Bioequivalence and Bioavailability Forum • RSAB model overspecified

Wolfgang
☆

Switzerland,
2013-08-05 16:37
(4300 d 03:45 ago)

Posting: # 11202
Views: 11,619

RSAB model overspecified [RSABE / ABEL]

Hi everyone,

For highly variable drugs, the reference-scaled average bioequivalence method is suggested by FDA.
They request that a "conventional approach" analysis be done with some specific code, containing a random statement for SAS Proc Mixed,
random tmt / sub=subject type=FA0(2);
But, in the case of a partially replicated design (RRT/RTR/TRR), this model is overspecified, which can lead to various error messages from SAS, to non-convergence, and to unstable results.
Has anyone thought about using type=FA0(1) instead? Which in this case amounts to the same model but correctly specified.
What would FDA's attitude be towards this?

jag009 ★★★ NJ, 2013-08-05 17:35 (4300 d 02:47 ago) @ Wolfgang Posting: # 11204 Views: 9,978	RSAB model overspecified Post reply
	Hi, I have experienced non-convergence issues a few months back with one of my datasets. I see if I an look up my original post. John Edit: Full quote removed. Please delete everything from the text of the original poster which is not necessary in understanding your answer; see also this post! [Helmut]

d_labes
★★★

Berlin, Germany,
2013-08-05 17:59
(4300 d 02:23 ago)

@ Wolfgang
Posting: # 11206
Views: 10,106

Proc Mixed FA0(1) for partial replicate

Post reply

Dear Wolfgang,

❝ For highly variable drugs, the reference-scaled average bioequivalence method is suggested by FDA. They request that a "conventional approach" analysis be done with some specific code, containing a random statement for SAS Proc Mixed,

❝ random tmt / sub=subject type=FA0(2);

❝ But, in the case of a partially replicated design (RRT/RTR/TRR), this model is overspecified, which can lead to various error messages from SAS, to non-convergence, and to unstable results.

Welcome to the club. We had some discussions about that topic here in the forum.
Look here or here or here for instance. But no satisfactory solution.

❝ Has anyone thought about using type=FA0(1) instead? Which in this case amounts to the same model but correctly specified.

Interesting! Can you please a bit more specific:

Why do you think that it is the same model?
Why do you think it is correctly specified? Seems to me a logical trap: same models, one correct, the other overspecified.

❝ What would FDA's attitude be towards this?

Duno since never had that idea :-D

. Also no experience with the FDA.
But I guess your proposal will be accepted if straight justified. The statisticians at FDA are those who :cool:

—
Regards,

Detlew

Wolfgang
☆

Switzerland,
2013-08-05 18:24
(4300 d 01:58 ago)

@ d_labes
Posting: # 11208
Views: 9,949

Proc Mixed FA0(1) for partial replicate

Post reply

❝ ❝ Has anyone thought about using type=FA0(1) instead? Which in this case amounts to the same model but correctly specified.

❝

❝ Interesting! Can you please a bit more specific:

Why do you think that it is the same model?
Why do you think it is correctly specified? Seems to me a logical trap: same models, one correct, the other overspecified.

Dear Detlew,
thanks for your comments.
If we look at the effects of the two random statements,
random tmt / sub=subject type=FA0(2); /* FDA */
random tmt / sub=subject type=FA0(1); /* new model */
then the corresponding G matrices are: (trying to display matrices in a linear form)
G1: 1st line: a², a*b; 2nd line: a*b, b²+c²
G2: 1st line: a², a*b; 2nd line: a*b, b²
(a, b, c vary freely, -1≤r≤1)

The repeated statement leads to the following R matrix:
For the RTRT/TRTR design, R=diag(X,Y,X,Y) for any subject of sequence RTRT
For the RTR/TRR/RRT design, R=diag(X,Y,X) for any subject of sequence RTR
where X and Y must be nonnegative numbers.

Within the same subject, we have, for the first random statement:

the variance of an R observation is a²+X
the variance of a T observation is b²+c²+Y
the covariance of two different R observations is a²
the covariance of two different T observations is b²+c²
the covariance of an R and a T observation is a*b

For the RTRT/TRTR design, this works well.

For the RTR/TRR/RRT design, however, the covariance of two different T observations (in the same subject; above in red) does not appear, and therefore the model is over-specified: the parameters c and Y appear only together, in the form c²+Y. As a consequence, Proc Mixed will often have difficulties in estimating the model, in particular it may fail to converge.

Remedy: we should replace c²+Y with a single value, i.e. delete either c or Y (or fix it at zero). Getting rid of Y does not seem to be easily possible, but c² can be removed by using FA0(1) instead of FA0(2).

