Bioequivalence and Bioavailability Forum

Hi Nastia,

❝ Let us consider the situation when CV of C_max and AUC are very close to each other, like 21% and 20%, and for the first stage the number of subjects (n₁=20) was sufficient for AUC, but not for C_max.

❝ […] even then the power for AUC for the second stage would be always enough. […]

Wow, you are a master of condensed R-code! Here my version:

library(PowerTOST)
library(Power2Stage)
delta <- 0.01
CV.lo <- seq(0.1, 0.3, 0.01)
CV.hi <- CV.lo + delta
res   <- data.frame(CV.lo = CV.lo, CV.hi = CV.hi,
                    n1 = NA, N = NA,
                    power.1 = NA, power.2 = NA)
for (j in seq_along(CV.lo)) {
  res$n1[j]      <- sampleN.TOST(CV = CV.lo[j],
                                 print=FALSE)[["Sample size"]]
  if (res$n1[j]  < 12) res$n1[j] <- 12 # acc. to guidelines
  temp           <- sampleN2.TOST(CV = CV.hi[j], n1 = res$n1[j])
  res$N[j]       <- sum(temp[7:8])     # N = n1 + n2
  res$power.1[j] <- signif(suppressMessages(
                             power.TOST(alpha = 0.0294,
                                        CV = CV.lo[j],
                                        n = res$N[j] - 1)), 4)
  res$power.2[j] <- signif(suppressMessages(
                             power.TOST(alpha = 0.0294,
                                        CV = CV.hi[j],
                                        n = res$N[j] - 1)), 4)
}
cat("delta", delta, "\n"); print(res, row.names = FALSE)

delta 0.01
 CV.lo CV.hi n1  N power.1 power.2
  0.10  0.11 12 12  0.9561  0.9168
  0.11  0.12 12 12  0.9168  0.8664
  0.12  0.13 12 12  0.8664  0.8079
  0.13  0.14 12 14  0.8836  0.8334
  0.14  0.15 12 14  0.8334  0.7778
  0.15  0.16 12 16  0.8470  0.7974
  0.16  0.17 14 18  0.8539  0.8086
  0.17  0.18 14 20  0.8566  0.8146
  0.18  0.19 16 22  0.8567  0.8171
  0.19  0.20 18 24  0.8549  0.8173
  0.20  0.21 20 26  0.8517  0.8157
  0.21  0.22 22 28  0.8475  0.8129
  0.22  0.23 22 30  0.8426  0.8091
  0.23  0.24 24 32  0.8370  0.8045
  0.24  0.25 26 34  0.8310  0.7994
  0.25  0.26 28 36  0.8246  0.7937
  0.26  0.27 30 40  0.8392  0.8109
  0.27  0.28 32 42  0.8315  0.8038
  0.28  0.29 34 44  0.8238  0.7965
  0.29  0.30 38 48  0.8336  0.8081
  0.30  0.31 40 50  0.8253  0.8001

delta 0.05
 CV.lo CV.hi n1  N power.1 power.2
  0.10  0.15 12 14  0.9826  0.7778
  0.11  0.16 12 16  0.9813  0.7974
  0.12  0.17 12 18  0.9789  0.8086
  0.13  0.18 12 20  0.9757  0.8146
  0.14  0.19 12 22  0.9718  0.8171
  0.15  0.20 12 24  0.9673  0.8173
  0.16  0.21 14 26  0.9621  0.8157
  0.17  0.22 14 28  0.9565  0.8129
  0.18  0.23 16 30  0.9503  0.8091
  0.19  0.24 18 32  0.9438  0.8045
  0.20  0.25 20 34  0.9368  0.7994
  0.21  0.26 22 36  0.9296  0.7937
  0.22  0.27 22 40  0.9348  0.8109
  0.23  0.28 24 42  0.9271  0.8038
  0.24  0.29 26 44  0.9192  0.7965
  0.25  0.30 28 48  0.9226  0.8081
  0.26  0.31 30 50  0.9144  0.8001
  0.27  0.32 32 52  0.9061  0.7922
  0.28  0.33 34 56  0.9084  0.8005
  0.29  0.34 38 60  0.9098  0.8071
  0.30  0.35 40 62  0.9014  0.7987

Is this what you mean?

❝ ❝ Take some Schützomycin?

❝

❝ Did you patent that? I gonna make a generic

Not mine. It was mentioned for the first time by ElMaestro back in 2010:

❝ ❝ Let's say we want to develop a generic of Schützomycin. The product is available in one strength, posology is 1 tablet daily. […] Schützomycin is a nice drug with little safety concern.

My claim seems to be unfounded.

❝ ❝ According to the Q&A:

stage, sequence, sequence × stage, subject(sequence × stage), period(stage), treatment.

❝ Are there any documents to refer which mention this model (excepting the answer on the EMA's web page?)

Made up out of thin air by the EMA. To quote myself¹

In none of the published procedures, a test for poolability was part of the simulations. Although statistical tests could be constructed comparing variances of stages, their precision is poor in such designs and should be applied with caution. Nonetheless, in 2013, the European Medicines Agency introduced an additional term sequence×stage to the statistical model. Since both sequence and stage are between-subject effects, the residual error (hence, the CI) should not be affected – which was recently demonstrated.²

Excerpt²

Special emphasis was also given to the significance (P value) of the additional term ‘sequence × stage’ used in the ANOVA model proposed by EMA. […]
In almost all situations, the significance of the ‘sequence × stage’ term was found to be nonsignificant (i.e. values were greater than 5%). Only when GMR was close to the limit of 1.25 can the significance obtain lower values than the significance level 5%, and thus, the ‘sequence stage’ effect was declared significant.
The overall performance in terms of percentage of BE acceptance of the TSD remains unaltered. Plausibly, no difference in both the df and the SS values is observed for the ‘residual error’; the ‘sequence × stage’ is in essence a between-subject factor, whereas BE assessment is based on CV_w.
In any case, the EMA guideline does not clarify what the consequence would be if the ‘sequence × stage’ is statistically significant.

You don’t have to be a rocket scientist to understand that. Why the EMA introduced it, remains a mystery. Maybe influenced by García-Arieta and Gordon?³

A term for the stage should be included in the ANOVA model. However, the guideline does not clarify what the consequence should be if it is statistically significant. In principle, the data sets of both stages could not be combined.

Well, stage  is  already a factor in all published methods (didn’t they read them?). Concerning “poolability” see above. Reminds me on Grizzle’s nonsense for crossovers “if the sequence-effect is significant, analyze data of the first period as a parallel design”.

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1. Schütz H. Two-stage designs in bioequivalence trials. Eur J Clin Pharmacol. 2015; 71:271–81. doi:10.1007/s00228-015-1806-2.
   
2. Karalis V, Macheras P. On the statistical model of the two-stage designs in bioequivalence assessment. J Pharm Pharmacol. 2013; 66:48–52. doi:10.1111/jphp.12164.
   
3. García-Arieta A, Gordon J. Bioequivalence requirements in the European Union: critical discussion. AAPS J. 2012; 14:738–48. doi:10.1208/s12248-012-9382-1.