ddubins
☆

Toronto,
2010-04-26 18:28

Posting: # 5241
Views: 30,452

## Reference scaled BE - FDA (and FARTSSIE update) [RSABE / ABEL]

Hi group,

The FDA just recently posted a draft individual product bioequivalence recommendation for progesterone capsules, which provides useful SAS code for their reference-scaled approach. It goes in to extremely useful detail:

This is a critical read for anyone using the FDA's reference-scaled approach.

I updated my sample size program, FARTSSIE, incorporating the EMEA reference-scaled and FDA reference-scaled methods into the "Bioequivalence, Replicate" spreadsheet:

http://individual.utoronto.ca/ddubins/FARTSSIE17.xls

Best,
-Dave Dubins

Edit: Files linked. Note: Oh, wow! [Helmut]

David Dubins, Ph.D., B.Eng.
Instructor and Laboratory Coordinator
Leslie Dan Faculty of Pharmacy
University of Toronto
d_labes
★★★

Berlin, Germany,
2010-04-27 06:58

@ ddubins
Posting: # 5243
Views: 27,968

## Reference scaled ABE - FDA

Hi Dave,

wow! Very valuable contribution!

Fascinating: The codes provided rely solely on suitably choosen intra-subject contrasts! Sometimes called Method of Moments if I'm not wrong. No explicit mixed effects modeling(!) with the exception of the "conventional" unscaled ABE analysis for a fully (4-period) replicate design.

BTW: What is missing is explicit code for ABE evaluation for the partial replicate design in case of sWR < 0.294 (CV = 30%): "The two one-sided test procedure must be used for PK parameters with sWR < 0.294."

BTW2: It is not clear why they recommend Proc MIXED in case of fully replicate design and Proc GLM in case of partial replicate. In both cases the call to these procedures have only sequence as fixed effect. They only serve as means to get the sequence "stratified" means of the corresponding inter-subject contrasts. Or is this meant with respect to the unscaled ABE evaluation?

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2010-04-27 11:39

@ d_labes
Posting: # 5247
Views: 28,015

## Reference scaled ABE - FDA

Dear dlabes and all,

» BTW2: It is not clear why they recommend Proc MIXED in case of fully replicate design and Proc GLM in case of partial replicate.

Perhaps they also could not specify a Mixed Model where a intra-Ref sigma could be derived where replicated Test is not available? I wonder if SAS has this ability for partial repliacted studies at all.

From the dataset DGLM1, calculate the following:
DGLM1: dfd=df;
s2wr=ms/2;

Do you know how/what/why? I am not readily able to understand where it comes from or why this would work, but I would think this is the modeled equivalent of the Swr calculation in step 1 (?).

Bonus questions:
1. Is the estimated Swr unbiased?
2. If yes, is it even reflecting max likelihood?

Best regards
EM - sailor and simulant.
d_labes
★★★

Berlin, Germany,
2010-04-27 12:30

@ ElMaestro
Posting: # 5248
Views: 27,884

## Factor 2 is unbiased

Dear sailor and simulant,

» From the dataset DGLM1, calculate the following:
» DGLM1: dfd=df;
» s2wr=ms/2;

»
» Do you know how/what/why?

Here the analysis covers the differences R1-R2 which has expected variance =2*sigma2WR.
The trick of the GLM call (diffs versus sequence) gives the MS of R1-R2 "stratified" over the sequences (double sum in the formula for s2WR in step 1 on page 2 divided by (n-m)). You can verify this by conventional calculation.

To be honest: I don't know exactly. But I conjecture: Yes in balanced design without missings.

Regards,

Detlew
d_labes
★★★

Berlin, Germany,
2010-05-04 10:10

@ ElMaestro
Posting: # 5279
Views: 28,055

## Biased bonus from MoM

Dear ElMaestro, dear All,

» Bonus questions:
» 1. Is the estimated Swr unbiased?
» 2. If yes, is it even reflecting max likelihood?

During wondering about your questions and fiddling with some numerical experiments I came across with a peculiarity I didn't expect.

