Bioequivalence and Bioavailability Forum

ElMaestro
★★★

Denmark,
2013-08-12 21:54
(4292 d 22:28 ago)

Posting: # 11264
Views: 11,524

3-per, 2-trt designs and estimation of intra-subj. variance [RSABE / ABEL]

Hi all,

I think it sounds fully plausible and intuitive to say that intrasubject variabilities for Test products cannot be estimated in TRR/RTR/RRT xover designs. We do not have replication of Test and this is the principal argument.

But is that really a reflection of the facts?

Well, due to the replication of Ref we can estimate s²_wR as well as s²_b, with w being 'within' and b being 'between'. I think it can be argued that in a crossover it makes no sense to discuss different betweens-variances between treatments.
Now, the variance of the observations done on test will be s²_wT + s²_b - this is what we will have on the diagonal of the covariance matrix V.
So, we can definitely estimate the sum (s²_wT + s²_b), and since the ref. replication allows us to get s²_b, I see no reason why we cannot estimate the elusive s²_wT.

Try to think about it without just blindly referencing what a guideline says.
Now butcher me, please. :pirate:

Edit: Category changed. [Helmut]

—
Pass or fail!
ElMaestro

Dr_Dan ★★ Germany, 2013-08-13 12:11 (4292 d 08:10 ago) @ ElMaestro Posting: # 11265 Views: 10,234	3-per, 2-trt designs and estimation of intra-subj. variance Post reply
	Hi ElMaestro Interesting considerations. However, I do not see the advantage of a TRR/RTR/RRT xover design. What do you think about TRT vs RTR? In this case you also have a 3-per, 2-trt design but you can estimate the intrasubject variability for the Test product, right? Kind regards Dan — Kind regards and have a nice day Dr_Dan

ElMaestro ★★★ Denmark, 2013-08-13 17:00 (4292 d 03:21 ago) @ Dr_Dan Posting: # 11270 Views: 10,195	3-per, 2-trt designs and estimation of intra-subj. variance Post reply
	Hi Dr_Dan, yes, RTR/TRT designs sound like a good concept. I'd be happy enough with that type of design. I think I will propose it next time I write a sc. advice briefing book. — Pass or fail! ElMaestro

Helmut ★★★ Vienna, Austria, 2013-08-13 20:23 (4291 d 23:59 ago) @ ElMaestro Posting: # 11271 Views: 10,221	TRT\|RTR rulez! Post reply
	Hi ElMaestro! I’m recommending this goodie for years. BTW, if you like a reference that’s a design the two Lászlós used in some of their papers / simulations. — 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

d_labes
★★★

Berlin, Germany,
2013-08-14 10:16
(4291 d 10:06 ago)

@ Helmut
Posting: # 11275
Views: 10,103

TRT|RTR rulez? Really?

Post reply

Dear Dan, ElMaestro and Helmut!

Rulez? Really?
Read chapter 9.3 of Chow and Liu, especially 9.3.4 "Assessment of Intra-Subject Variabilities". And if you have, read it again.

For me the consequence of this chapter is: While it is true that this design has the advantage of allowing the separate estimation of the intra-subject variabilities of Reference and Test, there is no valid way to evaluate T versus R in case of heteroscedasticity. And this renders the advantage useless.

A valid evaluation with GLM-ANOVA or even with intra-subject contrasts, which in other replicate designs is able to deal with different variabilities, has to assume s²_wT=s²_wR.

If this is also the case for evaluation via mixed model software is left for discussion. But at least in the EMA jurisdiction doesn't allow for it if we take the Q&A literally and translate 'not recommended' into 'not allowed'.

❝ BTW, if you like a reference that’s a design the two Lászlós used in some of their papers / simulations.

Which one do you mean?

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2013-08-14 13:29
(4291 d 06:53 ago)

@ d_labes
Posting: # 11278
Views: 10,204

TRT|RTR rulez? Really? Yes.

Post reply

Hi d_labes,

❝ Read chapter 9.3 of Chow and Liu, especially 9.3.4 "Assessment of Intra-Subject Variabilities". And if you have, read it again.

how was the birthday? Me and 17 friends thought of partycrashing at your home but due to an unfortunate event recently involving twenty ostriches, a nuclear power station and a barrel of rum we decided against it.

I don't really get all the points of C&L. If any :-D

❝ (...) there is no valid way to evaluate T versus R in case of heteroscedasticity. And this renders the advantage useless.

