## The unknown ρ [Power / Sample Size]

sorry for the late reply; busy with answering stupid deficiency letters…

❝ ❝ Oh dear, *that* one! Opening Pandora’s box.

❝ And my apologies for opening the Pandora's box! I will pretend that I never wrote this

❝ Here, I still am a bit confused. Since my alternative is T1=R AND T2=R, I need to carry out 2 tests. Both have to be rejected at a given significance (without adjustment, say 0.05) and jointly, these 2 tests have to achieve a given power (say 0.8). Here is where I am confused:

❝

❝ 1) The 2 tests must jointly have a power > 0.8. For illustration, if we assume independence of these 2 tests (which is clearly not the case but for the sake of illustration), we want to achieve the following:

❝ power(test1, test2) = power(test1) x power(test2) = 0.8

❝

❝ If we further assume equal power for test 1 and test 2, then we need sqrt(0.8) = 0.9 for each test. When computing the sample size in PowerTOST, I am wondering whether the power argument is for the joint tests, or for individual test. If it's specified at an individual level, the joint power is 0.8 x 0.8 = 0.64.

That’s a fallacy. I fell into this trap myself. See post dealing with overall power of multiple studies.

❝ 2) I understand that T1 and T2 are treated independently. But are they assumed to be independent from R, as well?

❝ Even if T1, T2 and R are pairwise independent, R appears in both test statistics so the two tests will show correlation.

Some, yes. To which degree is unknown. Consider this:

*AUC*is the PK metric of extent of absorption and

*C*

_{max}the one of rate of absorption. Where the former in comparative BA depends only on the fractions absorbed ƒ (we assume that the doses are identical and clearances are constant), the latter is a composite metric,

*i.e.*, depends on ƒ,

*k*

_{a}, and

*k*

_{e}. Hence, it is clear that they are correlated, but how much?

(Almost) everything is possible in

`PowerTOST`

.`library(PowerTOST)`

# CV and theta0 must be two-element vectors, values can be different

# below identical ones for simplicity

CV <- c(0.20, 0.20) # assumed CVs

theta0 <- c(0.95, 0.95) # assumed T/R-ratios

target <- 0.80

design <- "2x2"

rho <- seq(0.5, 1, 0.1)

res <- data.frame(rho = rho, n = NA_integer_, power = NA_real_)

for (j in seq_along(rho)) {

res[j, 2:3] <- sampleN.2TOST(CV = CV, theta0 = theta0, design = design,

targetpower = target, rho = rho[j],

print = FALSE)[7:8]

}

print(res, row.names = FALSE)

rho n power

0.5 24 0.82667

0.6 24 0.83335

0.7 22 0.80145

0.8 22 0.81385

0.9 22 0.82943

1.0 20 0.83606

❝ Does this correlation taken into account when computing the sample size?

❝

❝ In my naive opinion, I feel that, under the alternative which stipulates BE, the PK parameters should be correlated between treatments and this correlation should be taken into account.

As you rightly wrote, since R occurs in both tests, likely there is some correlation indeed. But (but!) from where and how will you get it?

In the exercise of two metrics we failed to establish a consistent value (see this lenghty thread for our desperate attempts)… With T1=R AND T2=R we are fishing in the dark. You would have to perform a couple of studies (at least three) to come up with an estimate of \(\rho\) which you could use in the sample size estimation of the fourth.

*Dif-tor heh smusma*🖖🏼 Довге життя Україна!

_{}

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

### Complete thread:

- Sample size for 3-way crossover chrisk 2022-08-09 18:55 [Power / Sample Size]
- Sample size for 3-way crossover Helmut 2022-08-10 11:30
- Sample size for 3-way crossover chrisk 2022-08-10 15:24
- AND or OR, that’s the question Helmut 2022-08-10 15:58
- AND or OR, that’s the question chrisk 2022-08-10 17:46
- AND or OR, that’s the question Helmut 2022-08-10 20:59
- AND or OR, that’s the question chrisk 2022-08-10 22:00
- AND or OR, that’s the question Helmut 2022-08-11 15:25
- AND or OR, that’s the question chrisk 2022-08-12 16:02
- The unknown ρHelmut 2022-08-17 15:04

- AND or OR, that’s the question chrisk 2022-08-12 16:02

- AND or OR, that’s the question Helmut 2022-08-11 15:25

- AND or OR, that’s the question chrisk 2022-08-10 22:00

- AND or OR, that’s the question Helmut 2022-08-10 20:59

- AND or OR, that’s the question chrisk 2022-08-10 17:46

- AND or OR, that’s the question Helmut 2022-08-10 15:58

- Sample size for 3-way crossover chrisk 2022-08-10 15:24

- Sample size for 3-way crossover Helmut 2022-08-10 11:30