Bioequivalence and Bioavailability Forum

Hi Tina,

in case you have , a gimmick which keeps the overall balance in most cases. Quick, dirty, and almost untested.

TSD <- function(method1, method2, n1, n1.1, n1.2, CV, do.2) {
  ############################################################
  # Modify the function only if you know what you are doing! #
  ############################################################
  require(PowerTOST)            # what else?
  n1.e   <- n1.1+n1.2           # stage 1: eligible subjects
  n1.ar  <- n1.2/n1.1           # stage 1: sequence allocation ratio
  do.r   <- abs((n1.e-n1)/n1)   # stage 1: drop-out rate
  if(!missing(do.2)) do.2 <- do.2/100 # expect. drop-out rate stage 2
  if(missing(do.2)) do.2  <- do.r     # apply 1st if not given
  CV     <- CV/100
  if (method1 == 1) {adj <- 0.0294; GMR <- 0.95; pwr <- 0.8}
  if (method1 == 2) {adj <- 0.0280; GMR <- 0.90; pwr <- 0.8}
  if (method1 == 3) {adj <- 0.0284; GMR <- 0.95; pwr <- 0.9}
  if (method1 == 4) {adj <- 0.0274; GMR <- 0.95; pwr <- 0.9}
  if (method1 == 5) {adj <- 0.0269; GMR <- 0.90; pwr <- 0.9}
  if (method2 == 1) me <- "exact"
  if (method2 == 2) me <- "nct"
  if (method2 == 3) me <- "shifted"
  nt     <- as.numeric(sampleN.TOST(alpha=adj, CV=CV, theta0=GMR,
              targetpower=pwr, method=me, print=FALSE)[7])
  n2.p   <- nt-n1.e                # preliminary stage 2 sample size
  n2.1   <- ceiling((nt/2-n1.1)/(1-do.2)) # adjust for drop-outs
  n2.2   <- ceiling((nt/2-n1.2)/(1-do.2)) # adjust for drop-outs
  n2     <- n2.1+n2.2              # dosed in stage 2
  n2.1e  <- round(n2.1*(1-do.2), 0)# stage 2: expected eligible subjects in seq. 1
  n2.2e  <- round(n2.2*(1-do.2), 0)# stage 2: expected eligible subjects in seq. 2
  n2.e   <- n2.1e+n2.2e            # stage 2: expected eligible subjects
  n2.ar  <- n2.2e/n2.1e            # stage 2: sequence allocation ratio
  ar     <- (n1.2+n2.2e)/(n1.1+n2.1e) # pooled data’s allocation ratio
  ifelse(ar == 1, bal <- "(balanced)", bal <- "(imbalanced)")
  sep    <- paste(paste0(rep("\u2500", 49), collapse=""), "\n")
  if(method2 > 1) me <- c(me, "t-distribution")
  cat("\n TSD-method:", method1,
  paste0("(\u03b1=", adj, ", GMR=", 100*GMR, "%, power=", 100*pwr, "%)\n"),
  "Sample size estimation:", me, "\n", sep,
  "Stage 1\n", sep,
  "randomized/dosed subjects …………………………………", n1, "\n",
  "eligible subjects (drop-outs)…………………………", n1.e, paste0("(", n1-n1.e,")"), "\n",
  "eligible subjects in sequences RT|TR ……", paste0(n1.1, "|", n1.2), "\n",
  "allocation ratio RT/TR …………………………………………", paste0("1:", signif(n1.ar, 4)), "\n",
  "drop-out rate  ………………………………………………………………", paste0(signif(100*do.r, 4), "%\n"), sep,
  "Interim analysis\n", sep,
  "relevant PK metrics’ maximum CV …………………", paste0(signif(100*CV, 4), "%\n"),
  "estimated total sample size ……………………………", as.numeric(nt), "\n", sep,
  "Stage 2\n", sep,
  "preliminary sample size ………………………………………", n2.p, "\n",
  "expected drop-out rate …………………………………………", paste0(signif(100*do.2, 4),"%\n"),
  "final sample size (adj. for drop-outs) ", n2, "\n",
  "randomized subjects in sequences RT|TR ", paste0(n2.1, "|", n2.2), "\n",
  "expected eligible subj. in seq. RT|TR …", paste0(n2.1e, "|", n2.2e), "\n",
  "allocation ratio RT/TR …………………………………………", paste0("1:", signif(n2.ar, 4)), "\n", sep,
  "Pooled data set\n", sep,
  "expected eligible subjects ………………………………", n1.e+n2.e, "\n",
  "expected eligible subj. in seq. RT|TR …", paste0(n1.1+n2.1e, "|", n1.2+n2.2e), "\n",
  "allocation ratio RT/TR …………………………………………", paste0("1:", signif(ar, 4)), bal, "\n\n")
}
################
# Stage 1 data #
################
method1 <- 1 # select from TSD-Methods 1–5
###################################################
#                                     GMR% power% #
# 1 Potvin et al. (2008) Methods B/C:  95    80   #
# 2 Montague et al. (2011) Method D:   90    80   #
# 3 Fuglsang (2013) Method B:          95    90   #
# 4 Fuglsang (2013) Method C1/D1:      95    90   #
# 5 Fuglsang (2013) Method C2/D2:      90    90   #
###################################################
method2 <- 1 # select from power-estimation Methods 1–3
#######################################
# 1 exact (Owen’s Q-function): best   #
# 2 noncentral t-distribution: better #
# 3 shifted t-distribution:    good   #
#######################################
n1   <- 24 # dosed subjects
n1.1 <- 11 # eligible subjects in sequence RT
n1.2 <- 7  # eligible subjects in sequence TR
CV   <- 23 # maximum CV in percent
TSD(method1, method2, n1, n1.1, n1.2, CV) # call the function

