Bioequivalence and Bioavailability Forum

Dear all,

the following may be a little bit of hair-splitting, but thus I'm known to you :

The formulas for the power and sample size are given in many references and obviously also implemented in software for power calculations and sample size estimation are formulated in terms of the total sample size.
Implicit assumption for these formulas is that the sequence groups are balanced, i.e. have the same number of subjects randomized to them. That would imply that for instance for the classical 2x2 cross-over design only even numbers of subjects are reasonable.

Nevertheless sometimes uneven numbers of subjects necessary to achieve a target power given the CV and true ratio are reported. See for instance Helmut's last famous lecture, slide 33. These may be in error or at least based on incorrect power values calculated via the formulas assuming a balanced design.

The key terms in the power calculations are the so called non-centrality parameters which are given in case of the classical 2x2 crossover with log-transformed values of th PK metrics as
  delta1 = (µT - µR - ln(lBE))/(se*sqrt(2/N))
  delta2 = (µT - µR - ln(uBE))/(se*sqrt(2/N))
with µT and µR the means for Test and Reference, se=sqrt(MSE), lBE and uBE the lower and upper acceptance ranges and N the total number of subjects.
If these parameters are to be calculated for unbalanced cross-over, i.e. different numbers of subjects in the sequence groups, the term sqrt(2/N) has to be replaced by
  sqrt((1/n1+1/n2)/2)
which gives different non-centrality parameters and in turn different power values if n1 not equal n2.

Here an example:
CV=10%, N=7 which can at its best (least unbalanced) realised by n1=3 and n2=4, whichever sequence TR or RT is 1 or 2 is unimportant.
# power.TOST also uses formulas based on assuming balanced sequence groups
# n is here the total number
> power.TOST(CV=0.1, n=7)
[1] 0.8625377
# modified code to account for "unbalancedness"
[1] 0.8560221

Ok, the difference is not that much and I must confess that I have not found any instance where the uneven number reported for 2x2 cross-over is invalidated by the power values calculated taking into account the "unbalancedness" if it is assumed only 1 subject.

But I think for cases where the power itself is of value, f.i. for the first step of the evaluation of a 2-stage design via Potvin's Method C where the power has to be checked, it can make a difference if the sample size is small compared to the imbalance.
# modified code to account for "unbalancedness"
# CV=10%, n1=2, n2=5 -> N=7
[1] 0.791085

Power: That which statisticians are always calculating but never have.
Stephen Senn