## AND or OR, that’s the question [Power / Sample Size]

Thanks again for your help on this, Helmut.

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.

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. 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.

Thanks again for your guidance!

❝ Oh dear, *that* one! Opening Pandora’s box. However, only relevant if one would assess the study with the ‘All at Once’ approach (*i.e.*, an ANOVA of pooled data). The jury left the courtroom 2½ years ago and hasn’t returned ever since…

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

❝ In the ‘Two at a Time’ approach T1 and T2 are treated independently. Hence, you have two separate distributions. You see that also in the evaluation, where you obtain two residual variance estimates.

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.

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. 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.

Thanks again for your guidance!

### 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 questionchrisk 2022-08-12 16:02
- The unknown ρ Helmut 2022-08-17 15:04

- AND or OR, that’s the questionchrisk 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