Bioequivalence and Bioavailability Forum

❝ Yes, as far as I understood, the paper uses ANOVA,…

❝ […] the “Two-at-a-Time” approach with unequal variances would have been the better choice (but again leading to the problem of contructing RCIs for the inverse normal combination test and the need to use Potvin's algorithms).

❝ Hope to see you again in person soon.

❝ hope you are doing fine and my post here fits to the subject.

❝ I`m wondering how to show a power achieved after completion of the first stage in parallel design using Power2Stage functions, according to Potvin C algo?

n1 <- c(50, 49) # 110 planned, 11 dropouts (different in the groups)
CV <- 0.45      # observed
PE <- 0.93      # observed (worse than the 0.95 assumed)
TSD.par(method = "B", CV, n1, PE)
Parallel design
Method B (‘Type 1’ TSD)
adj   = 0.0294 (both stages)
observed values in stage 1:
   CV = 0.45
   PE = 0.93
   n1 = 50 and 49 / per group (total 99)
interim power with adj 0.0294
not BE in stage 1, 94.12% CI: 78.85% - 109.69%
interim power = 0.4292, initiate stage 2 with 93 subjects

TSD.par(method = "C", CV, n1, PE)
Parallel design
Method C (‘Type 2’ TSD)
alpha = 0.05 (conditional on power)
adj   = 0.0294 (conditional)
observed values in stage 1:
   CV = 0.45
   PE = 0.93
   n1 = 50 and 49 / per group (total 99)
interim power with alpha 0.05
interim power = 0.5648
not BE in stage 1, 94.12% CI: 78.85% - 109.69%
initiate stage 2 with 93 subjects

n1 <- c(90, 89)
CV <- 0.45
PE <- 0.90
TSD.par(method = "B", CV, n1, PE)
Parallel design
Method B (‘Type 1’ TSD)
adj   = 0.0294 (both stages)
observed values in stage 1:
   CV = 0.45
   PE = 0.9
   n1 = 90 and 89 / per group (total 179)
interim power with adj 0.0294
not BE in stage 1, 94.12% CI: 79.66% - 101.69%
interim power = 0.7714, initiate stage 2 with 13 subjects

TSD.par(method = "C", CV, n1, PE)
Parallel design
Method C (‘Type 2’ TSD)
alpha = 0.05 (conditional on power)
adj   = 0.0294 (conditional)
observed values in stage 1:
   CV = 0.45
   PE = 0.9
   n1 = 90 and 89 / per group (total 179)
interim power with alpha 0.05
interim power = 0.8421
BE in stage 1, 90.00% CI: 80.94% - 100.08%

❝ When I do it for crossover study using interim.tsd.in function I can extract the value of Power Stage 1 component, but how about parallel study?

TSD.par <- function(method = "B", CV, n1, PE) {
  library(PowerTOST)
  library(Power2Stage)
  # values of Fuglsang’s paper; don’t change!
  alpha  <- 0.05   # for Method C in stage 1
  adj    <- 0.0294 # adjusted alpha (conditionally in Method C, both stages in B)
  # Note: theta0 and target are fixed
  #       CIs in % rounded to two deciaml places acc. to GLs
  theta0 <- 0.95
  target <- 0.80
  info <- paste("\nParallel design")
  if (method == "B") {
    info <- paste(info, "\nMethod B (‘Type 1’ TSD)",
                  "\nadj   =", adj, "(both stages)")
  } else {
    info <- paste(info, "\nMethod C (‘Type 2’ TSD)",
                  "\nalpha =", alpha, "(conditional on power)",
                  "\nadj   =", adj, "(conditional)")
  }
  info <- paste(info, "\nobserved values in stage 1:",
                "\n   CV =", CV,
                "\n   PE =", PE,
                "\n   n1 =", paste(n1, collapse = " and "), "/ per group",
                sprintf("(total %.0f)", sum(n1)),
                "\ninterim power with")
  ifelse (method == "B",
    info <- paste(info, "adj", adj, "\n"),
    info <- paste(info, "alpha", alpha))
  cat(info)
  # the sample size of stage 2 is the same for both methods
  n2 <- sampleN2.TOST(alpha = adj, CV = CV, n1 = sum(n1), theta0 = theta0,
                      targetpower = target, design = "parallel")[["Sample size"]]
  prt.CI <- function(alpha, CI, lf = TRUE) {
    tmp <- sprintf("%.2f%% CI: %.2f%% - %.2f%%",
                   100 * (1 - 2 * alpha), CI[["lower"]], CI[["upper"]])
    if (lf) tmp <- paste0(tmp, "\n")
    return(tmp)
  }
  if (method == "B") {
    # we check for BE with adjusted alpha first
    CI.B <- round(100 * CI.BE(alpha = adj, pe = PE, CV = CV,
                              n = n1, design = "parallel"), 2)
    if (CI.B[["lower"]] >= 80 & CI.B[["upper"]] <= 125) {
      cat("BE in stage 1,", prt.CI(adj, CI.B, TRUE))
    } else {                # failed
      pwrB <- power.TOST(alpha = adj, CV = CV, theta0 = theta0,
                         design = "parallel", n = n1)
      if (pwrB >= target) { # stop because sufficient power
        cat("not BE in stage 1,", prt.CI(adj, CI.B, FALSE),
            "\nstop, because interim power =", signif(pwrB, 4), "\n")
      } else {              # low power → stage 2
        cat("not BE in stage 1,", prt.CI(adj, CI.B, FALSE),
            paste0("\ninterim power = ", signif(pwrB, 4),
                   ", initiate stage 2 with ", n2, " subjects\n"))
      }
    }
  } else {
    # we check power with unadjusted alpha first
    pwrC <- power.TOST(alpha = alpha, CV = CV, theta0 = theta0,
                       design = "parallel", n = n1)
    cat("\ninterim power =", signif(pwrC, 4))
    if (pwrC >= target) {
      # check for BE with unadjusted alpha first
      CI.C1 <- round(100 * CI.BE(alpha = alpha, pe = PE, CV = CV,
                                 n = n1, design = "parallel"), 2)
      if (CI.C1[["lower"]] >= 80 & CI.C1[["upper"]] <= 125) { # pass
        cat("\nBE in stage 1,", prt.CI(alpha, CI.C1, TRUE))
      } else {                                                # fail
        cat("\nnot BE in stage 1,", prt.CI(alpha, CI.C1, TRUE))
      }
    } else {
      # sufficient power, check for BE with adjusted alpha
      CI.C2 <- round(100 * CI.BE(alpha = adj, pe = PE, CV = CV,
                                 n = n1, design = "parallel"), 2)
      if (CI.C2[["lower"]] >= 80 & CI.C2[["upper"]] <= 125) { # pass
        cat("\nBE in stage 1,", prt.CI(adj, CI.C2, TRUE))
      } else {                                                # fail → stage 2
        cat("\nnot BE in stage 1,", prt.CI(adj, CI.C2, FALSE),
            "\ninitiate stage 2 with", n2, "subjects\n")
      }
    }
  }
}