Bioequivalence and Bioavailability Forum

Dear All!

Today is my birthday and, in the tradition of the hobbits, here is my birthday gift for you - a long sermon.

During development of the functions for power/sample size for scaled ABE in PowerTOST I noticed that for the 2x3x3 crossover design using the EMA recommended evaluation the power values calculated via subject data sims in case of CVwT<CVwR were markedly higher, in case of CVwT>CVwR they were markedly lower compared to the simulations of the key statistics pe, mse and σ²_wR used for high-speed simulations.

To explore into this the question arose “How performs the EMA recommended evaluation of the replicate designs (“Use the same ANOVA model as for the classical 2x2x2 crossover”) for deciding ABE in terms of type I error (alpha), especially if the homoscedasticity assumption is not true?”
The following table summarizes the simulated alpha values via subject data:

CVwT   CVwR   pooled
               CV      n   'alpha' power.TOST
0.3    0.3    0.3     12   0.0440   0.0445
                      24   0.0505   0.0500
                      36   0.0500   0.0501
0.4    0.4    0.4     12   0.0164   0.0164
                      24   0.0483   0.0482
                      36   0.0500   0.0500
0.5    0.5    0.5     12   0.0027   0.0028
                      24   0.0323   0.0324
                      36   0.0484   0.0484
0.3    0.4    0.3690  12   0.0202   0.0251
                      24   0.0361   0.0495
                      36   0.0363   0.0500
0.3    0.5    0.4407  12   0.0068   0.0084
                      24   0.0267   0.0443
                      36   0.0285   0.0498
0.4    0.3    0.3359  12   0.0434   0.0354
                      24   0.0659   0.0499
                      36   0.0663   0.0500
0.5    0.3    0.3754  12   0.0319   0.0232
                      24   0.0757   0.0493
                      36   0.0783   0.0500
power.TOST results calculated with pooled CV (= mse2CV((CV2mse(CVwT) + 2* CV2mse(CVwR))/3))

Wow! While performing as expected if σ²_wT=σ²_wT, comparable to the evaluation via power.TOST(), the empirical alpha values via subject data simulations are much too conservative in case of CVwT<CVwR. In case of CVwT>CVwR they are too liberal up to a considerable alpha inflation!

This observation resembles well known results for one-way or two-way ANOVA, showing that the usual F-test for testing effects are no longer valid, may be too liberal or too conservative, if the assumption of equal variances is violated.

The only explanation could be that the distributional assumption (“mse is chi-squared distributed”) no longer holds if the homoscedasticity is not true. As far as I know there is no way out here since there is no solution to the question of the mse distribution within the crossover ANOVA for the case of heteroscedasticity, beside to use mixed model software. More over the EMA forced us to use this fixed effects ANOVA without allowing mixed models evaluation.

Thus we had to stuck with subject data simulations with the burden of very long simulation run-times if we wish to calculate empirical power / alpha for the EMA method within a 2x3x3 design with all bells and whistles.

So far so bad. Any body out there to prove me wrong?

BTW: Do you remember the regulatory body abandoning Potvin C for 2-stage studies because a maximum of empirical alpha's of 0.0510 were reported in the Potvin et.al. paper, claiming an alpha-inflation for that method ?