AngusMcLean
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USA,
2014-06-25 04:11
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Posting: # 13137
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 Sample size for replicate design partial AUC [Power / Sample Size]

We have early partial AUC data from a completed study and the CV (%) for the early partial AUC intra­subject between formulation variance is 30.5%. The reason we have measured the partial AUC is that we are required by the FDA to submit BE data on the partial AUC metric when performing a pivotal BE study.

The question arises if we should use a replicate study design. I am thinking "yes" and thinking in terms of a full replicate design comparing test and reference formulations.


I am thinking in terms of estimating sample size and power. Are there Tables available for estimating the sample size and power of such study designs I have information here only for nonrepliciate BE study designs.

Comments are most welcome on this topic,

Angus
Helmut
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2014-06-25 04:53
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@ AngusMcLean
Posting: # 13138
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 sampleN.RSABE()

Hi Angus,

❝ We have early partial AUC data from a completed study and the CV (%) for the early partial AUC intrasubject between formulation variance is 30.5%.


If this was a nonreplicated study CVintra is pooled from CVWR and CVWT.

❝ The reason we have measured the partial AUC is that we are required by the FDA to submit BE data on the partial AUC metric when performing a pivotal BE study.


Zolpidem, MPH?

❝ The question arises if we should use a replicate study design.


Are you thinking about RSABE? If CVintra ~ CVWR ~ CVWT with 30.5% you will save something in the sample size (example will follow).

❝ I am thinking "yes"…


Well, would be nice if the FDA accepts RSABE for this drug.

❝ …and thinking in terms of a full replicate design comparing test and reference formulations.


Nothing tells you more about the performance of formulations than a fully replicated design. I like it.

❝ I am thinking in terms of estimating sample size and power. Are there Tables available for estimating the sample size and power of such study designs.


Only one paper.* You need simulations, since there is no explicit formula for power of the mixed pro­ce­dure (no scaling for sWR <0.294, scaling ≥0.294, T/R within 0.8–1.25). The convergence is slow (the 10,000 sim’s of the paper are to few). In the meantime the two Lászlós themselves recommend the freeware R / package PowerTOST.

❝ I have information here only for nonrepliciate BE study designs.


OK, you can play around with what you got.
  • First let’s try unscaled ABE, CV 30.5%, T/R 0.90, 90% power, TRTR|RTRT:
    require(PowerTOST)
    sampleN.TOST(CV=0.305, theta0=0.9, targetpower=0.9, design="2x2x4", details=F)

    You should get:
    +++++++++++ Equivalence test - TOST +++++++++++
                Sample size estimation
    -----------------------------------------------
    Study design:  2x2x4 replicate crossover
    log-transformed data (multiplicative model)

    alpha = 0.05, target power = 0.9
    BE margins        = 0.8 ... 1.25
    Null (true) ratio = 0.9,  CV = 0.305

    Sample size (total)
     n     power
    56   0.902900


  • Now RSABE with FDA’s method:
    require(PowerTOST)
    sampleN.RSABE(CV=0.305, theta0=0.9, targetpower=0.9, design="2x2x4", details=F)

    You should get:
    ++++++++ Reference scaled ABE crit. +++++++++
               Sample size estimation
    ---------------------------------------------
    Study design:  2x2x4 (full replicate)
    log-transformed data (multiplicative model)
    1e+05 studies for each step simulated.

    alpha  = 0.05, target power = 0.9
    CVw(T) = 0.305; CVw(R) = 0.305
    Null (true) ratio = 0.9
    ABE limits / PE constraints = 0.8 ... 1.25
    Regulatory settings: FDA

    Sample size
     n    power
    44   0.90338

Why do we need only 44 subjects for RSABE when we need 56 for ABE? Since with a CV of 30.5% we are almost at exactly the borders of scaling, i.e., we have a ~50% chance that in the actual study the CV will be higher. More scaling, higher power, less subjects. ;-)


  • Tóthfalusi L, Endrényi. Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs. J Pharm Pharmaceut Sci. 2011;15(1):73–84. [image] open access.

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AngusMcLean
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USA,
2014-06-25 15:42
(3564 d 08:42 ago)

@ Helmut
Posting: # 13142
Views: 16,314
 

 sampleN.RSABE()

