## Lower bound for power of two combined TOST? [Power / Sample Size]

Dear All!

Occasionally we have discussed here the impact on overall power if we decide BE based on two (or more) PK metrics, combined via 'and'. For instance here.
PowerTOST has a function power.2TOST() to deal with that problem.
But crucial is here the correlation argument, which is difficult to estimate. See this lengthy thread.

Quite recently I discovered something in that direction in the book

Patterson, Jones
Bioequivalence and Statistics in Clinical Pharmacology,
Second Edition, CRC Press, Boca Raton 2017
Chapter 5.7 "Determining Trial Size", page 134

Quote:
It is assumed for the purposes of this discussion that within-subject variability estimates are available, for both AUC and Cmax, to determine the trial size. For this purpose the larger of the two pooled estimates is of primary interest in calculations, for obvious reasons (i.e., power will be greater, or alternatively the probability of a Type 2 error will be lower, for the endpoint with smaller variation). However, the degree of this increase should be estimated using appropriate code (just switching the estimate of variability) to ensure adequate overall power for the study, as it is known [918] that

Power >= PAUC + PCmax - (2 - 1)

where PAUC is the estimate of power for AUC and PCmax is the estimate of power for Cmax. In the event that the overall power falls below the desired level, sample size may be increased to compensate, resulting in the desired level of power. For example, if power for Cmax is 0.90, and for AUC 0.95, the resulting overall study power is at least
0.9 + 0.95 - 1 = 0.85.

They use that formula or an analogous one also in other context for combining powers. Search for the reference [918] in the Patterson/Jones book.

My question(s): Does anybody knows where that formula came from?
Does anybody own the reference and can enlighten me?

[918] Nauta, J. (2010) Statistics in Clinical Vaccine Trials. Springer, London.

Regards,

Detlew

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