Here part of my R code tried to implement the FDA method.
It assumes that the data.frame PKdata has subject, sequence, code, repl and the PK-metrics AUC, Cmax. Key is here the variable repl which is the replicate number within subject and code (T or R).
avar contains the name of the PK metric (log-transformed).
# tmt2=T1, ... based on replicate number PKdata$tmt2 <- paste(PKdata$code,PKdata$repl,sep="") # reshape to subject, sequence, T1, ..., R1, ... PKdata <- reshape(PKdata[c("subject","sequence","tmt2", avar)], direction="wide", timevar="tmt2", v.names=avar, idvar=c("subject","sequence")) # change names as f.i. logAUC.T1 to T1 names(PKdata) <- sub(paste(avar,".",sep=""),"",names(PKdata)) # get the T1, T2, ... and R's correspondingly Ts <- names(PKdata)[substring(names(PKdata),1,1)=="T"] Rs <- names(PKdata)[substring(names(PKdata),1,1)=="R"] # "basic estimator" aka within subject contrasts of T-R # attention: coercion to a vector if only one T or R! # rowMeans then does not work! if (length(Ts)>1) Tv <- rowMeans(PKdata[Ts], na.rm=TRUE) else Tv <- PKdata[Ts] if (length(Rs)>1) Rv <- rowMeans(PKdata[Rs], na.rm=TRUE) else Rv <- PKdata[Rs] # attention: in case of only one T,R the column TR will be auto-named T1 # if assignment PKdata$TR <- is used! PKdata["TR"] <- (Tv - Rv) # now we can make the 'Proc GLM' call # with standard contr.treatment we get not the wanted intercept! # with that the intercept is intercept+seq1 oc <- options("contrasts") options(contrasts=c("contr.sum","contr.poly")) mlm <- lm(TR ~ sequence, data=PKdata) # SAS gives us the same answer using Proc GLM with sequence as effect and ESTIMATE 'T-R' intercept 1; CIs <- data.frame(point=coef(mlm)[1],                   lower=confint(mlm,1,level=1-2*alpha)[1],                   upper=confint(mlm,1,level=1-2*alpha)[2]) if (logtrans) CIs <- exp(CIs) options(oc)  #restore
Note the attention: comments to see that R is also sometimes a hard to repressing beast.

Using the bear example dataset for replicate studies (4-period-2-sequence study with sequences RTRT/TRTR and 14 subjects balanced over sequence groups) I get:
       point      lower      upper   1.01479     0.99336     1.03667    Intra-subject contrasts (MoM) # compare to:   1.01479     0.95223     1.08146    lme() like bear   1.01479     0.95494     1.07839    SAS with FDA code Proc MIXED but CSH 

Unbiased estimate of point estimator T-R but much tighter confidence interval of MoM!
Whats going on here? As I have commented in the above code "The power to know" gives us the same results. Thus the implementation seems error free. Someone out there to prove me wrong?

Then I remembered Stephen Senn's example 7.3 ("Cross-over Trials in Clinical Research" page 237 ff) who showed a similar effect due to a negative correlation of the two within-subject contrasts T1-R1 and T2-R2 which lead to a negative subject by formulation interaction if the constraints that variance-covariance components must >0 are relaxed.
In testing this for the bear data set I got a correlation coefficient r=-0.9366165, a very strong negative correlation, why ever.
Proc Mixed with type=UN as covariance matrix gives then a negative SxF interaction and the confidence interval:
       point      lower      upper   1.01479     0.99336     1.03667    Proc Mixed, Type=UN
exactly the same as via the intra-subject contrasts.

BTW: Dear EM, just to cite you: "...you ask a question here in this forum to gain insight, but when you ask about X you get a story about Y ..."
Therefore here the results for the intra-individual variances (in the log domain):
         MoM     Proc Mixed FDA (REML) s2WT    0.030130  0.02010  s2WR    0.028125  0.01974

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2010-05-04 10:54

@ d_labes
Posting: # 5281
Views: 27,598

## Biased bonus from MoM

Nice post d_labes,

as usual I am able to understand about 1% or less of what is going on. I was never much into phrenology* but equipped with a brain the size of a walnut I am feeling a bit challenged on these matters.

