Bioequivalence and Bioavailability Forum

Hi Hötzi,

❝ Be assured of my love. Not just right now but at first sight.

Thank you.

OK, here goes:

1. In a recent previous thread dealing with RRT/RTR/TRR designs, I showed how it is entirely possible to work with the covariance matrix without summing it as ZGZt+R.
   I have been doing a little work on that for some time.
   
   
2. I have been lying awake for some time over s²WR and s²WB - what the fuck are these things?
   (They are between-subject variances, for .... yes, for what exactly, given that we have treatment as fixed effect in the BE model???)
   
   
3. Then s²BTR (also called other stuff) - what the f#ck is that between-thingy really?
   (It is the covariance of T with R, but what the heck is that in reality???? I am aware what a covariance is, how a covariance is defined (average sum of product of differences in expected minus observed values), but what the f#ck is s²TR at the end of the day from a practical perspective in a model with treatment?

It basically boils down to one single good question for me: What is the practical relevance of S²BR, s²BT, s²BTR, when the BE model already includes treatment as a fixed effect, and if I can't bend my head around that part, so how would I rather do it instead?

I actually started thinking it over, and I would much rather work with a single between-subject variance component. I can totally see how within subjects there will be a diffentiated variance according to the treatment measured, but across subjects, no I can't readily see why there would be a difference between S2BR, s2BT, s2BTR when treatment is modeled as fixed in the model.

So, and I had this idea a few years back, I think I hinted at it on a post in this forum but I can't find the link, in a replicated design, it makes great sense to me just to have s²WR, s²WT and a single component for the betweens. This is my little provocation.
I was happy to look up how Chow & Liu see it; see e.g. formula 2.5.1 and 9.1.1 in the 3rd edition: A single variance component between subjects is also suggested -or implied- there. This has a tremendously useful consequence for TRR/RTR/RRT designs: With these three components in V, and nothing else, V becomes invertible and the model has a quantifiable likelihood. And therefore, even though T is not replicated we can still estimate s²wt in a TRR/RTR/RRT design (!!!). But if you think about it, you will see it makes good sense (happy to take that discussion further if need be, but for starters: we never look directly at within subject contrasts when we fit a BE model, regardless of how replicated the design is, and regardless of whether we do mixed models or all fixed, just think about it).

Again, I am not saying anything here is compliant with a guideline/guidance. I am solely looking at the foundations of my own knowledge and not trying to build a house/walls/roof from it. The perspective from all this is potentially good news for those who are into inhaled products for FDA submission.

Here's my result with EMA's dataset II, if it is of interest:
 Var.Component  Ini.value      Value
         varWT 0.01240137 0.01833538
         varWR 0.01240137 0.01211072
          varB 0.03576077 0.04080577

Now, can you all show your love and repeat after me: "Anders, you are a friggin genius"