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Dr_Dan
Senior

2015-06-24 13:55

Posting: # 14974
Views: 14,096
 

 3 period full repli­cate [RSABE / ABEL]

Dear all
From a leading European regulatory authority I just recieved the information that a 3-period full replicate design (two sequences: T-R-T and R-T-R) for a bioequivalence study will not be accepted. The reason is that

for a full replicate design R as well as T need to be replicated and this is only possible with the res­pec­tive sequences (so TTRR etc.). A 3-period full replicate design would result only in two parallel sepa­rate groups (one which – apart from the integrated 2-period crossover– only get Test and the other only Reference); an evaluation would probably only possible as a general 2-period crossover study.

Please excuse the confusing explanation but the text set in italic is the verbal translation from what I received.
My interest is just of academic nature: What should one reply to this attitude?
Looking forward to your replies and to a fruitful discussion.
Kind regards
Dr_Dan


Edit: Category changed from Design Issues. I think this category fits better – especially if one is inter­ested in reference-scaling. [Helmut]

Kind regards and have a nice day
Dr_Dan
ElMaestro
Hero

Denmark,
2015-06-24 14:11

@ Dr_Dan
Posting: # 14975
Views: 12,881
 

 3 period full replicate

Hi Dr_Dan,

» My interest is just of academic nature: What should one reply to this attitude?

How about something like:

"From a statistical perspective the added value of a fully replicated trial, in contrast to a semi-replicated trial, is that it contributes direct information about the within-subject variability for the Test product. This information has, however, in its own right no regulatory use because:
  1. Population bioequivalence and individual bioequivalence are abandoned concepts.
  2. Reference-scaled bioequivalence relies solely on estimates of Test and Reference fixed effects along with within-subject variability associated with the Reference product (all of which are provided via a semi-replicated trial).
A fully replicated trial involves four periods with IMP administration to each subject, while a semi-replicated trial involves just three periods with IMP administration. From the perspective of the trial subject the burden associated with the fully replicated design is higher. Since the additional information obtained in a fully replicated trial does not contribute towards the average bioequivalence conclusion -regardless of whether scaling is actually applied or not- it is contended that the fully replicate design is associated with added risk but not additional benefit. In accordance with GCP clause 2.2 the fully replicated design therefore has no obvious merit."


:-)

if (3) 4

Best regards,
ElMaestro

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
Helmut
Hero
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Vienna, Austria,
2015-06-24 14:29

@ Dr_Dan
Posting: # 14976
Views: 13,008
 

 3 period full replicate

Dear Dan,

[image]this statement is outright bizarre (politely speaking). :not really:

» What should one reply to this attitude?

Suggest the “leading European regu­la­tory authority” to read some stuff (»Lesen bildet!«).

One of the authors (László Tóth­falusi) is a former member of the PKWP and an­other one (Alfredo García-Arieta) still is:


    References:
  1. 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. PMID 11474774
  2. Tóthfalusi L, Endrényi L. Limits for the Scaled Average Bioequivalence of Highly Variable Drugs and Drug Products. Pharm Res. 2003;20(3):382–9. PMID 12669957
  3. 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
  4. 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


Edit (after reading the authority’s statement again):

» A 3-period full replicate design would result only in two parallel separate groups (one which […] only get Test and the other only Reference) […]

I beg your pardon? :blind:
Sequence 1: TRT only Test?
Sequence 2: RTR only Reference?

» an evaluation would probably only possible as a general 2-period crossover study

What? Throw away the data of the third period? They don’t get the terminology right and mix up groups with sequences. If I follow this logic, they could also say:

A 2-period crossover design would result in two separate groups (one which gets Test and Reference in the order TR and the other one in the order RT); an evaluation would probably only possible as two separate paired designs – assuming no period effects.

Gimme a break.

Cheers,
Helmut Schütz
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d_labes
Hero

Berlin, Germany,
2015-06-24 14:38

@ Dr_Dan
Posting: # 14977
Views: 12,843
 

 3 period full replicate

Dear Dr_Dan!

» From a leading European regulatory authority I just recieved the information that a 3-period full replicate design (two sequences: T-R-T and R-T-R) for a bioequivalence study will not be accepted. The reason is that for a full replicate design R as well as T need to be repclicated and this is only possible with the respective sequences (so TTRR etc.) ...

Seems the story "Potvin C is not valid in Europe" is duplicated here on another field!

The full replicate 3-period design has the sequences TRT / RTR as is stated in your post and of course has T and R replicated. Not in each subject, but if balanced in half of ntotal each. And this design of course allows the evaluation of the intra-subject variabilities of T and R separately.

