Bioequivalence and Bioavailability Forum

Hi Mikalai,

❝ Are there any rules or recommendations for setting up the pre-specified limit (U) as a futility criterion?

Not really. You have to find a balance between the maximum study costs you are accepting to spend and the potential loss in power. Xu et al.* recommend a futility of 42 on n_total for CV ≤30% and 180 for CV >30%. Generally a small stage 1 sample size is not a good idea.

library(Power2Stage)
power.tsd.fC(method="B", alpha=c(0.0249, 0.0357), CV=0.25, n1=24,
             fCrit="CI", fClower=0.9374, max.n=42) # fixed GMR 0.95

TSD with 2x2 crossover
Method B: alpha (s1/s2) = 0.0249 0.0357
Interim power monitoring step included
Target power in power monitoring and sample size est. = 0.8
Power calculation via non-central t approx.
CV1 and GMR = 0.95 in sample size est. used
Maximum sample size max.n = 42
Futility criterion 90% CI outside 0.9374 ... 1.06678
BE acceptance range = 0.8 ... 1.25

CV = 0.25; n(stage 1) = 24; GMR = 0.95

1e+05 sims at theta0 = 0.95 (p(BE) = 'power').
p(BE)    = 0.83087
p(BE) s1 = 0.6057
Studies in stage 2 = 33.2%

Distribution of n(total)
- mean (range) = 27.9 (24 ... 42)
- percentiles
 5% 50% 95%
 24  24  42

power.tsd.fC(method="B", alpha=c(0.0249, 0.0357), CV=0.25, n1=24,
             fCrit="CI", fClower=0.9374, max.n=42, usePE=TRUE)
             # fully adaptive

TSD with 2x2 crossover
Method B: alpha (s1/s2) = 0.0249 0.0357
Interim power monitoring step included
Target power in power monitoring and sample size est. = 0.8
Power calculation via non-central t approx.
CV1 and PE1 in sample size est. used
Maximum sample size max.n = 42
Futility criterion 90% CI outside 0.9374 ... 1.06678
BE acceptance range = 0.8 ... 1.25

CV = 0.25; n(stage 1) = 24

1e+05 sims at theta0 = 0.95 (p(BE) = 'power').
p(BE)    = 0.87839
p(BE) s1 = 0.6057
Studies in stage 2 = 33.2%

Distribution of n(total)
- mean (range) = 30 (24 ... 42)
- percentiles
 5% 50% 95%
 24  24  42

Remember that if you deviate from one of the published methods (except by adding a futility which leads to early stopping) you have to assess the Type I Error. Fine with the setting above:

power.tsd.fC(method="B", alpha=c(0.0249, 0.0357), CV=0.25, n1=24,
             fCrit="CI", fClower=0.9374, max.n=42, usePE=TRUE,
             theta0=1.25)

TSD with 2x2 crossover
Method B: alpha (s1/s2) = 0.0249 0.0357
Interim power monitoring step included
Target power in power monitoring and sample size est. = 0.8
Power calculation via non-central t approx.
CV1 and PE1 in sample size est. used
Maximum sample size max.n = 42
Futility criterion 90% CI outside 0.9374 ... 1.06678
BE acceptance range = 0.8 ... 1.25

CV = 0.25; n(stage 1) = 24

1e+06 sims at theta0 = 1.25 (p(BE) = TIE 'alpha').
p(BE)    = 0.045069

The maximum inflation of the TIE is often observed at combinations of small n₁ and low CV. The minimum n₁ for Xu’s method is 18. With CV 10% we get a TIE of 0.035744.

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• Xu J, Audet C, DiLiberti CE, Hauck WW, Montague TH, Parr AF, Potvin D, Schuirmann DJ. Optimal adaptive sequential designs for crossover bioequivalence studies. Pharm Stat. 2016;15(1):15–27. doi:10.1002/pst.1721.