A little bit more extreme than my previous example:

TSD-method: 1 (α=0.0294, GMR=95%, power=80%)
Sample size estimation: exact
─────────────────────────────────────────────────
Stage 1
─────────────────────────────────────────────────
randomized/dosed subjects ………………………………… 24
eligible subjects (drop-outs)………………………… 18 (6)
eligible subjects in sequences RT|TR …… 11|7
allocation ratio RT/TR ………………………………………… 1:0.6364
drop-out rate  ……………………………………………………………… 25%
─────────────────────────────────────────────────
Interim analysis
─────────────────────────────────────────────────
relevant PK metrics’ maximum CV ………………… 23%
estimated total sample size …………………………… 30
─────────────────────────────────────────────────
Stage 2
─────────────────────────────────────────────────
preliminary sample size ……………………………………… 12
expected drop-out rate ………………………………………… 25%
final sample size (adj. for drop-outs)  17
randomized subjects in sequences RT|TR  6|11
expected eligible subj. in seq. RT|TR … 4|8
allocation ratio RT/TR ………………………………………… 1:2
─────────────────────────────────────────────────
Pooled data set
─────────────────────────────────────────────────
expected eligible subjects ……………………………… 30
expected eligible subj. in seq. RT|TR … 15|15
allocation ratio RT/TR ………………………………………… 1:1 (balanced)

Be sure to use the same sample size estimation method which you applied in the power estimation of the decision scheme. Available are "exact" (based on Owen’s Q function, preferred), "nct" (based on the noncentral t-distribution), and "shifted" (based on the shifted central t-distribution).
Sometimes crazy things happen. If your area was hit be a flu epidemic during the first stage, it might be reasonable to assume a lower drop-out rate for the second stage. Use the optional variable do.2 (expected drop-out rate in percent). Example: Method D, noncentral t, and a 10% drop-out rate:

TSD-method: 2 (α=0.028, GMR=90%, power=80%)
Sample size estimation: noncentral t-distribution
─────────────────────────────────────────────────
Stage 1
─────────────────────────────────────────────────
randomized/dosed subjects ………………………………… 24
eligible subjects (drop-outs)………………………… 18 (6)
eligible subjects in sequences RT|TR …… 11|7
allocation ratio RT/TR ………………………………………… 1:0.6364
drop-out rate  ……………………………………………………………… 25%
─────────────────────────────────────────────────
Interim analysis
─────────────────────────────────────────────────
relevant PK metrics’ maximum CV ………………… 23%
estimated total sample size …………………………… 60
─────────────────────────────────────────────────
Stage 2
─────────────────────────────────────────────────
preliminary sample size ……………………………………… 42
expected drop-out rate ………………………………………… 10%
final sample size (adj. for drop-outs)  48
randomized subjects in sequences RT|TR  22|26
expected eligible subj. in seq. RT|TR … 20|23
allocation ratio RT/TR ………………………………………… 1:1.15
─────────────────────────────────────────────────
Pooled data set
─────────────────────────────────────────────────
expected eligible subjects ……………………………… 61
expected eligible subj. in seq. RT|TR … 31|30
allocation ratio RT/TR ………………………………………… 1:0.9677 (imbalanced)