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Supplementary Material 1 to
Inflation of Type I Error in the Evaluation of Scaled Average
Bioequivalence, and a Method for its Control
Detlew Labes • Helmut Schütz
Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its
Control
Supplementary Material 1
The supplementary material contains detailed information about the methods employed and results.
All simulations were performed with a fixed seed
of R’s Mersenne-Twister pseudo-random number
generator to support reproducing our results. Thus,
the default setseed = TRUE in power.scABEL() and
sampleN.scABEL() was used. In the assessment of
convergence (Fig. 1) random seeds (created from
the current time and the process ID) were used.
Empiric type I errors (TIE) for a nominal α 0.05
obtained in 1,000,000 simulated BE studies of the
two-sequence full replicate designs RTRT | TRTR,
RTR | TRT, and the partial replicate design RRT |
RTR | TRR with sample sizes for a range of expected GMRs of 0.85 to 0.95, a range of target powers
of 80% to 90%, and a range of CVwR 25% to 42.5%
are presented in the Supplementary material 2,
Tables I a – VIII a.
Adjusted alphas required to maintain the consumer risk ≤ 0.05, resulting sample sizes and power
are presented in Tables I b – VIII b. 100,000 BE
studies were simulated for power and 1,000,000 in
updating the adjusting α for increased sample sizes.
A comparison of empiric type I errors obtained
by power.scABEL() to subject simulations via lm()
is given in Table IX and presented in Fig. 2.
An example (RTRT | TRTR design, expected
GMR 0.90, target power 80%; see also the respective entries in Table V and XI) demonstrates that
power converges substantially faster than the empiric type I error is presented in Fig. 1. Hence, the
chosen number of simulations is justified. Note that
due to the lower α the sample size has to be increased from 34 to 42 in order to maintain the target power.
Two examples of the application of PowerTOST
are given in the following.
Function scABEL.ad(): CVwR 0.30, n1 15, n2 17
(unbalanced), 4-period full replicate design, regulator EMA.
library(PowerTOST)
scABEL.ad(CV=0.30, n=c(15, 17), design="2x2x4",
regulator="EMA")
gives
2
+++++++++ scaled (widened) ABEL +++++++++
iteratively adjusted alpha
----------------------------------------Study design: 2x2x4 (RTRT|TRTR)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.
CVwR 0.3, n(i) 15|17 (N 32)
Nominal alpha
:
Null (true) ratio
:
Regulatory settings
:
Empiric TIE for alpha 0.0500 :
Power for theta0 0.900
:
Iteratively adjusted alpha
:
Empiric TIE for adjusted alpha:
Power for theta0 1.150
:
0.05
0.900
EMA (ABE)
0.08121
0.781
0.02865
0.0500
0.701
With an empiric type I error of 0.08121 the consumer risk is compromised. Cmax should be evaluated with the iteratively adjusted α of 0.02865 (i.e.,
via a 94.27% confidence interval).
Function sampleN.scABEL.ad(): expected CVwR
0.35, GMR 1.15, 4-period full replicate, desired
(target) power 80%, regulator EMA.
library(PowerTOST)
sampleN.scABEL.ad(CV=0.35, theta0=1.15,
design="2x2x4", targetpower=0.8,
regulator="EMA", details=TRUE)
gives
+++++++++ scaled (widened) ABEL +++++++++
Sample size estimation
for iteratively adjusted alpha
----------------------------------------Study design: 2x2x4 (RTRT|TRTR)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.
Expected CVwR 0.35
Nominal alpha
:
Null (true) ratio :
Target power
:
Regulatory settings:
Switching CVwR
:
Regulatory constant:
Expanded limits
:
Upper scaling cap :
PE constraints
:
n 56, nomin. alpha:
n
64,
0.05
1.150
0.8
EMA (ABEL)
30%
0.760
0.7723...1.2948
CVwR 0.5
0.8000...1.2500
0.0500 (power 0.8036),
TIE: 0.0651
adj. alpha: 0.03621 (power 0.8093),
TIE: 0.05000
With nominal α a sample size of 56 would be required to achieve 80.4% power. However, the consumer risk is not maintained (TIE 0.0651). With the
iteratively adjusted α of 0.03621 the required sample size would increase to 64 (power 80.9%). Since
study costs are almost linear related to sample size
an increase of approximately 14% should be expected.
3
Figure 1
Labes and Schütz
Convergence of simulations; expected GMR 0.90, target power 80%. Empiric type I error (left
panels) and power (right panels). Nominal α 0.05 (top panels) and adjusted α 0.0283 (bottom
panels). Ten runs with different seeds of the Mersenne-Twister pseudo-random number generator. In each run 10,000 –1,000,000 simulations for the TIE and 1,000 – 100,000 for power.
Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its
Control
Supplementary Material 1
Figure 2
4
Comparison of empiric type I error for the 216 explored scenarios obtained by power.scABEL()
and subject simulations via lm(). Orthogonal regression (left panel) and Bland-Altman plot
(right panel).