» Therefore here the results for the intra-individual variances (in the log domain):
» MoM    Proc Mixed FDA (REML)
» s2WT    0.030130  0.02010
» s2WR    0.028125  0.01974

I think ultimately this may come down to what is going in the covariance matrix structure for PROC MIXED. In this post you used both CSH and UN if I get it right. I am not an SAS user but I think you can ask the beast to spit out the G and R matrices (from V=ZGZt+R). I think these output should be compared with the info extracted from R's lme object.
A VERY annoyong/inconvenient fact in my opinion is that I don't think there is any proper way of getting to V when using R. I simply cannot find an extractor method or anything that resembles. If you know of one I would love to learn it (getVarCov doesn't do the job).

Are you sure, btw, that the FDA proposal is really an MoM approach? As far as I recall MoM methods are based on solving equations (as in X+5=8, then what is X? -and so forth in a slightly more complicated fashion when multiple equations are involved), if I get it right rendering the solutions unbiased (?) but not necessarily max. likelihood. Isn't the FDA approach just a 'manual' way of getting to the sigma's that serious playtime with PROC MIXED would or should give?

Last, to address my own bonus question partially, I think there could be a practical difference between the FDA approach and the way by a Mixed Model. They may achieve exactly the same in the cases where there are no missing values. But if we have a subject who has an N.A. for one of the two Ref values, then certainly the mixed model output will be reflecting max likelihood, when I guess the linear model will not because such a subject will have to be discarded for the fit.

Best regards
EM.

*From the Simpsons, season 7, episode 8:
Mr Burns: Who could forget such a monstrous visage? She has the sloping brow and cranial bumpage of the career criminal.
Smithers: Sir? Phrenology was dismissed as quackery 160 years ago.
Mr Burns: Of course you'd say that...you have the brainpan of a stagecoach tilter!

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2010-05-04 12:40

@ ElMaestro
Posting: # 5285
Views: 27,570

## Mixed MoM

Dear EM,

» Nice post d_labes,

Thanx for the flowers.

» Last, to address my own bonus question partially, I think there could be a practical difference between the FDA approach and the way by a Mixed Model. They may achieve exactly the same in the cases where there are no missing values.

That was also my opinion before. But what I liked to illustrate with my post is that this is not always true. We have here no missing values, balance also. But very different results of the first part of the scaled average BE criterion.

BTW: I'm not sure if MoM is a misnomer. But in the FDA guidance the analogous evaluation if the IBE and PBE criterion via sample statistics is named such.

BTW2: I'm not sure where the bear dataset comes from and if the described phenomenon is real or relies on an artificial dataset.

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2010-05-04 13:08

@ d_labes
Posting: # 5286
Views: 27,556

## Mixed MoM

Dear d_labes,

» That was also my opinion before. But what I liked to illustrate with my post is that this is not always true. We have here no missing values, balance also. But very different results of the first part of the scaled average BE criterion.

I agree with this, sorry if I did not explain myself correctly, my proposal was that the sigma-values are (must be) depending on the way the covariance matrix V is specified. Thus, it would be interesting to see the R and G's for the CSH and UN fits.

Best regards
EM.

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2010-05-05 08:42

@ ElMaestro
Posting: # 5292
Views: 27,745

## R and G, CSH and UN

Dear Grosser Meister,

My Power Beast told me:
CSH fit:
               Estimated G Matrix  Row  Effect  code  subject    Col1      Col2    1  code    R      1       0.005653  0.005251    2  code    T      1       0.005251  0.004878        Estimated R matrix for subject 1  Row    Col1      Col2     Col3     Col4    1   0.01974    2            0.01974    3                     0.02010    4                              0.02010 
UN fit:
               Estimated G Matrix  Row  Effect  code  subject   Col1      Col2    1  code    R      1                0.01202    2  code    T      1       0.01202        Estimated R matrix for subject 1  Row       Col1     Col2     Col3    Col4    1     0.02732    2              0.02732    3                       0.02476    4                                0.02476 
Seems the R matrix is sorted by R1, R2, T1, T2.

The G matrix of the UN fit is rather strange I think.

