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

Hi Chris,

❝ Thanks again, Helmut, for sharing your wisdom so quickly and in details.

When you know something, say what you know.
When you don’t know something, say that you don’t know.
That is knowledge.
Confucius

I know much less about anything than I know about something. Wisdom is not my thing.

❝ Since the alternative is based on T1 = R OR T2 = R, …

Are you trying to confuse me? In your OP you stated:

❝ ❝ ❝ I want to demonstrate that both test drugs T1 and T2 are BE to the reference R.

That’s an AND-conjunction, right?

❝ … I am attempted to think that a larger sample size is required for the alternative T1 = R AND T2 = R. Is that indeed the case?

Correct! In an OR-conjunction in your case you get at least two chances (either T1 or T2 passes or both). In the most simple case* you have to use Bonferroni’s adjustment
$$\alpha_\text{adj}=\alpha/k\small{,}\tag{1}$$ where $$\alpha$$ is the nominal level of the test and $$\small{k}$$ the number of tests. Then the family-wise error rate is controlled with
$$FWER=1-\left(1-\alpha_\text{adj}\right)^k\tag{2}$$
You have to adjust only for T1 | T2 = R (two tests).

❝ If so, would you have any recommendation on how to approach the sample size estimation?

See the last paragraph of my previous post. Call the script with alpha = 0.025.

Reference                : C
Tests                    : A, B
Sequences                : ABC, ACB, BAC, BCA, CAB, CBA
Subjects per sequence    : 7 | 7 | 7 | 7 | 7 | 7 (balanced)
Estimated sample size    : 40
Achieved power           : 0.8159
Adjustment to obtain period-balance of IBDs
Adjusted sample size    : 42
Achieved power          : 0.8353
Randomized               : 2022-08-10 15:56:23 CEST
Seed                     : 9874408

treatment metric theta0 CV   n  power
A         Cmax   0.94   0.25 42 0.8353
A         AUC    0.95   0.20 42 0.9731
B         Cmax   0.96   0.23 42 0.9489
B         AUC    0.97   0.18 42 0.9980

• There are more powerful approaches (e.g., Bonferroni-Holm, hierarchical testing,…) but the chances are close to nil that assessors dealing with BE know/understand them.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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