## Use of lower bound for power of two combined TOST [Power / Sample Size]

Dear Detlew,

interesting observation. Thanks for pointing this out.

The formula is indeed very general and no assumptions are needed. Which makes it also (maximally) conservative.

Yes, the text reads as if they would recommend this calculation. However, I have mixed feelings here. The endpoints AUC and Cmax are typically highly correlated* and therefore a multiplicity adjustment regarding Power is not needed (it often suffices to just use the higher variability; or more generally speaking: calculate sample size for AUC and Cmax separately and then take the higher one). They even write this in Section 3.6:
No adjustment is made for multiplicity of endpoints AUC and Cmax , and the larger variance of logAUC or logCmax is typically used in the power sample size calculations.

To me such a formula (or another concept of overall power) makes sense in case we have for example 2 analytes. One should then adjust power regarding this multiple comparison. Often it is not clear how they are correlated and then I tend to assume they are independent to be on the safe side. Thus the overall power formula is power(analyte 1) * power(analyte 2) (in contrast to the formula from the book). It is interesting that even this rather conservative assumption of complete independence is slightly better in terms of higher power / requiring less subjects (as compared to the formula from the book). (I have no general proof of it at hand, but for the examples it was true).

» Seems to me that the authors recommend to use the lower bound as criterion for setting the targetpower, i.e. if an overall power of 0.8 is aimed for, the powers for the two metrics have to be chosen such that the lower bound >= 0.8.

Maybe I missed it but I don't see this statement explicitly. Can you point me to the direction? In case of the analytes example my personal target power would still be 90%. For sensitivity scenarios we can aim for 80 (ish).

* Maybe this is not true after all, then the approach of taking an overall power would make sense to me.

Best regards,
Ben. Ing. Helmut Schütz 