So what's the problem of the "leading European regulatory authority"?
Especially in the light of ElMaestro's remark that only the intra-subject variability of R is of any regulatory concern if scaled ABE is aimed for :confused:.

Regards,

Detlew
Dr_Dan
Senior

2015-06-24 16:43

@ d_labes
Posting: # 14978
Views: 12,787
 

 3 period full replicate

Dear Detlew

» The full replicate 3-period design has the sequences TRT / RTR as is stated in your post and of course has T and R replicated. Not in each subject, but if balanced in half of ntotal each. And this design of course allows the evaluation of the intra-subject variabilities of T and R separately.

I think the question is whether a replication in only half of ntotal is sufficient in order to scale or ???

» So what's the problem of the "leading European regulatory authority"?
» Especially in the light of ElMaestro's remark that only the intra-subject variability of R is of any regulatory concern if scaled ABE is aimed for :confused:.

With regard to ElMaestro's reply: in a former statement the "leading European regulatory authority" had no problem with full replicate design (=4 periods and 6 possible sequences) and half replicate designs (3 periods and 3 sequences). A full replicate 3-period design would not work according to their argumentation since a 3 period design implicity needs 3 sequences.
I do not know why. Maybe you have an idea?
Kind regards
Dr_Dan

Kind regards and have a nice day
Dr_Dan
Helmut
Hero
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Vienna, Austria,
2015-06-24 18:00

@ Dr_Dan
Posting: # 14979
Views: 12,964
 

 3 period full replicate

Dear Dan et alii,

» I think the question is whether a replication in only half of ntotal is sufficient in order to scale or ???

Yes it is. Sometimes (!) you’ll need (slightly) more subjects.

     TRR|RRT|RTR   TRT|RTR  
 CV    n   power   n   power
0.30  27  0.8257  26  0.8191
0.35  30  0.8381  28  0.8037
0.40  27  0.8004  30  0.8156
0.45  30  0.8430  30  0.8064
0.50  30  0.8260  32  0.8174
0.60  36  0.8192  36  0.8013
0.70  45  0.8113  46  0.8187
0.80  57  0.8223  56  0.8155
1.00  78  0.8137  76  0.8025


Note that the partial replicate is a crappy design and sample size estimations are approximate at its best. For details see Detlew’s tractatus in your R-installion folder
…\library\PowerTOST\Implementation_scaledABE_sims.pdf
See also this thread.

» in a former statement the "leading European regulatory authority" had no problem with full replicate design (=4 periods and 6 possible sequences)…

Four periods and six sequences? Common are two (RTRT|TRTR). Four were explored in the dark ages of PBE/IBE (RTRT|TRTR|RTTR|TRRT). The analysis was tricky. Of course you could go even beyond that: RTRT|TRTR|RTTR|TRRT|RRTT|TTRR. Have you really done that?

» … and half replicate designs (3 periods and 3 sequences).

Sure. That’s what they seem to believe to be the only possible one.

» A full replicate 3-period design would not work according to their argumentation since a 3 period design implicity needs 3 sequences.

Crap.

» I do not know why.

You are not alone.

» Maybe you have an idea?

Maybe because the TRT|RTR is not mentioned in the Q&A? Probably the “leading European regulatory authority” didn’t realize that the two designs given there are nothing more than examples.

To illustrate [sic] these approaches, […] data from a four-period unbalanced study […] and data from a three-period balanced study […] were analysed.


BTW, I disagree with our Captn’s arguments.

» » From the perspective of the trial subject the burden associated with the fully replicated design is higher.

Only if we compare a 4-period design to 3-period designs. The burden of particular subjects in the partial replicate and the fully replicated 3-period is identical. Given the table above there are cases where less subjects are required in the fully replicated if compared to the partial replicate.

» » Since the additional information obtained in a fully replicated trial does not contribute towards the average bioequivalence conclusion […] it is contended that the fully replicate design is associated with added risk but not additional bene­fit. In accordance with GCP clause 2.2 the fully replicated design therefore has no obvious merit.

Well, that would imply:
  • RTRT|TRTR (which is specifically mentioned in the Q&A) should be avoided.
  • Only the TRR|RRT|RTR is acceptable?

Cheers,
Helmut Schütz
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The quality of responses received is directly proportional to the quality of the question asked. ☼
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ElMaestro
Hero

Denmark,
2015-06-24 18:11

@ Helmut
Posting: # 14980
Views: 12,742
 

 3 period full replicate

Hi Hötzi and all,

» BTW, I can’t follow our Captn’s arguments.