Can you explain me why in words that are adapted to people with walnut-sized brains such as myself?

❝ A valid evaluation with GLM-ANOVA or even with intra-subject contrasts, which in other replicate designs is able to deal with different variabilities, has to assume s²_wT=s²_wR.

I am quite sure that it is possible to make GLM work ad modum EMA's recent adventures with fixed effects, by tampering a bit with the data sets and factor inclusions, allowing extractions of individual variances.

Anways, we are now digressing from the original topic. To which I'd still appreciate some responses. :-D

—
Pass or fail!
ElMaestro

d_labes
★★★

Berlin, Germany,
2013-08-14 15:34
(4291 d 04:48 ago)

@ ElMaestro
Posting: # 11284
Views: 10,112

TRT|RTR rulez? Really? No.

Post reply

Dear Öberster größter Meister,

❝ ❝ Read chapter 9.3 of Chow and Liu, especially 9.3.4 "Assessment of Intra-Subject Variabilities". And if you have, read it again.

❝ ...

❝ I don't really get all the points of C&L. If any :-D

Did you follow my advice? Read again? If yes, read again. If no, don't worry longer about that issue :-D

❝ ❝ (...) there is no valid way to evaluate T versus R in case of heteroscedasticity. And this renders the advantage useless.

❝

❝ Can you explain me why in words that are adapted to people with walnut-sized brains such as myself?

An initiated must speak in words that are not understandable by ordinary people, not to mention people with walnut-sized brains :cool:

.
The key is hidden in Formulas 9.3.21 and 9.3.22 showing that the expected variances of the sequence groups TRT and RTR are not the same. Thus you can't pool them without the danger of too liberal or too conservative results after doing so. You had to use a method which accounts for this heteroscedasticity. AFAIK there is no in the context we discuss here, except eventually a real mixed model analysis.

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2013-08-14 16:25
(4291 d 03:56 ago)

@ d_labes
Posting: # 11286
Views: 10,153

TRT|RTR rulez? Really? No.

Post reply

Dear d_labes,

❝ The key is hidden in Formulas 9.3.21 and 9.3.22 showing that the expected variances of the sequence groups TRT and RTR are not the same. Thus you can't pool them without the danger of too liberal or too conservative results after doing so. You had to use a method which accounts for this heteroscedasticity. AFAIK there is no in the context we discuss here, except eventually a real mixed model analysis.

Thank you.
Yes, I realise that the error variance in TRT and RTR can differ but does that leave us really helpless? From the two sequences we get two equations with two unknowns -eqn. 9.3.21- from which we can solve explicitly for s²_wR and s²_wT. We could plug those into 9.2.17 and get 90% CI's.
Am I overlooking something?

—
Pass or fail!
ElMaestro

d_labes ★★★ Berlin, Germany, 2013-08-14 18:35 (4291 d 01:46 ago) @ ElMaestro Posting: # 11287 Views: 10,101	TRT\|RTR Lost in formulas. Post reply
	Dear EM, ❝ Am I overlooking something? Duno. Lost in formulas. But if it were that easy I think C&L would have described this. — Regards, Detlew

Helmut
★★★

Vienna, Austria,
2013-08-14 15:44
(4291 d 04:37 ago)

@ d_labes
Posting: # 11285
Views: 10,223

TRT|RTR rulez? References

Post reply

Dear Detlew!

❝ Rulez? Really?

❝ Read chapter 9.3 of Chow and Liu, especially 9.3.4 "Assessment of Intra-Subject Variabilities". And if you have, read it again.

Ok, promised.

❝ ❝ BTW, if you like a reference that’s a design the two Lászlós used in some of their papers / simulations.

❝

❝ Which one do you mean?

Likely all of them. I guess they started with simulations in 2001^1,2 (only 2- and 4-period studies) and referred back to these papers in all followings. In 2007³ László Tóthfalusi recommended TRTR|RTRT and TRT|RTR. Interesting enough no details were given until 2009⁴:

“The power curves were obtained by simulating 10 000 three-period bioequivalence trials with a design of TRT/RTR and 36 subjects.”