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2010-05-05 09:49

@ d_labes
Posting: # 5294
Views: 27,652

## A sailor gets confused

Dear d_labes,

» My Power Beast told me (...)
» Seems the R matrix is sorted by R1, R2, T1, T2.

Thanks for all this.
I think I understand the R's, but I would like your help understanding how the G's from SAS can be read.

The reason why I am somewhat bewildered is as follows:
If you have say m observations on n subjects and subject is the only random effect*, then I would expect the Z matrix to have exactly m rows and n columns; can be parameterised by one column for per subject, i.e. no intercept, or can be specified with an intercept and same total number of cols. Same thing, just different parameterisation.
(I find it easier to grasp the former, where one column corresponds to one subject and hence no intercept).
Now, for ZGZt we know that "Multiplying an m×n matrix with an n×p matrix results in an m×p matrix", so as our Z is m×n, G must be n×n, Zt is n×m and law an order is restored in V being m×m.
Am I mistaking something here?
If not, then G in your above post was 2×2, so obviously SAS expects the user to be able to figure out the rest. For matrix R, SAS spat out just a block, so the entire R can be constructed by just extending that diagonal down all subjects, but for G I am not sure how to expand the thing so that ZGZt comes into existence. Can you tell?

Is there a way you can ask the power to spit out V too for these models?

I have previous to this post looked at various webpages dealing with definitions of R and G's because getVarCov in the R software also spits out G in this odd fashion. The good places to start are in my opinion here and here. I also have the book by Pinheiro and Bates. unfortunately, in order to understand anything in that book you need an IQ of at least 200 and it does not cover this aspect at all.

Best regards
EM.

*: More random effects = ultimately more columns in Z.

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2010-05-05 12:49

@ ElMaestro
Posting: # 5300
Views: 27,614

## Confused V/Z matratzen and tips

Dear confused sailor,

willingly I do you the favor:

CSH fit:
          Estimated V Matrix for subject 1  Row        Col1        Col2        Col3        Col4    1     0.02540    0.005652    0.005251    0.005251    2    0.005652     0.02540    0.005251    0.005251    3    0.005251    0.005251     0.02498    0.004878    4    0.005251    0.005251    0.004878     0.02498 
UN fit:
          Estimated V Matrix for subject 1  Row        Col1        Col2        Col3        Col4    1     0.02732                 0.01202     0.01202    2                 0.02732     0.01202     0.01202    3     0.01202     0.01202     0.02476    4     0.01202     0.01202                 0.02476
Same block for other subjects.

Unfortunately I have not found an option to give you directly the Z matrix to temper your confusion.
My suggestion: Go to sea and let the storm blow your brain clear .

For 1 subject I suppose (if the four values are sorted by tmt and replicate number):
       random tmt Tmt   R    T R1    1    0 R2    1    0 T1    0    1 T2    0    1
According to your clear and concise memo of the rules of matrix multiplication this gives a 4x4 matrix for Z*G*Zt. Hope this helps.

Another tip: Have a look into

Helen Brown, Robin Prescott
"Applied Mixed Models in Medicine"
Second edition, John Wiley & Sons 2006
ISBN: 0-470-02356-2

There are a lot of matrix specifications in it.

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2010-05-05 19:32

@ d_labes
Posting: # 5305
Views: 27,462

## superb post!

Dear d_labes,

This is incredibly informative - thanks a lot. CSH provides a reasonable parametrisation of V, in my opinion, but I just can't say the same about UN.
Obs 1 & Obs 2 (and 3 & 4, etc) must co-vary the way the data were ordered.

I think I will bet my money on CSH for future applications ("but the guideline says....").

Best regards and again thanks.

EM.

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2010-05-06 09:00

@ ElMaestro
Posting: # 5311
Views: 27,490

## Gambling with UN

Dear ElMaestro,

» I think I will bet my money on CSH for future applications ("but the guideline says....").

Don't be so lightheaded with your money. The first million is the difficilest .

For the ABE evaluation (Proc MIXED code) the FDA statistics guideline recommends FA0(2). And then "... In the Random statement, TYPE=FA0(2) could possibly be replaced by TYPE=CSH. This guidance recommends that TYPE=UN not be used, as it could result in an invalid (i.e., not nonnegative definite) estimated covariance matrix ...". Sic!

