Bioequivalence and Bioavailability Forum

Dear Osama and neds,

I updated the development version 1.5.2.9000 of PowerTOST on GitHub (it’s not on CRAN yet). If you want to give it a try:

install.packages("remotes")
remotes::install_github("Detlew/PowerTOST")

Some examples in the following. Business as usual (ABE):

sampleN.TOST(CV = 0.29, theta0 = 0.9, design = "2x2x4")

+++++++++++ Equivalence test - TOST +++++++++++
            Sample size estimation
-----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)

alpha = 0.05, target power = 0.8
BE margins = 0.8 ... 1.25
True ratio = 0.9,  CV = 0.29

Sample size (total)
 n     power
38   0.814460

Note that we are close to the switching CV_wR 30%. What about ABEL?

sampleN.scABEL(CV = 0.29, theta0 = 0.9, design = "2x2x4",
               regulator = "EMA", details = FALSE)

+++++++++++ scaled (widened) ABEL +++++++++++
            Sample size estimation
   (simulation based on ANOVA evaluation)
---------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1e+05 studies for each step simulated.

alpha  = 0.05, target power = 0.8
CVw(T) = 0.29; CVw(R) = 0.29
True ratio = 0.9
ABE limits / PE constraint = 0.8 ... 1.25
Regulatory settings: EMA

Sample size
 n     power
34   0.8095

Lower sample size than for ABE but – as usual – we would face an inflated Type I Error of 0.07095.

Using the new argument regulator = "GCC".

sampleN.scABEL(CV = 0.29, theta0 = 0.9, design = "2x2x4",
               regulator = "GCC", details = FALSE)

+++++++++++ scaled (widened) ABEL +++++++++++
            Sample size estimation
   (simulation based on ANOVA evaluation)
---------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1e+05 studies for each step simulated.

alpha  = 0.05, target power = 0.8
CVw(T) = 0.29; CVw(R) = 0.29
True ratio = 0.9
ABE limits / PE constraint = 0.8 ... 1.25
Regulatory settings: GCC

Sample size
 n     power
28   0.8014

Even lower sample size than for ABEL because sometimes the CV is misclassified and widened limits of 75.00–133.33% are applied. What about the Type I Error for this approach?

power.scABEL(CV = 0.29, theta0 = 1.25, design = "2x2x4",
             n = 28, regulator = "GCC")
[1] 0.14573

Nasty. Due to the misclassification a huge inflation of the TIE (patient’s risk more than twice of ABEL).
Iteratively adjust α:

scABEL.ad(CV = 0.29, theta0 =0.9, design = "2x2x4",
          n = 28, regulator = "GCC")

+++++++++++ scaled (widened) ABEL ++++++++++++
         iteratively adjusted alpha
   (simulations based on ANOVA evaluation)
----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.

CVwR 0.29, CVwT 0.29, n(i) 14|14 (N 28)
Nominal alpha                 : 0.05
True ratio                    : 0.9000
Regulatory settings           : GCC (ABE)
Switching CVwR                : 0.3
BE limits                     : 0.8000 ... 1.2500
PE constraints                : 0.8000 ... 1.2500
Empiric TIE for alpha 0.0500  : 0.14573
Power for theta0 0.9000       : 0.801
Iteratively adjusted alpha    : 0.01102
Empiric TIE for adjusted alpha: 0.05000
Power for theta0 0.9000       : 0.602

Substantial loss in power due to evaluation by the 100(1–2×0.01102)=97.796% CI.
Increase the sample size to maintain power (show progress of iterations):

sampleN.scABEL.ad(CV = 0.29, theta0 = 0.9, design = "2x2x4",
                  regulator = "GCC", progress = TRUE, details = TRUE)

+++++++++++ scaled (widened) ABEL ++++++++++++
            Sample size estimation
        for iteratively adjusted alpha
   (simulations based on ANOVA evaluation)
----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.

Assumed CVwR 0.29, CVwT 0.29
Nominal alpha      : 0.05
True ratio         : 0.9000
Target power       : 0.8
Regulatory settings: GCC (ABE)
Switching CVwR     : 0.3
BE limits          : 0.8000 ... 1.2500
PE constraints     : 0.8000 ... 1.2500
Progress of each iteration:

n  28, nomin. alpha: 0.05000 (power 0.8014), TIE: 0.1457

Sample size search and iteratively adjusting alpha
n  28,   adj. alpha: 0.01102 (power 0.6016), rel. impact on power: -24.94%
n  48,   adj. alpha: 0.00488 (power 0.7372)
n  46,   adj. alpha: 0.00529 (power 0.7274)
n  48,   adj. alpha: 0.00488 (power 0.7372)
n  50,   adj. alpha: 0.00452 (power 0.7461)
n  52,   adj. alpha: 0.00419 (power 0.7550)
n  54,   adj. alpha: 0.00389 (power 0.7629)
n  56,   adj. alpha: 0.00359 (power 0.7703)
n  58,   adj. alpha: 0.00333 (power 0.7787)
n  60,   adj. alpha: 0.00312 (power 0.7850)
n  62,   adj. alpha: 0.00289 (power 0.7924)
n  64,   adj. alpha: 0.00268 (power 0.7989)
n  66,   adj. alpha: 0.00251 (power 0.8058), TIE: 0.05000
Compared to nominal alpha's sample size increase of 135.7% (~study costs).