Tóthfalusi L, Endrényi L, Midha KK, Rawson MJ, Hubbard JW. Evaluation of the Bioequivalence of Highly-Variable Drugs and Drug Products. Pharm Res. 2001;18(6):728–33.
Tóthfalusi L, Endrényi L. Limits for the Scaled Average Bioequivalence of Highly Variable Drugs and Drug Products. Pharm Res. 2003;20(3):382–9.
L Tóthfalusi L. Scaled Average Bioequivalence to Evaluate Bioequivalence of Highly Variable Drugs. Presentation at the informa conference “Dissolution Testing, Bioavailability & Bioequivalance”, 24 May 2007, Budapest, Hungary
Tóthfalusi L, Endrényi L, García-Arieta A. Evaluation of Bioequivalence for Highly Variable Drugs with Scaled Average Bioequivalence. Clin Pharmacokinet. 2009;48(11):725–43. doi 10.2165/11318040-000000000-00000

—
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

d_labes
★★★

Berlin, Germany,
2013-08-14 21:12
(4290 d 23:10 ago)

@ Helmut
Posting: # 11288
Views: 10,087

References László & László

Post reply

Dear Helmut!

❝ Likely all of them ...

Thank you for the excellent history. I must confess that I had overlooked up to now this detail of the papers of the two Lászlós. In my mind I had remembered only 2x2 and 2x2x4 designs.
But always they have mainly simulated under the assumption CVwT=CVwR, I think. Right?

Meanwhile I have some sims underway which (hopefully) will underpin my criticism of the TRT/RTR design.

Have a nice evening :party:

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-08-17 19:33
(4288 d 00:48 ago)

@ d_labes
Posting: # 11300
Views: 10,004

Heteroscedasticity

Post reply

Dear Detlew!

❝ But always they have mainly simulated under the assumption CVwT=CVwR, I think. Right?

Duno, but I think so as well.

❝ Meanwhile I have some sims underway which (hopefully) will underpin my criticism of the TRT/RTR design.

Kudos for your pioneering work on exploring heteroscedasticity. :-D

Overall I must confess that I’m a little bit lost. What do we gain from the overspecified partial replicate, if in some cases our softwares’ implementation of (in)famous FDA’s code does not converge, yield different results, or we have to retreat to strange workarounds?

—
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

d_labes
★★★

Berlin, Germany,
2013-08-29 18:45
(4276 d 01:37 ago)

(edited on 2013-08-30 09:23)
@ Helmut
Posting: # 11387
Views: 10,306

Heteroscedasticity: TRT|RTR rulez!

Post reply

Dear Helmut! Dear All!

❝ ❝ Meanwhile I have some sims underway which (hopefully) will underpin my criticism of the TRT/RTR design.

❝

❝ Kudos for your pioneering work on exploring heteroscedasticity. :-D

Thanks for the praise in advance. Let's see if I'm worth the "Vorschuss-Lorbeeren".

Going the long walk and simulating ABE type I error for the 3-period design with the sequences TRT and RTR via subject data sims (1 value after ~24 h!) I was more and more frustrated: No alpha-inflation, no too conservative alphas far and wide! Although the Oracle Chow&Liu had forecasted their certain occurrence!

But finally arriving in the green realms of FDA/ISC I got the following, resorting to heavy imbalance (forgot my Schützomycin dose :-D

CVwT  CVwR   n1   n2   EMA     FDA

                       ANOVA   ISC

---------------------------------------

0.5   0.2   18   18   0.0509   0.0496^*   

            21   15   0.0509   0.0559^*

            15   21   0.0507   0.0430^*

            24   24   0.0505   

---------------------------------------

0.5   0.3   18   18   0.0503   0.0502

            19   17            0.0516

            21   15   0.0502   0.0541

            15   21   0.0503   0.0459

---------------------------------------

0.2   0.5   18   18   0.0509   

            21   15   0.0511   

            15   21   0.0506      

            24   24   0.0505   

---------------------------------------

0.3   0.5   18   18   0.0505   0.0504

            19   17            0.0490

            21   15   0.0505   0.0469

            15   21   0.0503   0.0542

---------------------------------------

n1 = n(TRT), n2 = n(RTR)

1E6 sims, ^*1E5 sims only

There is no hint of an alpha-inflation or too conservative alpha values if one uses the EMA recommended evaluation (same ANOVA as for the classical 2x2x2 crossover), regardless of grade of heteroscedasticity and unbalancedness analyzed!

The evaluation via intra-subject contrasts also do not show noticeable deviations from the nominal level 0.05 as long as the design is balanced or only slightly unbalanced.
Alpha-inflation or too conservative type I error values are only observed if the sequence groups are strongly unbalanced.