But if we look at the results via intra-subject contrast for the Scaled ABE evaluation it seems that this estimation method is implicit an UN evaluation.

What I really don't understand in case of a full replicate study (4-period study) why not use the results of the ABE code and then do the calculations for the scaled BE criterion?! All necessary terms are at hand, point estimate of T-R with CI (or SD), intra-subject sWR.

Ok, for the extra-reference design we don't have a working code via Proc MIXED.

Regards,

Detlew
yjlee168
★★

Kaohsiung, Taiwan,
2010-05-05 07:12

@ d_labes
Posting: # 5291
Views: 27,571

## Mixed MoM

Dear D. Labes,

The dataset (2x2x2 crossover, single-dosed) in bear was from a real BE study. Others, such as parallel or replicate crossovers, were artificial, as far as I can remember. Thanks.

» BTW2: I'm not sure where the bear dataset comes from and if the described phenomenon is real or relies on an artificial dataset.

All the best,
-- Yung-jin Lee
bear v2.8.6:- created by Hsin-ya Lee & Yung-jin Lee
Kaohsiung, Taiwan http://pkpd.kmu.edu.tw/bear
KR
★

India,
2010-04-27 11:14
(edited by Jaime_R on 2010-04-27 11:50)

@ ddubins
Posting: # 5246
Views: 27,887

## Reference scaled BE - FDA (and FARTSSIE update)

Dear Dave,

It is really a valuable and appreciable work done. Thanks.

However, in "Bioequivalence, Replicate" sheet, I have one concern: FDA recommends 3 period, 3 sequence crossover study but there is only one option of 3 period, 2 sequence (ABB/BAA) in the excel sheet which is not matching with the sequences given in the FDA recommendation. So in this case,the sample size calculated from the xls sheet and the sample size required for FDA recommended reference scaled study will remain the same ? Same case is for 4 period replicate study also.

Kindly provide your valuable comment on the same.

Thanks once again.

KR

Helmut
★★★

Vienna, Austria,
2010-04-27 12:58

@ ddubins
Posting: # 5249
Views: 28,031

## Reference scaled BE - EMA

Dear Dave & all,

THX again for the update!

<nitpicking>

What do you think about the switching condition thetas (aka regulatory constant k)?

</nitpicking>

EMA did not give details of the calculation and (for simplicity?) used k=0.760. Actually it's based on the regulatory standardized variation sigma0 (i.e., the limit for HVDs/HVDPs). For CVWR=30% we get sigma0=sqrt[ln(0.3²+1)]=0.293560379... and thetas=k=ln(1.25)/sigma0=0.760128298... Backcalculating from 0.760 one would get a limit for HVDs/HVDPs of 30.005%.
This explains small discrepancies between tables of Tóthfalusi et al. (2009), EMA (2010), and exact values.

        Tóthfalusi      EMA GL         exact CV (%)   L     U      L      U       L      U  30     80.0 125.0   80.00 125.00   80.00 125.00  35     77.2 129.5   77.23 129.48   77.23 129.49  40     74.6 134.0   74.62 134.02   74.61 134.02  45     72.2 138.6   72.15 138.59   72.15 138.60  50     69.8 143.2   69.84 143.19   69.83 143.20  55     67.7 147.8   like CV 50 %   67.65 147.81  60     65.6 152.4   like CV 50 %   65.61 152.42

In other words, what value would you use? 0.760 because stated as such in the guideline, or the exact value based on CVWR 30%? It's similar to the limit for NTIDs: L=0.9 and U=1/L=1.1... (not 1.1111).

@Dave: For Method A – Direct Expansion of BE limits (outdated in the EU soon, but still applicable in other countries) the limits are L=0.75 and U=1/L=1.3 (not 1.33 as in cell Y10).

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
ddubins
☆

Toronto,
2010-04-27 15:52

@ Helmut
Posting: # 5250
Views: 28,224

## Reference scaled BE - EMA

Hi everyone,

Thanks for noticing this post The reason I came across this is because about a year ago, a number of people asked me how to run the stats for scaled BE (FDA), and I wrote the FDA to ask them how they did it. Yesterday, the FDA called me to notify me of the progesterone recommendations.