I was too fast, I thought full replicate = four periods, and semi-replicate = three periods (RTR, RRT, TRR) but I see now this was not the designs discussed here. I did not consider the option of full replicate in three periods.
Consider my post above a highly relevant answer to a question that no-one asked, please :-D

Who's got some publications about the performance of RTR / TRT? Do they exist?

if (3) 4

Best regards,
ElMaestro

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
Helmut
Hero
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Vienna, Austria,
2015-06-24 18:25

@ ElMaestro
Posting: # 14981
Views: 12,893
 

 3 period full replicate

Ahoy!

» I was too fast,

A privilege of the youth. :-D

Note that the BE-GL does not state a specific design, only:

If an applicant suspects that a drug product can be considered as highly variable […], a replicate cross-over design study can be carried out.
For the acceptance interval to be widened the bioequivalence study must be of a repli­cate design where it has been demonstrated that the within-subject variability for Cmax of the reference com­pound in the study is >30%.
It is acceptable to apply either a 3-period or a 4-period crossover scheme in the repli­cate design study.


» Who's got some publications about the performance of RTR / TRT? Do they exist?

See the ones mentioned above. AFAIK, the last one was the basis for EMA’s approach; the paper introduced the term “ABEL” = Average Bioequivalence with Expanding Limits. All (!) simu­­la­tions were performed for TRT|RTR (for an example plot see this post). The TRT|RTR is also mentioned in FDA’s Guidance (Appendix C, Table 3) in the context of IBE. Write a letter to László Endrényi asking him what he thinks about the partial replicate. :pirate:
Honestly, I don’t know why the partial replicate entered the European scene. Just because it was examined by the FDA before?


Edit: After the concept paper on HVDs/HVDPs* was published in 2006, I suggested at con­fe­ren­ces that agencies should collect real data and explore the performance of different me­thods. The data exist, since the pre-specied widening of the acceptance range for Cmax to 0.75–1.3333 also required a replicate design. Reaction: Nada, Zero, Τίποτα. The same people don’t get­ting tired to say “I don’t believe in simulations”, in the case of refe­rence-scaling do exactly this – rely entirely on them. OK, fine with me. But the sim’s were done for 3- and 4-period fully repli­cated designs!

@Dan: Ask the agency for the background of their statement.
  • If they carried out simulations with the partial replicate demonstrating that it performs “better” than the 3-period full replicate (ha-ha!), they should publish these results.
  • If this completely unjustified statement is the agency’s policy, don’t select them as a RMS. What about choosing Spain or Hungary?

  • See its discussion points. One of them:
    • Decide what to do if the within-subject variance ratio shows that the test product is more variable than the reference product.
    Only possible in a full replicate design, right?
    The concept paper was more or less written single-handed by László Tóth­falusi and the only one ever (in an Orwellian manner) deleted in October 2007 from EMA’s website.

‘Who controls the past,’ ran the Party slogan,
‘controls the future: who controls the present controls the past.’

George Orwell (Nineteen Eighty-Four, 1949; Chapter 3)


Cheers,
Helmut Schütz
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Science Quotes
d_labes
Hero

Berlin, Germany,
2015-06-25 08:52

@ Helmut
Posting: # 14983
Views: 12,590
 

 3 period full replicate

Dear Helmut et al.

I think it's a waste of time to discuss this bizarre opinion of a "leading European regulatory authority" in detail.
Scientifically there is IMHO not a single argument to underpin this opinion.
All details in Dan's quote are simply not true. Full stop.

Unfortunately it seems that scientific arguments don't play a big role in some regulators "opinions". And much, much more unfortunately regulators don't have to justify their judgements :crying:.

So what's left? Your tip: "Don’t select them as a RMS!"
Another possibility would be to reactivate our Ol' pirate as a regulator :-D

BTW: The handling of the Concept paper on HVDs/HVDPs is revealing ...

Regards,

Detlew
ElMaestro
Hero

Denmark,
2015-06-25 10:37

@ d_labes
Posting: # 14985
Views: 12,542
 

 3 period full replicate

Haha,

» So what's left? Your tip: "Don’t select them as a RMS!"
» Another possibility would be to reactivate our Ol' pirate as a regulator :-D

I am already a bit reactivated :waving: but not in a role where I write the assessment reports. Perhaps that day will arrive, though... :pirate:

if (3) 4

Best regards,
ElMaestro

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
d_labes
Hero

Berlin, Germany,
2015-06-25 11:20

@ ElMaestro
Posting: # 14988
Views: 12,574
 

 OT

Dear ElMaestro,

» I am already a bit reactivated ...