Thus this design is much more ‘friendly’ in respect to heteroscedasticity than the 2x3x3 design in the EMA evaluation. See this post of mine.

Some simulations of power of scaled ABE show that even for heavy unbalanced designs the influence of this observed behaviour is only small.

Conclusion 1: TRT|RTR rulez! :-D

Conclusion 2: Expect an update of PowerTOST in the near future.

—
Regards,

Detlew

ElMaestro ★★★ Denmark, 2013-08-29 19:50 (4276 d 00:31 ago) @ d_labes Posting: # 11389 Views: 9,839	Heteroscedasticity: TRT\|RTR rulez! Post reply
	Hi d_labes, ❝ The evaluation via intra-subject contrasts also do not show noticeable deviations from the nominal level 0.05 as long as the design is balanced or only slightly unbalanced. Am still trying to get to terms with all this. How is the evaluation done? Which model is fitted? — Pass or fail! ElMaestro

d_labes
★★★

Berlin, Germany,
2013-08-30 11:16
(4275 d 09:06 ago)

@ ElMaestro
Posting: # 11392
Views: 9,827

Sims & evaluation

Post reply

Dear ElMaestro

❝ How is the evaluation done? Which model is fitted?

EMA ANOVA:

Sims according to the 'all effects fixed' doctrine as discussed here.
Evaluation of the replicate design with the same ANOVA as for the 2x2x2 crossover. Point estimator and its 90% CI via LSMeans difference.
ABE decision via 90% CI's.

FDA ISC:

Sims with the full mixed model underlying the Proc MIXED code of the FDA. We had it already here in the forum but couln't find at the moment in the meanwhile tons of posts. For a short description see Chow&Liu Chapter 20.5.7.1.. The subject-by-treatment interaction was assumed zero.
Calculate appropriate intra-subject contrast to obtain individual T-R. Analyzed them by an ANOVA with sequence as the solely effect. The intercept of this ANOVA gives the point estimator T-R and its 90% CI.
ABE decision via the 90% CI's.

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2013-08-31 16:11
(4274 d 04:10 ago)

(edited on 2013-08-31 17:04)
@ d_labes
Posting: # 11398
Views: 9,895

Sims & evaluation

Post reply

Dear d_labes,

❝ EMA ANOVA:Sims according to the 'all effects fixed' doctrine as discussed here.

❝ • Evaluation of the replicate design with the same ANOVA as for the 2x2x2 crossover. Point estimator and its 90% CI via LSMeans difference.

OK... lemmefink....if you apply one lm (or one anova) then it must imply that you are working with one error term. Where TRT and RTR imply two different error mean squares, as you referenced earlier, I think we cannot do both TRT data and RTR data in one model if the model has only one error term. It must be either one model with (at least) two error terms (aka. mixed) or two models with one error term in order to cover both groups.

—
Pass or fail!
ElMaestro

d_labes
★★★

Berlin, Germany,
2013-08-31 19:20
(4274 d 01:02 ago)

@ ElMaestro
Posting: # 11399
Views: 9,758

Evaluation subtleties

Post reply

Dear ElMaestro,

❝ OK... lemmefink. ...

❝ Where TRT and RTR imply two different error mean squares, as you referenced earlier, I think we cannot do both TRT data and RTR data in one model if the model has only one error term.

My simulations show: Yes we can! EMA ANOVA don't show any sign of alpha inflation or too conservative type I error which one would expect if pooling is not allowed. Astonishing for me.
FDA ISC works also as long as the designs are balanced or only slightly balanced.

❝ It must be either one model with (at least) two error terms (aka. mixed) or two models with one error term in order to cover both groups.

Both nice ideas. If you have worked them out lemme know :cool:

—
Regards,

Detlew

3-per, 2-trt designs and estimation of intra-subj. variance [RSABE / ABEL]

3-per, 2-trt designs and estimation of intra-subj. variance

3-per, 2-trt designs and estimation of intra-subj. variance

TRT|RTR rulez!

TRT|RTR rulez? Really?

TRT|RTR rulez? Really? Yes.

TRT|RTR rulez? Really? No.

TRT|RTR rulez? Really? No.

TRT|RTR Lost in formulas.

TRT|RTR rulez? References

References László & László

Heteroscedasticity

Heteroscedasticity: TRT|RTR rulez!

Heteroscedasticity: TRT|RTR rulez!

Sims & evaluation

Sims & evaluation

Evaluation subtleties