Runtime    : 79 seconds
Simulations: 97,500,000

Note that the TIE depends strongly on the sample size. Hence, in every step we have to adjust α as well. OK, with 66 subjects we achieve the target power but it comes with a price, namely evaluation by a 99.498% CI…

Inspect the plots of this post again. If the true CV_wR > 30% it might misclassified as well but this time towards the conventional limits and the TIE is not inflated. Hence, we get the same sample size by

sampleN.scABEL(CV = 0.31, theta0 = 0.9, design = "2x2x4",
               regulator = "GCC", details = FALSE)

+++++++++++ scaled (widened) ABEL +++++++++++
            Sample size estimation
   (simulation based on ANOVA evaluation)
---------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1e+05 studies for each step simulated.

alpha  = 0.05, target power = 0.8
CVw(T) = 0.31; CVw(R) = 0.31
True ratio = 0.9
ABE limits / PE constraint = 0.8 ... 1.25
Widened limits = 0.75 ... 1.333333
Regulatory settings: GCC

Sample size
 n     power
28   0.8135

and

sampleN.scABEL.ad(CV = 0.31, theta0 = 0.9, design = "2x2x4",
                  regulator = "GCC", details = FALSE)

+++++++++++ scaled (widened) ABEL ++++++++++++
            Sample size estimation
        for iteratively adjusted alpha
   (simulations based on ANOVA evaluation)
----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.

Assumed CVwR 0.31, CVwT 0.31
Nominal alpha      : 0.05
True ratio         : 0.9000
Target power       : 0.8
Regulatory settings: GCC (ABEL)
Switching CVwR     : 0.3
Regulatory constant: 0.97998
Widened limits     : 0.7500 ... 1.3333
PE constraints     : 0.8000 ... 1.2500

n  28, nomin. alpha: 0.05000 (power 0.8135), TIE: 0.0263
No inflation of the TIE expected; hence, no adjustment of alpha required.

Inflation of the Type I error in different approaches (2-sequence, 4-period full replicate designs):

1. Conventional ABE with fixed limits never exceeds nominal α. Since TOST is not a most powerful test, for high CVs combined with small sample sizes, the TIE will be below nominal α. All is good.
   
2. For the EMA’s ABEL the maximum inflation occurs at CV_wR 30%. If CV_wR increases, the TIE decreases since the probability of a misclassification decreases as well. Starting with the upper scaling cap at CV_wR 50% limits are fixed and the TIE is driven by the conservatism of TOST – together with the PE-constraint. However, even for very high CVs (not shown) the TIE doesn’t exceed nominal α.
   
3. Similar for Health Canada’s ABEL though the minimum TIE is observed at it’s upper cap of ~57.38%.
   
4. For the GCC’s widened limits huge inflation of the TIE if CV_wR ≤30% (the highest of all approaches). Strong dependency on the sample size. Behaves with increasing CVs like TOST.
   
5. Huge inflation of the TIE for the FDA’s RSABE with implied limits if CV_wR <30%. Moderate to extremely conservative otherwise. That’s the model all (‼) authors (except ones of the FDA¹) considered the applicable one since products are approved according to this model and not according to f.
   
6. Lower inflation of the TIE by the FDA’s RSABE “desired consumer risk model”.¹ No more than a mathematical pre­stidigitation and called even “hocus pocus” by some.² Here the maximum inflation of the TIE occurs at ~25.4%.
   

---

1. Davit BM, Chen ML, Conner DP, Haidar SH, Kim S, Lee CH, Lionberger RA, Makhlouf FT, Nwa­kama PE, Patel DT, Schuirmann DJ, Yu LX. Implementation of a Reference-Scaled Average Bioequivalence Approach for Highly Variable Generic Drug Products by the US Food and Drug Administration. AAPS J. 2012: 14(4); 915–24. doi:10.1208/s12248-012-9406-x. PMC Free Full text.
   
2. Detlew Labes, László Endrényi, myself…

---

Edit 2021-01-18: PowerTOST 1.5-3 on CRAN.