Define a bit. Is it like a bit pregnant? :-D

Regards,

Detlew
ElMaestro
Hero

Denmark,
2015-06-25 11:54
(edited by ElMaestro on 2015-06-25 12:16)

@ d_labes
Posting: # 14989
Views: 12,596
 

 OT

Dear d_labes,

» Define a bit. Is it like a bit pregnant? :-D

Yes.
Of course there are rules for Conflicts of Interest. And they need to be observed and that is just ... not a problem at all despite the bureaucracy.

The real challenge is that bureaucracy aside, in certain parts of the system (and pardon me, I will not define what that means in relation to this post) what matters is possibly not whether you are competent but whether you have the right friends. I have several times read reg. 2010/1235 (and of course also 2001/83 in its entirety) and so far without identifying clauses that allow interpersonal relations to prevail over science. Odd.

Let it be said, that since I prefer not to define "system" above it is entirely possible that I am just reading the wrong legal framework. Besides, I am quite possibly not very competent myself.

So I guess the answer is yes I am a little bit pregnant.

Would like to add: There are many people on the inside who are really trying very hard and very wholeheartedly to keep the administration principles scientific. I feel we owe them our expression of gratitude – often they do not get the thanks they deserve.

if (3) 4

Best regards,
ElMaestro

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
d_labes
Hero

Berlin, Germany,
2015-06-25 13:51

@ ElMaestro
Posting: # 14990
Views: 12,551
 

 OT

Dear ElMaestro,

» So I guess the answer is yes I am a little bit pregnant.

Many beer drinking people suffer from this. Especially man. Earlier or later ... :-D

» ... There are many people on the inside who are really trying very hard and very wholeheartedly to keep the administration principles scientific. I feel we owe them our expression of gratitude – often they do not get the thanks they deserve.

Full ACK!

Regards,

Detlew
Helmut
Hero
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Vienna, Austria,
2015-06-26 13:51

@ Dr_Dan
Posting: # 14993
Views: 12,666
 

 CVs (borderline OT)

Dear Dan,

» […] the question is whether a replication in only half of ntotal is sufficient in order to scale or ???

Adding to this post in the spirit of this thread: We could calculate the width of the confidence interval of the CV taking into account the sample size to achieve the desired power. Try this one (with the default T/R 0.9 and target power 0.8)…

library(PowerTOST)
CV       <- c(seq(0.3, 0.5, 0.05), seq(0.6, 0.8, 0.1), 1)
res      <- matrix(nrow=length(CV), ncol=10, byrow=T,
              dimnames=list(NULL, c("CV%",
                "2×3×3: n |", "df |", "width",
                "2×2×3: n |", "df |", "width",
                "2×2×4: n |", "df |", "width")))
res[, 1] <- 100*CV
side     <- "2-sided"
for(j in 1:length(CV)) {
  res[j, 2] <- sampleN.scABEL(CV=CV[j], design="2x3x3", details=F, print=F)[[8]]
  res[j, 3] <- 2*res[j, 2]-3
  res[j, 4] <- diff(range(CVCL(CV=CV[j], df=res[j, 3], side=side)))
  res[j, 5] <- sampleN.scABEL(CV=CV[j], design="2x2x3", details=F, print=F)[[8]]
  res[j, 6] <- 2*res[j, 5]-3
  res[j, 7] <- diff(range(CVCL(CV=CV[j], df=res[j, 6], side=side)))
  res[j, 8] <- sampleN.scABEL(CV=CV[j], design="2x2x4", details=F, print=F)[[8]]
  res[j, 9] <- 3*res[j, 8]-4
  res[j,10] <- diff(range(CVCL(CV=CV[j], df=res[j, 9], side=side)))
}
print(as.data.frame(round(res, 4)), row.names=F)


… which gives

 CV%  2×3×3: n | df | width  2×2×3: n | df | width  2×2×4: n | df | width
  30        27   51  0.1267        26   49  0.1294        18   50  0.1280
  35        30   57  0.1415        28   53  0.1471        20   56  0.1428
  40        27   51  0.1749        30   57  0.1646        20   56  0.1662
  45        30   57  0.1889        30   57  0.1889        20   56  0.1908
  50        30   57  0.2144        32   61  0.2066        22   62  0.2048
  60        36   69  0.2426        36   69  0.2426        24   68  0.2446
  70        45   87  0.2619        46   89  0.2587        30   86  0.2635
  80        57  111  0.2757        56  109  0.2783        38  110  0.2770
 100        78  153  0.3197        76  149  0.3242        52  152  0.3208


Since the partial replicate (2×3×3) and the 3-period full replicate (2×2×3) have the same degrees of freedom (2n–3), the winner (i.e., narrower width of the CI) is always the one with the larger sample size. The “best” designs (highest dfs ⇒ narrowest CI) formated in green (including the 4-period full replicate, where df = 3n–4).
But as Detlew correctly pointed out, the “precision” of the CV is not a requirement of the GL and all designs perform pretty similar.

