d_labes
★★★

Berlin, Germany,
2018-07-25 12:40
(1890 d 10:57 ago)

Posting: # 19096
Views: 5,472

## 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  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  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?

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

Regards,

Detlew
ElMaestro
★★★

Denmark,
2018-07-25 14:07
(1890 d 09:30 ago)

@ d_labes
Posting: # 19097
Views: 4,897

## Lower bound for power of two combined TOST?

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.

Pass or fail!
ElMaestro
d_labes
★★★

Berlin, Germany,
2018-07-25 16:46
(1890 d 06:50 ago)

@ ElMaestro
Posting: # 19100
Views: 4,702

## Lower bound for power of two combined TOST?

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.

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,
2018-07-26 02:07
(1889 d 21:29 ago)

@ d_labes
Posting: # 19102
Views: 4,862

## Appendix D

Dear Detlew,

I hope Jozef won't be agry with that screenshot: Kind regards,
Mittyri
d_labes
★★★

Berlin, Germany,
2018-07-26 15:41
(1889 d 07:56 ago)

@ mittyri
Posting: # 19103
Views: 4,821

## Appendix D

Dear mittyri,

THX!
Seems there is no "validity within a set of strict conditions", but rather general validity.
The astonishing fact that the right-hand 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,
2018-07-26 16:29
(1889 d 07:08 ago)

@ d_labes
Posting: # 19104
Views: 4,714

## Use of lower bound for power of two combined TOST

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=0.9805344
lbound = 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=0.9595363
lbound = 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
★

2018-08-06 20:11
(1878 d 03:26 ago)

@ d_labes
Posting: # 19154
Views: 4,534

## Use of lower bound for power of two combined TOST

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.
d_labes
★★★

Berlin, Germany,
2018-08-07 13:50
(1877 d 09:46 ago)

@ Ben
Posting: # 19157
Views: 4,427

## Use of lower bound for power of two combined TOST

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  Ing. Helmut Schütz 