ddubins ☆ Toronto, 20100426 18:28 Posting: # 5241 Views: 30,452 

Hi group, The FDA just recently posted a draft individual product bioequivalence recommendation for progesterone capsules, which provides useful SAS code for their referencescaled approach. It goes in to extremely useful detail: http://www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/UCM209294.pdf This is a critical read for anyone using the FDA's referencescaled approach. I updated my sample size program, FARTSSIE, incorporating the EMEA referencescaled and FDA referencescaled 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, 20100427 06:58 @ ddubins Posting: # 5243 Views: 27,968 

Hi Dave, wow! Very valuable contribution! Fascinating: The codes provided rely solely on suitably choosen intrasubject 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 (4period) replicate design. BTW: What is missing is explicit code for ABE evaluation for the partial replicate design in case of s_{WR} < 0.294 (CV = 30%): "The two onesided test procedure must be used for PK parameters with s_{WR} < 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 intersubject contrasts. Or is this meant with respect to the unscaled ABE evaluation? — Regards, Detlew 
ElMaestro ★★★ Belgium?, 20100427 11:39 @ d_labes Posting: # 5247 Views: 28,015 

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 intraRef sigma could be derived where replicated Test is not available? I wonder if SAS has this ability for partial repliacted studies at all. But what about this (page 5, emphasis in red by me): 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, 20100427 12:30 @ ElMaestro Posting: # 5248 Views: 27,884 

Dear sailor and simulant, » But what about this (page 5, emphasis in red by me): » 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 R1R2 which has expected variance =2*sigma^{2}_{WR}. The trick of the GLM call (diffs versus sequence) gives the MS of R1R2 "stratified" over the sequences (double sum in the formula for s^{2}_{WR} in step 1 on page 2 divided by (nm)). You can verify this by conventional calculation. Since I have answered your question I'm exempted from answering the bonus questions . To be honest: I don't know exactly. But I conjecture: Yes in balanced design without missings. — Regards, Detlew 
d_labes ★★★ Berlin, Germany, 20100504 10:10 @ ElMaestro Posting: # 5279 Views: 28,055 

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 PKmetrics 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 (logtransformed). # tmt2=T1, ... based on replicate number Note the attention: comments to see that R is also sometimes a hard to repressing beast. Using the bear example dataset for replicate studies (4period2sequence study with sequences RTRT/TRTR and 14 subjects balanced over sequence groups) I get:
point lower upper Unbiased estimate of point estimator TR 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 ("Crossover Trials in Clinical Research" page 237 ff) who showed a similar effect due to a negative correlation of the two withinsubject contrasts T1R1 and T2R2 which lead to a negative subject by formulation interaction if the constraints that variancecovariance 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 exactly the same as via the intrasubject 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 intraindividual variances (in the log domain): MoM Proc Mixed FDA (REML) — Regards, Detlew 
ElMaestro ★★★ Belgium?, 20100504 10:54 @ d_labes Posting: # 5281 Views: 27,598 

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 intraindividual variances (in the log domain): » MoM Proc Mixed FDA (REML) » s^{2}_{WT} 0.030130 0.02010 » s^{2}_{WR} 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=ZGZ^{t}+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, 20100504 12:40 @ ElMaestro Posting: # 5285 Views: 27,570 

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?, 20100504 13:08 @ d_labes Posting: # 5286 Views: 27,556 

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 sigmavalues 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, 20100505 08:42 @ ElMaestro Posting: # 5292 Views: 27,745 

Dear Grosser Meister, My Power Beast told me: CSH fit:
Estimated G Matrix UN fit:
Estimated G Matrix 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?, 20100505 09:49 @ d_labes Posting: # 5294 Views: 27,652 

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 ZGZ^{t} 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, Z^{t} 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 ZGZ^{t} 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, 20100505 12:49 @ ElMaestro Posting: # 5300 Views: 27,614 

Dear confused sailor, willingly I do you the favor: CSH fit: Estimated V Matrix for subject 1 UN fit: Estimated V Matrix for subject 1 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 According to your clear and concise memo of the rules of matrix multiplication this gives a 4x4 matrix for Z*G*Z^{t}. 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: 0470023562 There are a lot of matrix specifications in it. — Regards, Detlew 
ElMaestro ★★★ Belgium?, 20100505 19:32 @ d_labes Posting: # 5305 Views: 27,462 

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 covary 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, 20100506 09:00 @ ElMaestro Posting: # 5311 Views: 27,490 