(Note: both models are mathematically correct, one is just overspecified which makes it much more difficult to estimate and less stable numerically. A better version of Proc Mixed might detect this itself and handle it correctly.)

Hope I have explained it in an understandable way?
Best regards,
Wolfgang Seewald

ElMaestro
★★★

Denmark,
2013-08-05 19:19
(4300 d 01:03 ago)

@ Wolfgang
Posting: # 11211
Views: 9,899

Proc Mixed FA0(1) for partial replicate

Post reply

Hello Wolfgang,

❝ Within the same subject, we have, for the first random statement:

❝ – the variance of an R observation is a²+X

❝ – the variance of a T observation is b²+c²+Y

❝ – the covariance of two different R observations is a²

❝ – the covariance of two different T observations is b²+c²

❝ – the covariance of an R and a T observation is a*b

❝

❝ For the RTRT/TRTR design, this works well.

I am no statistician, so I don't know what I am talking about here. That holds true for a lot of my life's other elements, too.
Can we possibly discuss the model on basis of within(w)- and between(b)-sigmas?

I would think that V (=ZGZ^t+R) should look like this:

the variance of an R observation is s²_b,R+s²_w,R
the variance of a T observation is s²_b,T+s²_w,T
the covariance of R observations is s²_b,R
the covariance of T observations is s²_b,T
optional: the covariance of a mixed observation is "q", which expresses a correlation of the between-sigmas.

No?

—
Pass or fail!
ElMaestro

Wolfgang
☆

Switzerland,
2013-08-06 10:35
(4299 d 09:47 ago)

@ ElMaestro
Posting: # 11219
Views: 10,126

Proc Mixed FA0(1) for partial replicate

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❝ I would think that V (=ZGZ^t+R) should look like this:

❝ – the variance of an R observation is s²_b,R+s²_w,R

❝ …

❝ – optional: the covariance of a mixed observation is "q", which expresses a correlation of the between-sigmas.

❝

❝ No?

Hello ElMaestro,
this is correct although the last line should be: covariance of an R and T observation is s_b,R*s_b,T*q, if q is the correlation.
However, this is the parameterization used with type=CSH, which FDA suggests as an alternative to type=FA0(2).
The two models are equivalent, if you identify a² with s²_b,R, b²+c² with s²_b,T, q with b/sqrt(b²+c²), X with s²_w,R, Y with s²_w,T.
When the design is partially replicated, s²_b,T and s²_w,T cannot be estimated, only their sum can be, leading to overspecification and convergence problems.
With this notation, the remedy would formally be equivalent to setting q=1 – but this is more difficult to justify, and also I would not know how to specify it in SAS.
So I guess we have to stick to FA0(1) and the corresponding notation.

ElMaestro
★★★

Denmark,
2013-08-06 12:34
(4299 d 07:49 ago)

@ Wolfgang
Posting: # 11222
Views: 9,823

Proc Mixed FA0(1) for partial replicate

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Hi Wolfgang,

❝ With this notation, the remedy would formally be equivalent to setting q=1 - but this is more difficult to justify, and also I would not know how to specify it in SAS.

❝ So I guess we have to stick to FA0(1) and the corresponding notation.

If I recall correctly, and mind you I could be completely wrong, you can omit the consideration of correlated random effects and still get the right estimates of the s²_w,r, s²_b,r and compound sum for test.
Effectively, in my naïve notation where I would write q=rs_b,r s_b,t it would correspond to setting r=0; then the restricted max likelihood of the optimised V matrix changes, but the determination of s²_w,r and s²_b,r (and s²_w,t+s²_b,t, where applicable) do not.
This might still be in line with the random effects model, if I get Chow & Liu right. After all, the model does not rely on correlated random effects but justs allows to calculate correlations if we find it appropriate.

Not a statistician and not a SAS user so don't know any syntax that fits anything.

—
Pass or fail!
ElMaestro

d_labes ★★★ Berlin, Germany, 2013-08-06 12:30 (4299 d 07:53 ago) @ Wolfgang Posting: # 11221 Views: 9,786	Proc Mixed FA0(1) for partial replicate Post reply
	Dear Wolfgang! Thank you. Your explanation seems logical to me. Seems this is the straightforward solution we have searched for . At least in SAS. For another attempt see the SAS code in this thread. Will try it the next time numerically. Do you eventually know if your proposal is also implementable in R? — Regards, Detlew

Helmut
★★★

Vienna, Austria,
2013-08-07 17:13
(4298 d 03:10 ago)

@ d_labes
Posting: # 11235
Views: 9,965

FA0(1) for partial replicate in Phoenix

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Dear Detlew!

❝ Will try it the next time numerically.