BTW, comparing variances (additionally to means) would need much higher sample sizes. This was one of the reasons why IBE was abandoned and the discussion point of the concept paper (see the end of this post) likely was not followed any further.
The comparison of standard errors entered through the back-door in FDA’s RSABE for NTIDs. Not only the acceptance range is scaled – which would mean 12 subjects for a CV of 7% (AR ~0.9–1.11), T/R 0.975, power 0.8…

library(PowerTOST)
CV <- 0.07
L  <- exp(-log(1/0.9)/0.1*sqrt(log(0.1^2+1)))
U  <- exp(+log(1/0.9)/0.1*sqrt(log(0.1^2+1)))
sampleN.TOST(CV=CV, theta0=0.975, theta1=L, theta2=U, details=F)


… but the ratio of σWT/σWR must not exceed 2.5. Try

library(PowerTOST)
CV <- 0.07
sep2s <- function(CV, ratio=1) { # split CV to s-ratio
  sp  <- CV2mse(CV) # pooled s²
  # (1) (s²T + s²R)/2 = s²
  # (2) sT/sR = ratio
  # solve (1+2) for sT, sR
  swt <- sqrt(2)*sqrt(sp)*ratio/sqrt(ratio^2+1)
  swr <- sqrt(2)*sqrt(sp)/sqrt(ratio^2+1)
  r   <- se2CV(c(swt, swr))
  r
}
sampleN.NTIDFDA(theta0=0.975, CV=sep2s(CV=CV, ratio=1.0))
sampleN.NTIDFDA(theta0=0.975, CV=sep2s(CV=CV, ratio=2.0))
sampleN.NTIDFDA(theta0=0.975, CV=sep2s(CV=CV, ratio=0.5))


Due to the additional requirement and different implied limits (based on CVWR) the sample size almost doubles to 22 – even if CVs are equal (σWT = σWR = 0.0699). If the ratio with 2 is close to the limit (σWT = 0.0884, σWR = 0.0442), we would need 142! Only if the test is much better (reversed ratio), the penalty almost vanishes (14 subjects).

  swt    swr  swt/swr  CVwt   CVwr  CVwt/CVwr  impl. limits   n  power
0.0699 0.0699   1.0   0.0700 0.0700   1.000   0.9290…1.0764  22 0.8299
0.0884 0.0442   2.0   0.0886 0.0442   2.003   0.9545…1.0477 142 0.8058
0.0442 0.0884   0.5   0.0442 0.0886   0.499   0.9110…1.0977  14 0.8478


Of course you could as well assume that only the CVWT changes:

  swt    swr  swt/swr  CVwt   CVwr  CVwt/CVwr  impl. limits   n  power
0.1398 0.0699   2.0   0.1405 0.0700   2.007   0.9290…1.0764 128 0.8016
0.0350 0.0699   0.5   0.0350 0.0700   0.500   0.9290…1.0764  16 0.8284

Cheers,
Helmut Schütz
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The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
d_labes
Hero

Berlin, Germany,
2015-07-02 09:09

@ Helmut
Posting: # 15014
Views: 11,951
 

 CVs (R, T) from pooled CV

Dear Helmut!

» library(PowerTOST)
» CV <- 0.07
» sep2s <- function(CV, ratio=1) { # split CV to s-ratio
»   sp  <- CV2mse(CV) # pooled s²
»   # (1) (s²T + s²R)/2 = s²
»   # (2) sT/sR = ratio
»   # solve (1+2) for sT, sR
»   swt <- sqrt(2)*sqrt(sp)*ratio/sqrt(ratio^2+1)
»   swr <- sqrt(2)*sqrt(sp)/sqrt(ratio^2+1)
»   r   <- se2CV(c(swt, swr))
»   r
» } ...

Sometimes it would be helpful to RTFM :-D.
Try
library(PowerTOST)
?CVp2CV
.

Note that the ratio is here the ratio of variances.

Regards,

Detlew
Helmut
Hero
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Homepage
Vienna, Austria,
2015-07-02 13:03

@ d_labes
Posting: # 15016
Views: 11,935
 

 Answering an un­asked ques­tion

Dear Detlew!