I'm by no means a statistician, but I will try to address some of the questions as I understand them.

D. Labes:
BTW: What is missing is explicit code for ABE evaluation for the partial replicate design in case of sWR < 0.294 (CV = 30%): "The two one-sided test procedure must be used for PK parameters with sWR < 0.294."

The explicit code for this is on the last page (page 9) of the progesterone guidance. It's just the unscaled approach. You can use that method for both 4-period and 3-period designs.

BTW2: It is not clear why they recommend Proc MIXED in case of fully replicate design and Proc GLM in case of partial replicate. In both cases the call to these procedures have only sequence as fixed effect. They only serve as means to get the sequence "stratified" means of the corresponding inter-subject contrasts. Or is this meant with respect to the unscaled ABE evaluation?

I don't know why they go back and forth on this. Up until this point I have only ever seen MIXED used in any replicate design, and to be perfectly honest this nuance along with the contrasts is completely above my head. When one method crashed when I was in CRO land, we would use the other and write a protocol deviation for the statistical analysis.

ElMaestro:
Bonus questions: Pass - I have no idea.

KR: I had this concern myself, since the FDA design 3-period replicate only repeats the test formulation. I emailed my sample size guru and oracle, Stephen Julious, back in February to confirm. Stephen said,

"It would not effect the sample size but I would push back on the sequences they recommend as you are not replicating the test treatment (I assume T and R mean Test and Reference) only the Reference which we are least interested in. The “standard” design is TRR RTT"

I had a sort of logical overload thinking about this problem because the TRR/RRT trial will have the same number of direct T-R and R-T comparisons as a 2-way. Picture smoke coming out of my ears from over-heating. In the end, I just put it away when I received Stephen's reply. After all, there has to be some advantage in sample size if you run another period.

Helmut:
In general, I would always take the guidance literally, instead of scientifically. For instance, when the EMEA general BA/BE guidance said:
"The interval must be prospectively defined e.g. 0.75-1.33" I never took the reciprocal of 0.75 to give 1.33 repeating, even though in theory that would make the most sense. This is for the sole reason that you never know when you are going to hit THAT regulator - the type A that doesn't care what makes scientific sense and takes every digit literally. So when the new guidance says 0.760, even though you and I both know that it doesn't back-calculate to a perfect 30.0% CV, guess what I would recommend using?

This problem pops up again and again. Look at the older Canadian guidelines for narrow therapeutics:

http://www.hc-sc.gc.ca/dhp-mps/alt_formats/pdf/prodpharma/applic-demande/guide-ld/bio/critical_dose_critique-eng.pdf

The requirements are 90.0-112.0%, not 90.0-111.11 repeating. They didn't even round down, they rounded UP. So I became a to-the-letter kind of guy, despite my academic side crying "unfair! unfair!". I learned to suppress those little hairs standing up at the back of my neck.

One thing I was happy about with the FDA progesterone recommendations was that the FDA specified 0.294 for sig_wr, because a few of their previous publications recommended using 0.25, which didn't map close to the 30% intraCV cutoff. I can drop the extra decimals (0.293560379) and still sleep ok. I presented this discrepancy in a talk at Washington State University, and it's almost worth flying back to Pullman to tell them the end of the story. Well, almost. There were 3 stop overs to that trip, and it took almost 12 hours to get there!!!! Maybe I'll just email.

Glad to hear from you guys,
-Dave

David Dubins, Ph.D., B.Eng.
Instructor and Laboratory Coordinator
Leslie Dan Faculty of Pharmacy
University of Toronto
d_labes
★★★

Berlin, Germany,
2010-04-28 07:47

@ ddubins
Posting: # 5252
Views: 27,876

## Partial replicate design and FDA ABE code

Hi Dave,

» » D. Labes:
» » BTW: What is missing is explicit code for ABE evaluation for the partial replicate design ...
»
» The explicit code for this is on the last page (page 9) of the progesterone guidance. It's just the unscaled approach. You can use that method for both 4-period and 3-period designs.
(underlined by me)

I do not believe that. We have experimented with the FDA code on partial replicate design here in the forum and got some strange results. See f.i. here and here.
IMHO this is due to the fact that the intra-subject variance for Test is not identifiable in this design.

