to the interval inclusion rule, conventionally using an ordinary 1-2*alpha CI.

The Berger, Hsu alternative CI+ is another CI (an 1-alpha CI) constructed based on TOST and gives the same result if used as decision rule for stating BE.

Code to show this via simulations:
power.sim2x2 <- function(CV, GMR, n, nsims=1E5, alpha=0.05, lBEL=0.8, uBEL=1.25,
                         CItype=c("usual", "alternative"), setseed=TRUE, details=FALSE)
{
  ptm <- proc.time()
  if(setseed) set.seed(123456)
  # check CItype
  CI <- match.arg(CItype)
  # n is total if given as simple number
  # to be correct n must then be even!

  if (length(n)==1) {
    nsum <- n
    fact <- 2/n
  }
  if (length(n)==2) {
    nsum <- sum(n)
    fact <- 0.5*sum(1/n)
  }
  mse    <- log(1.0 + CV^2)
  df     <- nsum-2
  tval   <- qt(1-alpha,df)
  # Attention! With nsims=1E8 memory of my machine (4 GB) is too low
  # Thus work in chunks of 1E6 if nsims>1E6.

  chunks <- 1
  ns     <- nsims
  if (nsims>1e6) {
    chunks <- trunc(nsims/1E6)
    ns     <- 1E6
    if (chunks*1e6!=nsims){
      nsims <- chunks*1e6
      warning("nsims truncated to", nsims)
    }
  }
  BEcount <- 0
  sdm     <- sqrt(fact*mse)
  mlog    <- log(GMR)
  for (i in 1:chunks)
  {
    # simulate sample mean via its normal distribution
    means  <- rnorm(ns, mean=mlog, sd=sdm)
    # simulate sample mse via chi-square distribution of df*mses/mse
    mses   <- mse*rchisq(ns,df)/df
    hw     <- tval*sqrt(fact*mses)
    lCL <- means - hw
    uCL <- means + hw
    if(CI=="alternative"){
      # CI+, i.e. expand CI to contain GMR=1
      lCL <- ifelse(lCL<0, lCL, 0)
      uCL <- ifelse(uCL>0, uCL, 0)
    }
    # point  <- exp(means)
    lCL    <- exp(lCL)
    uCL    <- exp(uCL)
    # standard check if CI of GMR (usual or alternative)
    # is in 0.8-1.25 (without rounding CI limits)

    BE     <- (lBEL<=lCL & uCL<=uBEL)
    BEcount <- BEcount + sum(BE)
  }
  if (details) {
    cat(nsims,"sims. Time elapsed (sec):\n")
    print(proc.time()-ptm)
  }
  BEcount/nsims
}


And now try:
power.sim2x2(CV=0.2, n=24, GMR=0.95, CItype="usual")
[1] 0.89572
power.sim2x2(CV=0.2, n=24, GMR=0.95, CItype="alternative")
[1] 0.89572

Quod errat demonstrandum. At least for CV=0.2, n=24 and GMR=0.95 :-D.

BTW: It's not necessary to rely on simulations. The EMA BSWP f.i. doesn't like simulations :angry:. You may consider different cases for lCL, uCL to show that the alternative CI+ gives always the same decision as the usual 1-2*alpha CI.
F.i.:
If the usual CI is within the acceptance range (with upper CL below 1) -> decide BE
Then the alternative CI with upper CL set to 1 is also in the acceptance range -> decide BE.
And so on for other constellations.

Regards,

Detlew

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