» Sometimes it would be helpful to RTFM :-D.

Oh, I did. You will note that I hacked your code of CVp2CV() for my function sep2s(). :-D

» Note that the ratio is here the ratio of variances.

Exactly. But I was interested in the ratio of standard errors instead.

library(PowerTOST)
sep2s <- function(CV, ratio=1) {
  sp  <- CV2mse(CV)
  swt <- sqrt(2)*sqrt(sp)*ratio/sqrt(ratio^2+1)
  swr <- sqrt(2)*sqrt(sp)/sqrt(ratio^2+1)
  r   <- se2CV(c(swt, swr))
  r
}
CV    <- 0.07
ratio <- 2
r1    <- CVp2CV(CV=CV, ratio=ratio)
mse1  <- CV2mse(r1)
se1   <- CV2se(r1)
r2    <- sep2s(CV=CV, ratio=ratio)
mse2  <- CV2mse(r2)
se2   <- CV2se(r2)
cat("Function CVp2CV()\n",
"CV", CV, "split based on the variance ratio:", signif(r1, 3), "\n",
"variances:", signif(mse1, 3), "ratio:", signif(mse1[1]/mse1[2], 3), "\n",
"standard errors:", signif(se1, 3), "ratio:", signif(se1[1]/se1[2], 3),
"\nFunction sep2s()\n",
"CV", CV, "split based on the SE ratio:",  signif(r2, 3), "\n",
"variances:", signif(mse2, 3), "ratio:", signif(mse2[1]/mse2[2], 3), "\n",
"standard errors:", signif(se1, 3), "ratio:", signif(se2[1]/se2[2], 3), "\n")


Which gives:

Function CVp2CV()
 CV 0.07 split based on the variance ratio: 0.0809 0.0571
 variances: 0.00652 0.00326 ratio: 2
 standard errors: 0.0807 0.0571 ratio: 1.41
Function sep2s()
 CV 0.07 split based on the SE ratio: 0.0886 0.0442
 variances: 0.00782 0.00196 ratio: 4
 standard errors: 0.0807 0.0571 ratio: 2


Cheers,
Helmut Schütz
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d_labes
Hero

Berlin, Germany,
2015-07-02 14:21

@ Helmut
Posting: # 15020
Views: 11,924
 

 Answer to an un­asked answer

Dear Helmut!

Beg me pardon, seems a copy-paste devil:
» CV 0.07 split based on the SE ratio: 0.0886 0.0442
»  variances: 0.00782 0.00196 ratio: 4
»  standard errors: 0.0807 0.0571 ratio: 2 (?)

What about:
library(PowerTOST)
CV    <- 0.07
ratio <- 2    # ratio of se's
# ratio of variances is simply ratio of se's squared
r1  <- CVp2CV(CV=CV, ratio=ratio^2)
CV2se(r1)


Gives standard errors:
[1] 0.0884356 0.0442178

Regards,

Detlew
Helmut
Hero
avatar
Homepage
Vienna, Austria,
2015-07-02 14:33

@ d_labes
Posting: # 15021
Views: 11,943
 

 Bloody typo!

Dear Detlew!

» Beg me pardon, seems a copy-paste devil:
» » CV 0.07 split based on the SE ratio: 0.0886 0.0442
» » variances: 0.00782 0.00196 ratio: 4
» » standard errors: 0.0807 0.0571 ratio: 2 (?)

Oops. Correct my code above from signif(se1, 3) to signif(se2, 3) which gives now:

standard errors: 0.0884 0.0442 ratio: 2


Your solution is elegant as ever. I noticed the squared-stuff but failed to figure out how to code it. Would have saved me time (instead of solving these equations – rusty algebra). ;-)

Cheers,
Helmut Schütz
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zizou
Regular

Plzeň, Czech Republic,
2015-07-05 19:20

@ Helmut
Posting: # 15041
Views: 11,816
 

 CVs (borderline OT)

Hi everybody and nobody.

Firstly I have not many experiences with replicate designs, so I may be wrong.

But I have doubts about the degrees of freedom in the post above. When we are calculating the within-subject variability of the reference product (CVWR). The ANOVA model is quite reduced, as EMA stated in Questions & Answers (with errors in red):

The following code removes all the test data from the data-set and then fits a model where the residual variance corresponds to the within subject variance for the test product.

data var;
set replicate;
if formulation='R';
run;

proc glm data=var;
class subject period sequence;
model logDATA= sequence subject_(sequence) period;
run;


Of course there si no formulation effect. (For TRT/RTR design there is even no sequence effect because for this design R is replicated only in one sequence.)