BTW: Regarding the sample size question of my very dear friend KR have a look into this tread.

Regards,

Detlew
Helmut
★★★

Vienna, Austria,
2010-04-28 12:46

@ ddubins
Posting: # 5258
Views: 27,752

## Reference scaled BE - EMA

Dear Dave!

» In general, I would always take the guidance literally, instead of scientifically.

I never did that (and will not do so in the future).

» For instance, when the EMEA general BA/BE guidance said: "The interval must be prospectively defined e.g. 0.75-1.33" I never took the reciprocal of 0.75 to give 1.33 repeating, even though in theory that would make the most sense. This is for the sole reason that you never know when you are going to hit THAT regulator - the type A that doesn't care what makes scientific sense and takes every digit literally.

Well I always stated the limits based on the acceptable difference (). For the ‘old’ extended range it’s 25%. So in my protocols I gave L=1- and U=1/L. Never got a deficiency letter.

» So when the new guidance says 0.760, even though you and I both know that it doesn't back-calculate to a perfect 30.0% CV, guess what I would recommend using?

The back-calculated CV was just an example. I would recommend to use the exact value.

» This problem pops up again and again. Look at the older Canadian
» guidelines for narrow therapeutics:
» The requirements are 90.0-112.0%, not 90.0-111.11 repeating. They didn’t even round down, they rounded UP. So I became a to-the-letter kind of guy, despite my academic side crying “unfair! unfair!”. I learned to suppress those little hairs standing up at the back of my neck.

Right, it’s also in the recent draft. I asked Eric Ormsby about that at the last BioInternational Conference and he said they have suggested this value for simplicity. I guess Eric has a low opinion about the IQ of people in the pharmaceutical industry.

» […] There were 3 stop overs to that trip, and it took almost 12 hours to get there!!!!

Come on. Last week I stranded in Brussels (you know, the volcano-business) and it took me five hours queuing to get a train ticket and finally 21 hours (including change trains six times) to go back to Vienna.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
d_labes
★★★

Berlin, Germany,
2010-05-04 12:04

@ ddubins
Posting: # 5284
Views: 27,687

## Switching - null sWR

Dear Dave, dear All,

» One thing I was happy about with the FDA progesterone recommendations was that the FDA specified 0.294 for sig_wr, because a few of their previous publications recommended using 0.25, which didn't map close to the 30% intraCV cutoff.

Please note sigmaS=0.294 is only the switching variability i.e. from which on the scaled ABE criterion is allowed if sWR>sigmaS. In their SAS code the regulatory constant s0 is still 0.25.
See f.i. on page 6 and 8 of the progesterone draft guidance:
  theta=((log(1.25)/0.25)**2
(=0.89257422)

This leads to the discontinuity at CV=30% and the inflation of the consumer's risk around CV=30% as described by:
L. Endrenyi and L. Tothfalusi
"Regulatory Conditions for the Determination of Bioequivalence of Highly Variable Drugs"
J. Pharm. Pharmaceut. Sci. (www.cspsCanada.org) 12 (1): 138 - 149, 2009

Regards,

Detlew
d_labes
★★★

Berlin, Germany,
2010-05-06 11:26

@ ddubins
Posting: # 5312
Views: 27,669

## The unknown x

Dear Dave, dear All,

this question is not solely to you Dave, but I must have an entry point.

After looking closer at the SAS code I'm sitting here and wondering about the analysis of the Iij which covers the first part

(YT-YR)2

of the linearized scaled ABE criterion.
Does anyone know what this ominous x in the code is and where it comes from .

"[snip] -----
From the dataset IGLM2 (or IOUT2), calculate the following:
IGLM2: pointest=exp(estimate);        x=estimate**2-stderr**2;        boundx=(max((abs(LowerCL)),(abs(UpperCL))))**2;
[/snip] -----".

See on page 5 or 8 of the FDA Progesterone DRAFT guidance.