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 intrasubject 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 (4period 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 TR with CI (or SD), intrasubject s_{WR}. Ok, for the extrareference design we don't have a working code via Proc MIXED. — Regards, Detlew 
yjlee168 ★★ Kaohsiung, Taiwan, 20100505 07:12 @ d_labes Posting: # 5291 Views: 27,571 

Dear D. Labes, The dataset (2x2x2 crossover, singledosed) 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,  Yungjin Lee bear v2.8.6: created by Hsinya Lee & Yungjin Lee Kaohsiung, Taiwan http://pkpd.kmu.edu.tw/bear Download link (updated) > here 
KR ★ India, 20100427 11:14 (edited by Jaime_R on 20100427 11:50) @ ddubins Posting: # 5246 Views: 27,887 

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 Edit: Full quote removed (like in almost all your replies). Please delete anything from the text of the original poster which is not necessary in understanding your answer; see also this post! [Jaime] 
Helmut ★★★ Vienna, Austria, 20100427 12:58 @ ddubins Posting: # 5249 Views: 28,031 

Dear Dave & all, THX again for the update! <nitpicking> What do you think about the switching condition theta_{s} (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 sigma_{0} (i.e., the limit for HVDs/HVDPs). For CV_{WR}=30% we get sigma_{0}=sqrt[ln(0.3²+1)]=0.293560379... and theta_{s}=k=ln(1.25)/sigma_{0}=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 In other words, what value would you use? 0.760 because stated as such in the guideline, or the exact value based on CV_{WR} 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, 20100427 15:52 @ Helmut Posting: # 5250 Views: 28,224 

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 onesided 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 4period and 3period 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 intersubject 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 3period 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 TR and RT comparisons as a 2way. Picture smoke coming out of my ears from overheating. 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. 0.751.33" 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 backcalculate 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.hcsc.gc.ca/dhpmps/alt_formats/pdf/prodpharma/applicdemande/guideld/bio/critical_dose_critiqueeng.pdf The requirements are 90.0112.0%, not 90.0111.11 repeating. They didn't even round down, they rounded UP. So I became a totheletter 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, 20100428 07:47 @ ddubins Posting: # 5252 Views: 27,876 

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 4period and 3period 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 intrasubject 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, 20100428 12:46 @ ddubins Posting: # 5258 Views: 27,752 

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. 0.751.33" 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 backcalculate to a perfect 30.0% CV, guess what I would recommend using? The backcalculated 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: » http://www.hcsc.gc.ca/dhpmps/alt_formats/pdf/prodpharma/applicdemande/guideld/bio/critical_dose_critiqueeng.pdf » The requirements are 90.0112.0%, not 90.0111.11 repeating. They didn’t even round down, they rounded UP. So I became a totheletter 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 volcanobusiness) 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, 20100504 12:04 @ ddubins Posting: # 5284 Views: 27,687 

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 sigma_{S}=0.294 is only the switching variability i.e. from which on the scaled ABE criterion is allowed if s_{WR}>sigma_{S}. In their SAS code the regulatory constant s_{0} is still 0.25. See f.i. on page 6 and 8 of the progesterone draft guidance: theta=((log(1.25)/0.25)**2 (=0.8925742^{2}) 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 Edit: online resource added. [Helmut] — Regards, Detlew 
d_labes ★★★ Berlin, Germany, 20100506 11:26 @ ddubins Posting: # 5312 Views: 27,669 

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 I_{ij} which covers the first part (Y_{T}Y_{R})^{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); [/snip] ". See on page 5 or 8 of the FDA Progesterone DRAFT guidance. — Regards, Detlew 
jdetlor ☆ 20100928 15:22 @ d_labes Posting: # 5943 Views: 26,979 

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**2stderr**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 ((Y_{T}Y_{R})^{2})). If we simply square the log difference, we actually have (Y_{T}Y_{R})^{2})) + sigma_d^{2} (As shown here) Approximation I from the Howe paper referenced in the Progesterone guidance requires the first moment for each part of the referencescaled 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 intrasubject 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, 20101001 10:14 @ jdetlor Posting: # 5970 Views: 26,846 

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 ((Y_{T}Y_{R})^{2})). If we simply square the log difference, we actually have (Y_{T}Y_{R})^{2})) + sigma_d^{2} (As shown here) Thanks for your opinion. 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 (µ_{T}µ_{R})^{2} is also part of the IBE criterion) there is not any hint of such a 'correction'. . — Regards, Detlew 
d_labes ★★★ Berlin, Germany, 20160921 11:22 @ Qilex Posting: # 16658 Views: 17,299 

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. 1516, 2016. Quote from the Handouts:
— Regards, Detlew 