So I think, the degrees of freedom incoming to R function CVCL() should be changed (as shown in rows below with "-->"):

TRT|RTR
Degrees of Freedom of ANOVA for 2x2x3 replicate crossover design:
   Source of variation   df            df from ANOVA with reference data only
   Formulations          1             NA
   Periods               2             1
   Subjects              n – 1         n/2 – 1
      Sequences          1             NA
      Subjects(SEQ)      n – 2         NA
   Error                 2n – 3  -–>  n/2 – 1
   Total                 3n – 1        n – 1

TRR|RTR|RRT
Degrees of Freedom of ANOVA for 2x3x3 replicate crossover design:
   Source of variation   df            df from ANOVA with reference data only
   Formulations          1             NA
   Periods               2             2
   Subjects              n – 1         n – 1
      Sequences          2             2
      Subjects(SEQ)      n – 3         n – 3
   Error                 2n – 3  -–>  n – 2
   Total                 3n – 1        2n – 1

TRTR|RTRT
Degrees of Freedom of ANOVA for 2x2x4 replicate crossover design:
   Source of variation   df            df from ANOVA with reference data only
   Formulations          1             NA
   Periods               3             3
   Subjects              n – 1         n – 1
      Sequences          1             1
      Subjects(SEQ)      n – 2         n – 2
   Error                 3n – 4  -–>  n – 3
   Total                 4n – 1        2n – 1


No questions about df of designs (namely df of MSE: 2n-3 for 2x2x3, 2n-3 for 2x3x3, 3n-4 for 2x2x4). I'm only pointing to degrees of freedom used for function CVCL().

I don't know how much it will change the width of CI of CVWR, so I am not making conclusions.

But in general, TRT|RTR design looks interesting, we get estimates of test and reference variability (each from half of subjects). But I am not convinced if there is right to compare (everyone can see) which variability T or R is higher, when it is comparison of CVWR from the first n/2 group of subjects (one sequence) and CVWT from the second n/2 group (second sequence, i.e. different subjects) - difference of T and R variability can be caused by different subjects - it looks like little bit as comparison of results from two studies performed in the same time/place/conditions with different subjects. Of course this comparison is not the goal of average BE, where we calculate GMR + 90% CI, but still many readers want to compare variabilities T versus R (maybe). For me this design looks like a "demo version" of the full replicate design TRTR|RTRT with only half of subjects for expanding Cmax acceptance limits.

Best regards,
zizou
d_labes
Hero

Berlin, Germany,
2015-07-06 11:10

@ zizou
Posting: # 15042
Views: 11,702
 

 Degrees of freedom 3-period full replicate

Dear Zizou!

Did you check the output from your SAS code against your degrees of freedom tables?

Here an example:
Scott Patterson and Byron Jones
"Bioequivalence and Statistics in Clinical Pharmacology"
Chapman and Hall/CRC Press: Boca Raton, London and New York, 2005

/*EXAMPLE 4.2*/
74 subjects, 3 periods, sequences TRR/RTT, unfortunately unbalanced with 35/39 subjects
PK metric Cmax without missings

Your SAS code according to the EMA crippled model with Proc GLM gives:
                   degrees of freedom
source                type I  type III
subject                73       72
  -sequence             1        0
  -subject(sequence)   72       72
period                  1        1
error                  34       34
total                 108      108

The type III df=0 for sequence or the changing df for subjects effects illustrates the typical mess with the EMA model.

Some differences to yours w.r.t. subject and total :cool:.

Fortunately the df's for the intra-subject variability of the reference don't play a role in the EMA recommended method for scaling of the BE acceptance ranges.

BTW: Your df's are correct for sequence balanced designs, if you use only the data for subjects with replicates of the Reference or if you evaluate via intra-subject contrasts like in the FDA recommended evaluation according to the progesterone guidance.

Regards,

Detlew
ElMaestro
Hero

Denmark,
2015-07-06 11:59

@ d_labes
Posting: # 15043
Views: 11,699
 

 Degrees of freedom 3-period full replicate

Hi d_labes,

» Your SAS code according to the EMA crippled model with Proc GLM gives:
»                    degrees of freedom
» source                type I  type III
» subject                73       72
»   -sequence             1        0
»   -subject(sequence)   72       72
» period                  1        1
» error                  34       34
» total                 108      108

» The type III df=0 for sequence or the changing df for subjects effects illustrates the typical mess with the EMA model.

