Bioequivalence and Bioavailability Forum

Hi Tina,

IMHO that’s a matter of taste. As an appetizer I can give you my procedure:

• Make sure that the clinical database is reviewed, locked, and can be reopened for adding data of the second stage. The last point is important. I once had to deal with a CRO telling me that they cannot open the database again and offered me preliminary data. That’s not acceptable.
  
• Make sure that you get final results from the analytical site. It’s not necessary to have an analytical report, but the data have to be reviewed and released by the QAU.
  
• Merge the data (administrations times, actual sampling times, concentrations, randomization) and perform NCA/biostats of the first stage.
  
• Based on the outcome (and following the decision tree of the chosen TSD method):
  • If BE is shown for all PK metrics, report it to the sponsor (formal written communication). The study will be stopped and all reports (clinical, analytical, statistical, and integrated) shall be prepared.
    
  • If not, continue:
• Perform the sample size estimation (n_total) based on the highest CV of relevant PK metrics.
  
• Calculate the preliminary sample size of the second stage n₂ = n_total – n₁.
  
• Based on the drop-out rate observed in stage 1, increase n₂ accordingly in order to maintain the desired power.¹
  
• The published methods were validated for balanced studies only. In order to prevent questions from nitpicking assessors, consider adjusting for imbalance in the second stage.¹ The wording I use in my protocols is:
  “If an unequal number of subjects in the respective sequences drop out in the first stage resulting in an imbalanced design, in the optional second stage subjects will be unequally randomized to sequences in order to counteract imbalance.”
  
• Communicate the sample size with the sponsor in writing.² The second stage will be initiated. Wait for the data. Merge them with the first data set. Business as usual.

All of the above should be unambiguously stated in the clinical study protocol and the SAP.

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1. Example:
     RT  TR       subjects / sequence
     12  12  24   n₁ dosed
     11  10  21   n₁ eligible
             14%  drop-out rate
             30   n_total (based on CV_max)
              9   preliminary n₂ (n_total – n₁)
             11   final n₂ (adjusted for drop-out rate)
      5   6  11   randomized/dosed in stage 2 (imbalanced to keep overall balance)
      4   5   9   expected eligible subjects in stage 2
     15  15  30   expected pooled data set (hopefully balanced)
   
2. CVs of relevant PK metrics and calculated sample size of the second stage. IMHO no need for a full report. An e-mail requesting acknowledgment is sufficient.

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)

Hi Tina,

❝ […] 2 stage design […] proceeding to a full sample size. In the current case, it is proceeding to the N planned for the next cohort.

Reading your post again I’m confused. Did you plan for a fixed sample size of the second stage?