## ‘alpha’ of scaled ABE? [RSABE / ABEL]

Dear All!

We had already from time to time the question which combination of CVwR and GMR apply to find alpha ≤ 0.05 from power calculations. No one was really sure. Except the statement that power calculations at the conventional BE limits 1.25 (or 0.8) as for the ABE test will not be appropriate.

A rather natural choice IMHO would be to calculate the power at the border(s) of the widened BE acceptance limits.
Thus I employed power.scABEL() and power.RSABE() from R-package PowerTOST.

Here the results for each method on its own:
GMR=1.25 if CV<=0.3 or GMR=exp(0.8925742*CV2se(CV)) if CV>0.3 for the FDA RSABE
GMR=1.25 if CV<=0.3 or GMR=exp(0.7601283*CV2se(CV)) if CV>0.3 for the EMA scABEL and cap on if CV>0.5

    CV      GMR pBE_ABEL     GMR2 pBE_RSABE  0.200 1.250000 0.050220 1.250000  0.050153  0.225 1.250000 0.050632 1.250000  0.051969  0.250 1.250000 0.053143 1.250000  0.061009  0.275 1.250000 0.061006 1.250000  0.082647  0.300 1.250000 0.076460 1.250000  0.116120 ┐  0.302 1.251782 0.075453 1.301734  0.048548 ┘ FDA discontinuity  0.325 1.272352 0.067721 1.326887  0.047741  0.350 1.294853 0.064379 1.354484  0.046156  0.375 1.317485 0.062291 1.382326  0.043941  0.400 1.340231 0.059506 1.410391  0.041041  0.425 1.363072 0.055282 1.438658  0.037799  0.450 1.385991 0.049795 1.467104  0.034401  0.475 1.408971 0.043420 1.495709  0.031150  0.500 1.431997 0.036960 1.524452  0.028097  0.525 1.431997 0.040644 1.553312  0.025383  0.550 1.431997 0.043412 1.582270  0.022875  0.575 1.431997 0.045423 1.611308  0.020699  0.600 1.431997 0.046647 1.640406  0.018660  0.625 1.431997 0.047379 1.669548  0.016948  0.650 1.431997 0.047754 1.698717  0.015457  0.675 1.431997 0.047698 1.727897  0.014135  0.700 1.431997 0.047192 1.757073  0.012861  0.725 1.431997 0.046282 1.786230  0.011799  0.750 1.431997 0.045020 1.815356  0.010865  0.775 1.431997 0.043338 1.844438  0.010057  0.800 1.431997 0.041320 1.873464  0.009425

Interesting! At least for me .

Both methods start for low CV's with pBE ~ 0.05. Classical ABE evaluation controls alpha≤0.05 .
Approaching CV=0.3 the probability of (falsely?) accepting BE raises, for the FDA method up to 0.116. This is understandable since a increasing part of our simulated studies will have sample CV's >0.3 which are then evaluated with widened limits or downscaled linear scaled ABE criterion.
After/at CV=0.3 pBE for the FDA method drops sharply to or below 0.05 due to the known discontinuity in the widened BE limits and becomes with rising CV lower and lower (dominated by the point estimator criterion? conservative?). For the EMA method the pBE stays above 0.05 up to CV=0.45. The cap at CV≥0.5 then prevents the decrease for higher CV's seen in the FDA method. pBE first rises slightly with increasing CV. This behaviour (power increases with increasing CV) we have already seen here. But then a decrease is seen (due to the point estimator criterion?).

If we really estimating here the type I error then both methods are cases of an alpha inflation!? The FDA method in the vicinity of CV≤0.3 and the EMA method also but additionally up to CV=0.45. And the 'alpha inflation' is much more pronounced as that what the EMA lead to question Potvin method C.

Regards,

Detlew