## Inflation of the TIE as well [RSABE / ABEL]

Hello everybody and nobody.

» you are modifying the common decision scheme. Why?
» Since you are making essentially the same decisions though in a different order, you gain nothing.

Exactly...

» Well, that’s the problem with ABEL (and the FDA’s RSABE as well). There is no Null Hypothesis.

and exactly again!

In standard ABE, null hypothesis (bioinequivalence) and alternative hypothesis (bioequivalence) are known. The BE is defined using the acceptance limits 80-125% for 90% CI of GMR.
The TIE (Type I Error) is the probability (alpha) of rejection of true null hypothesis by test. (It's clear but problematic when null hypothesis is unknown and even more problematic in mentioned different order - see below.)

TIE inflation for ABEL/RSABE with widening the acceptance limits (because of floating acceptance limits) is described in different topics many times.

The suggested different order of analysis in ABEL/RSABE is only a bad attempt to hide the increased patient's risk.

When we apply the standard limits of 80-125% in the first step and conclude BE using 90% CI - end of analysis ... hey, it looks like standard study with TIE <=5%.
But it is only the part of the decision tree.

It reminded me the Monty Hall problem with intuitive 50–50 chance. When we look only at a part of the game, it seems really as 50-50.

Back to the bioequivalence:
Can we say after the conclusion of BE in the first step that there was no option to wide the acceptance limits?
If yes - Ok.
If no - There was additional probability to conclude BE. TIE inflation is obvious when you look at the whole plan.
The simulations would certainly confirm TIE inflation. Keep in mind that the observed GMR and intra-subject CV of reference are only estimates, we don't know the true (population) values. Imagine the border case example with true GMR 80% (and true intra-subject CV of R 35%), 5% of studies will pass the acceptance criteria 80-125% in the first step and another few % of studies in the second step (after widening of limits). With true GMR 80%, study was performed and we were lucky in observed values, so we concluded BE in the first step. But we can't say that TIE was <=5%!

This different order is more controversial than the standard order according the guidelines. Null hypothesis is set in the first step of different order. BE is evaluated - concluded or not. When the test failed to conclude bioequivalence, the new null hypothesis will be tested with different definition of BE. BE is evaluated again - concluded or not.
Standard order is clear, you construct the null hypothesis and test it. With different order we can fail to demonstrate BE from the first test, oh no..., so we redefine bioequivalence, i.e. new hypothesis, new test (comparison), we have the second chance.

Btw. When I was starting to write the post I thought that order doesn't change the TIE, but now...
I think, the different order has higher TIE than the standard order.
(Helmut was faster with similar example.)
Example 2x2x4 replicate design with the same sample size n=24 for both orders: assumed GMR 90%, assumed intra-subject CV of R 50% (for higher intra-subject CVs there will be higher probability of two tests in different order).
• standard order
• One test with acceptance limits 69.84-143.19% or narrower (it depends on observed value), TIE <5% (due to scaling cap and GMR restriction) (let's denote TIE as TIE1)
• different order
• The first test with acceptance limits 80-125%, lower power than in standard order (with the same number of subjets) "partial" TIE 4.8%
The probability of the end after the first test is only 29% in this example, i.e. 71% probability that the second test will be done.
• The second test with acceptance limits 69.84-143.19% or narrower (it depends on observed value), TIE <5% (due to scaling cap and GMR restriction) (the same as TIE1)
In total, aggregate TIE is 4.8% + 0.71*TIE1 (aggregate TIE ?, probably higher then 5%).
It really looks like as the order changes the TIE - the reason is of course in the change of BE definition (i.e. null hypothesis) for the first test.

Moreover the first test is totally irrelevant if assessed as the whole. If the first test is BE (the second test would always confirm that), if not, only the second test would be decisive. So equal result of study you receive by testing only the second test, so I don't see any reason for using the different order - no pros, possible TIE inflation as cons.

Best regards,
zizou