Yes please. I tried John’s nasty data set (log-transformed data in full precision) where we ran into convergence problems and got different results for RSABE and ABE (we tried SAS and PHX; complete data set and after exclusion of incomplete data).
When I set up FA0(1) in PHX’ LME I got…
ERROR 11021: Variance matrix is not positive definite in QA.
… which is an indication that the initial variance estimates (by methods of moments) don’t “work”. I tried different starting values to no avail.

Now for the surprise: PHX’ BE-model (which supposedly sits ‘on top’ of LME) converged without a warning. I got

PE 101.0213% (90% CI: 80.7733–126.3450%) s²_WR 0.42100 (CV_WR 72.35%)

… matching Shuanghe’s and your’s in SAS with FDA’s FA0(2)-parameterization.
After exclusion of incomplete data (subjects 8, 14, 17, and 31)

PE 91.2423% (90% CI: 74.0705–112.3950%) s²_WR 0.40652 (CV_WR 70.82%)

… different to yours’.
Amazing.

—
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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d_labes
★★★

Berlin, Germany,
2013-08-08 11:42
(4297 d 08:40 ago)

@ Helmut
Posting: # 11246
Views: 9,967

FA0(1) for partial replicate in SAS

Post reply

Dear Helmut!

With John’s nasty data set I got:
- all data

PE= 101.0213% (90% CI: 80.7733–126.3450%) s²_WR= 0.4210 (CV_WR= 72.35%)

… matching yours and the former evaluation with FA0(2) to full precision. [image]

- complete cases only

PE= 90.5032% (90% CI: 73.5610–111.3474%) s²_WR= 0.4001 (CV_WR= 70.14%)

… different from yours [image]

but matching nearly to full precision the former FA0(2) evaluation (differences in the CI only after the second decimal). [image]

No hint of any convergence problems, but in both cases in the log
NOTE: Estimated G matrix is not positive definite.
Not quite sure if this is a problem.

What puzzles me a little bit is that we have an estimate of s²_WT in the model (via the repeated/grp=tmt sub=subject; statement) also Wolfgang claimed, correctly I think, that "... s²_b,T and s²_w,T cannot be estimated, only their sum." But have to think about that ...

—
Regards,

Detlew

lhughes
☆

USA,
2013-08-30 17:17
(4275 d 03:06 ago)

@ Helmut
Posting: # 11395
Views: 9,607

FA0(1) for partial replicate in Phoenix

Post reply

Hi Helmut!

You have:

❝ When I set up FA0(1) in PHX’ LME I got…

❝ ERROR 11021: Variance matrix is not positive definite in QA.

❝ Now for the surprise: PHX’ BE-model (which supposedly sits ‘on top’ of LME) converged without a warning. I got

PE 101.0213% (90% CI: 80.7733–126.3450%) s²_WR 0.42100 (CV_WR 72.35%)

… matching Shuanghe’s and your’s in SAS with FDA’s FA0(2)-parameterization.

❝ After exclusion of incomplete data (subjects 8, 14, 17, and 31)

PE 91.2423% (90% CI: 74.0705–112.3950%) s²_WR 0.40652 (CV_WR 70.82%)

… different to yours’.

I was able to run both LinMix and Bioeq in Phoenix and their results matched. Also when I excluded the four subjects with incomplete data, I got:
PE= 90.503 90% CI: 73.561–111.347
so the same as what Detlew posted in his next post with the SAS results. I emailed you a project.

Linda

Helmut
★★★

Vienna, Austria,
2013-08-30 19:07
(4275 d 01:15 ago)

@ lhughes
Posting: # 11396
Views: 9,625

FA0(1) in PHX: works (& more questions)

Post reply

Hi Linda!

❝ I was able to run both LinMix and Bioeq in Phoenix and their results matched. Also when I excluded the four subjects with incomplete data, I got:

❝ PE= 90.503 90% CI: 73.561–111.347

❝ so the same as what Detlew posted in his next post with the SAS results. I emailed you a project.

THX, confirmed. I screwed up the setup (correct BE but wrong LME) – therefore the “difference”. :crying:

So is FA0(1) the solution? Is FDA’s code – calling for FA0(2) – mandatory (aka “Holy Grail”)? As Wolfgang pointed out above in the general BE guidance FDA stated:

In the Random statement, TYPE=FA0(2) could possibly be replaced by TYPE=CSH.

As we have seen in the “nonconvergent” data sets this doesn’t help at all. “Could possibly be replaced by” FA0(1) as well?
Or abandon the partial replicate completely and opt for TRT|RTR – which seems to “work” according to Detlew’s findings?

To make a long story short: What will hurt FDA more or less likely accept?

Using the partial replicate (= guidance) and switch to FA0(1) (≠ guidance), or
Using a fully replicated 3-period design (≠ guidance) and keep FA0(2) (= guidance)?

—
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