d_labes ★★★ Berlin, Germany, 20180725 10:40 Posting: # 19096 Views: 1,885 

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 withinsubject 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 least0.9 + 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 
ElMaestro ★★★ Denmark, 20180725 12:07 @ d_labes Posting: # 19097 Views: 1,721 

Hi d_labes, » Power >= PAUC + PCmax  (2  1) where PAUC is the estimate of power for AUC and PCmax is the estimate of power for Cmax.Surely that's either wrong or an approximation that has some validity within a set of strict conditions. I have not read that work or reference 918, but it looks outright wrong. With that equation power could become negative, go figure. By the way, the term pooled in the first sentence is also weird. The variances estimates for Cmax and AUCt are not "pooled", but I think they meant that the pool is just the dichotomous set of two variances. — I could be wrong, but... Best regards, ElMaestro 
d_labes ★★★ Berlin, Germany, 20180725 14:46 @ ElMaestro Posting: # 19100 Views: 1,679 

Dear ElMaestro, » » Power >= PAUC + PCmax  (2  1) where PAUC is the estimate of power for AUC and PCmax is the estimate of power for Cmax.» Surely that's either wrong or an approximation that has some validity within a set of strict conditions. I have not read that work or reference 918, but it looks outright wrong. Here we are two. You may read under the link https://books.google.de/books?id=e3byCOxj8FoC&pg=PA45 some more sparse details. Unfortunately the Appendix D with a proof of that formula is not available in Gooooogle books.» With that equation power could become negative, go figure. One nitpicking: The lower bound of power could become negative if both powers are <0.5, values no one would use in estimating sample sizes. If such a bound is reasonable is of course highly questionable. — Regards, Detlew 
mittyri ★★ Russia, 20180726 00:07 @ d_labes Posting: # 19102 Views: 1,687 

Dear Detlew, I hope Jozef won't be agry with that screenshot: — Kind regards, Mittyri 
d_labes ★★★ Berlin, Germany, 20180726 13:41 @ mittyri Posting: # 19103 Views: 1,616 

Dear mittyri, THX! Seems there is no "validity within a set of strict conditions", but rather general validity. The astonishing fact that the righthand side may become negative lies in the fact that instead of subtracting Pr(E1 ∪ E2), which is within 0 ... 1, the upper bound of that probability is used. — Regards, Detlew 
d_labes ★★★ Berlin, Germany, 20180726 14:29 @ d_labes Posting: # 19104 Views: 1,622 

Dear All! Let me ask another question regarding the quote from the Patterson/Jones book: 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. Example with both metrics with equal variabilities (CV of both metrics 0.25): sampleN.TOST(CV=0.25, targetpower=0.9) gives n=38 and power=0.908890.power.TOST(CV=0.25, n=38) gives also power=0.908890.lbound = 0.908890 + 0.908890 1 = 0.81778 Example with different variabilities (CVs 0.2 and 0.25): sampleN.TOST(CV=0.25, targetpower=0.9) gives n=38 and power=0.908890.power.TOST(CV=0.2, n=38) gives power=0.9805344lbound = 0.908890 + 0.9805344 1 = 0.8894244 Here we could set the targetpower for the first step lower than 0.9: sampleN.TOST(CV=0.25, targetpower=0.85) gives n=32 power=0.857257.power.TOST(CV=0.2, n=32) gives power=0.9595363lbound = 0.857257 + 0.9595363 1 = 0.8167933 Do I understand that paragraph correct or read to much into it? If I'm correct, do you think that such an approach is resonable? — Regards, Detlew 
Ben ★ 20180806 18:11 @ d_labes Posting: # 19154 Views: 1,471 

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 [531], 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. 
d_labes ★★★ Berlin, Germany, 20180807 11:50 @ Ben Posting: # 19157 Views: 1,414 

Dear Ben, nice to hear from you again. » interesting observation. Thanks for pointing this out. You are welcome. » ... » » 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? You are correct that there is no statement explicitly. The wording of the whole paragraph is not that clear understandable, to say it politely. Maybe I read to much into it. But the 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" smells for me in that direction. — Regards, Detlew 