I would like to offer an alternative view here:
DF=0 for sequence may be correct cf. the ordinary interpretation of type III SS and the use of the bogus statement.

What I absolutely do not get is that 72+1+34 is supposed to give 108? Or what am I overlooking, perhaps this is just the power to confuse.

if (3) 4

Best regards,
ElMaestro

"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
d_labes
Hero

Berlin, Germany,
2015-07-06 13:53

@ ElMaestro
Posting: # 15044
Views: 11,657
 

 Degrees of freedom 3-period full replicate

Dear ElMaestro!

» What I absolutely do not get is that 72+1+34 is supposed to give 108?

Approximately :-D
This is one of the deep secrets of [image] my little statistical understanding can't understand. But it must be true since SAS says so :cool:.

» ... perhaps this is just the power to confuse.

Absolutely correct.

Regards,

Detlew
zizou
Regular

Plzeň, Czech Republic,
2015-07-06 17:25

@ d_labes
Posting: # 15045
Views: 11,690
 

 Degrees of freedom 3-period full replicate

Dear Detlew,

I didn't try the code, so it was only what I expect in result (my mistake - I didn't write that for 2x2x3 replicate design I thought to filter data to only those used in GLM).

For 3-period 2-sequence replicate design with R replicated only in one sequence I would use code:
data var;
set replicate;
if (formulation='R' & sequence='TRR');
run;

proc glm data=var;
class subject period;
model logDATA= suject period;
run;

(Instead of the code which I copied from EMA Q&A in my previous post. I think there is no reason to have sequence in the GLModel for this case. And I am used to see degrees of freedom in ANOVA table which are giving us with real number of cases in evaluation: (number of subjects in evaluation-1) and (number of values in evaluation-1) in total)
In the example, due to we have 35 subjects in sequence with replicated R, I am expecting:
   source    df
   period     1
   subject   34
   Error     34
   Total     69

However, it does not affect anything important.

But if 72+1+34 results in total 108 (similar as 2+2=?, I think 5 is a very very good estimate, ... in accordance with the principles of doublethink "Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once"), I have no power to know (same as Winston Smith has).

Btw. you are right about I did not include balance/unbalance option to table of dfs in my previous post (so it's true only for cases with no dropouts there).
Helmut
Hero
avatar
Homepage
Vienna, Austria,
2015-07-23 23:54

@ Dr_Dan
Posting: # 15139
Views: 11,130
 

 The almighty oracle has spoken!

Dear Dan et alii,

today the EMA published Rev. 12 of the Q&A-document (dated 25 June). See #23

Suitability of a 3-period replicate design scheme for the
demonstration of within-subject variability for Cmax

Essentially it says that CVwR is a “key parameter” and according to the GL a minimum of 12 subjects is needed for a “valid” BE study. In other words – and contrary to what the “leading European regulatory authority” told you – TRT|RTR is indeed acceptable if (‼) at least 12 subjects complete the RTR sequence.

Given the commonly applied T/R-ratio of 0.90 for HVD(P)s and ≥80% power this issue is practically not rele­vant. Full throttle for any study – unless you expect a lot (≥42%) of dropouts!

CVwR   N  nRTR  do.ratemax (%)
 25   42   21      42.9
 30   50   25      52.0
 40   40   20      47.8
 50   42   21      42.9
 60   48   24      50.0
 70   60   30      60.0
 80   74   37      67.6


[image]

[image]

Parturient montes, nascetur ridiculus mus. Horace (Ars poetica, Ep.II.3, 139)

Cheers,
Helmut Schütz
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The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
d_labes
Hero

Berlin, Germany,
2015-07-24 08:23

@ Helmut
Posting: # 15140
Views: 10,968
 

 Almighty oracle's mysterious saying

Dear Helmut!

» Parturient montes, nascetur ridiculus mus. Horace (Ars poetica, Ep.II.3, 139)

Full ACK :-D

As every Pythia's saying: Mysterious!
Nowhere in the scABEL method there is an term utilizing the 'uncertainty' of the variability of the reference.
Contrary to the FDA RSABE which has such a term.
That's the reason why both 3-period designs (TRR|RTR|RRT or TRT|RTR) require similar numbers of subjects.

And deeming 12 subjects as sufficient to 'control' that 'uncertainty' is at least strange (politely spoken):
library(PowerTOST)
CVCL(CV=0.3, df=12-2, "2-sided")
 lower CL  upper CL
0.2072921 0.5513299

From no widening up to cap on widening!

But the net effect is of course pleasing :cool:.

Regards,

Detlew
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