Regards,

Detlew
jdetlor
☆

2010-09-28 15:22

@ d_labes
Posting: # 5943
Views: 26,979

## The unknown x

Dear d_labes and all

» Does anyone know what this ominous x in the code is and where it comes from .
»
» "[snip] -----
» From the dataset IGLM2 (or IOUT2), calculate the following:
» IGLM2: pointest=exp(estimate);
»        x=estimate**2-stderr**2;
»        boundx=(max((abs(LowerCL)),(abs(UpperCL))))**2;
» [/snip] -----".
»
» See on page 5 or 8 of the FDA Progesterone DRAFT guidance.

I saw this and thought I'd post my opinion.

I believe the calculated x value represents the 'true' value of squared difference ((YT-YR)2)). If we simply square the log difference, we actually have
(YT-YR)2)) + sigma_d2
(As shown here)

Approximation I from the Howe paper referenced in the Progesterone guidance requires the first moment for each part of the reference-scaled criterion (labeled as x and y in the FDA SAS code), thus the variability must be subtracted to obtain this first moment for the squared difference.

This issue can also be highlighted through a simulation where random full (or partial) replicate results of the log difference are squared. Averaging over all simulations, the estimate of the squared log difference is the squared 'true' log difference (as specified in the input simulation parameters) plus the twice the intra-subject variability. In this case it is the variance of the log difference and not the squared standard error of the log difference because we are simulating the overall distribution of the log difference.

J. Detlor
d_labes
★★★

Berlin, Germany,
2010-10-01 10:14

@ jdetlor
Posting: # 5970
Views: 26,846

## Unknown x corrected?

Dear J. Detlor,

» I saw this and thought I'd post my opinion.
»
» I believe the calculated x value represents the 'true' value of squared difference ((YT-YR)2)). If we simply square the log difference, we actually have (YT-YR)2)) + sigma_d2 (As shown here)

But I'm not quite sure if this applies in our context.
What makes me curious about this is that in the various papers about scaled ABE and also about Individual BE (where (µTR)2 is also part of the IBE criterion) there is not any hint of such a 'correction'.
.

Regards,

Detlew
Qilex
☆

France,
2016-09-21 10:25

@ d_labes
Posting: # 16656
Views: 17,329

## Unknown x corrected?

Hi Detlew

» But I'm not quite sure if this applies in our context.
» What makes me curious about this is that in the various papers about scaled ABE and also about Individual BE (where (µTR)2 is also part of the IBE criterion) there is not any hint of such a 'correction'.
» .

Any update on the "corrected" x value? The two Lászlós presented a method to "calculate the confidence limits of the reference-scaled SABE by modifying the method of Hyslop et al". The latter method "utilizes Howe's approximation I". The FDA draft guidance on progesterone states that "the method of obtaining the upper confidence bound is based on Howe’s Approximation I".

1. Hyslop T, Hsuan F, Holder DJ. A small sample confidence interval approach to assess individual bioequivalence. Stat Med 2000;19(20):2885–2897.
2. Tothfalusi L, Endrenyi L, Arieta AG. Evaluation of bioequivalence for highly variable drugs with scaled average bioequivalence. Clin Pharmacokinet 2009;48(11):725–743.
3. Tothfalusi L, Endrenyi L, Midha KK, et al. Evaluation of the bioequivalence of highly-variable drugs and drug products. Pharm Res 2001;18(6):728–733.

Philippe
d_labes
★★★

Berlin, Germany,
2016-09-21 11:22

@ Qilex
Posting: # 16658
Views: 17,299

## Official unknown x

Dear Philippe,

» Any update on the "corrected" x value?

The first "official" statement regarding the "unknown x" is from a presentation of Donald Schuirmann held on "The Global Bioequvalence Harmonization Initiative: EUFEPS/AAPS Second International Conference", Washington, Sep. 15-16, 2016.

Quote from the Handouts:
• The statistical approach we use is very similar to that proposed by Tothfalusi, Endrenyi, et al. 2001, with a minor difference (use of an unbiased estimator for T – µR)2.)
• SAS code to implement the procedure was included in the Draft Guidance on Progesterone (2010).
Thus the user Detlor seems right in his explanation years ago.

Regards,

Detlew
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