Many Thanks Helmut;
It was a non-replicated study design. We are considering RSABE according to the recent FDA presentations and papers. The full replicate design is indeed what I favor.
I did a sample size calculation for the non replicate design (usual cross over design) using the Excel spreadsheet FARTSIE from David Dubins, I have found this spreadsheet to be most useful in the past. Using a desired power of 0.9, with CV of 30.05 and an anticipated ratio of 0.93. I get 67 subjects and this is more than your estimate. I wonder why?
{I must try the freeware package you recommend}. I do have a Study Size program from Sweden, but have not used it for this project as yet. It seems to be an excellent program. Update using Study Size for the above sample size calculation again I get 67 subjects.
There is a reference I have found that appears to be most useful since it tabulates the simulations for sample size including the EMEA and FDA’s approach to highly variable drugs.
Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs
Laszlo Tothfalusi1 and Laszlo Endrenyi2
1 Semmelweis University, Department of Pharmacodynamics, Budapest, Hungary.
2 University of Toronto, Department of Pharmacology and Toxicology, Toronto, ON, Canada, J Pharm Pharmaceut Sci (www.cspsCanada.org) 15(1) 73 - 84, 2012
When I consult this paper I see for the fully replicate design with 4 periods that for CV of 30% and power 0.9 with GMR ratio of 0.9 that 38 subjects are estimated.
So I am thinking ~ 40 subjects is needed. Any thought are welcome,

Angus
Helmut
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2014-06-25 17:02
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@ AngusMcLean
Posting: # 13144
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 sampleN.RSABE()

Hi Angus,

❝ I did a sample size calculation for the non replicate design (usual cross over design) using the Excel spreadsheet FARTSIE from David Dubins…


A member of the Forum since 2007. ;-)

❝ I have found this spreadsheet to be most useful in the past.


Yep, AFAIK it is still widely used.

❝ Using a desired power of 0.9, with CV of 30.05 and an anticipated ratio of 0.93. I get 67 subjects and this is more than your estimate. I wonder why?

  • In your OP you stated a CV of 30.5% – not 30.05%. I assumed a T/R-ratio of 0.90 – not 0.93.
  • The setup for conventional unscaled 2×2 ABE in FARTSSIE doesn’t help too much.
    • In [Bioequivalence, Crossover] I get n=67 with 90.11% power. Note that this would be an imbalanced design. You should round up to n=68 – as PowerTOST would do:
      sampleN.TOST(CV=0.3005, theta0=0.93, targetpower=0.9, design="2x2", details=F)
      +++++++++++ Equivalence test - TOST +++++++++++
                  Sample size estimation
      -----------------------------------------------
      Study design:  2x2 crossover
      log-transformed data (multiplicative model)

      alpha = 0.05, target power = 0.9
      BE margins        = 0.8 ... 1.25
      Null (true) ratio = 0.93,  CV = 0.3005

      Sample size (total)
       n     power
      68   0.904946

      We could calculate power for n=67 as well:*
      power.TOST(CV=0.3005, theta0=0.93, n=67, design="2x2")
      [1] 0.9011207

      Hey, confirmed FARTSSIE’s 90.11%… BTW, PowerTOST by default uses the exact method, where­as FARTSSIE uses an approximation based on the noncentral t-dis­tri­bution. Let’s check that:*
      power.TOST(CV=0.3005, theta0=0.93, n=67, design="2x2", method="nct")
      [1] 0.9011207

      OK, the approximation works.
      Compared to a 2×2 crossover in a fully replicated design with n/2 subjects we have the same number of treatments and can expect the same power. Therefore, for un­scaled ABE we get 67/2=33.5 → 34 subjects. Let’s check that:
      sampleN.TOST(CV=0.3005, theta0=0.93, targetpower=0.9, design="2x2x4", details=F)
      +++++++++++ Equivalence test - TOST +++++++++++
                  Sample size estimation
      -----------------------------------------------
      Study design:  2x2x4 replicate crossover
      log-transformed data (multiplicative model)

      alpha = 0.05, target power = 0.9
      BE margins        = 0.8 ... 1.25
      Null (true) ratio = 0.93,  CV = 0.3005

      Sample size (total)
       n     power
      34   0.906647

      Bingo!
    • In FARTSSIE’s [Bioequivalence, Replicate] [4 period replicate (ABBA/BAAB)] [Method C2 - FDA Approach] I get n=34 with 90.12% power.
      Note 1: FDA want RTRT|TRTR not TRRT|RTTR.
      Note 2: For RSABE and (EMA’s ABEL) simulations are mandatory. Both FARTSSIE and Study Size don’t work!
  • With your values the code for RSABE is
    sampleN.RSABE(CV=0.3005, theta0=0.93, targetpower=0.9, design="2x2x4", details=F)
    giving…
    ++++++++ Reference scaled ABE crit. +++++++++
               Sample size estimation
    ---------------------------------------------
    Study design:  2x2x4 (full replicate)
    log-transformed data (multiplicative model)
    1e+05 studies for each step simulated.

    alpha  = 0.05, target power = 0.9
    CVw(T) = 0.3005; CVw(R) = 0.3005
    Null (true) ratio = 0.93
    ABE limits / PE constraints = 0.8 ... 1.25
    Regulatory settings: FDA

    Sample size
     n    power
    30   0.91495

    Now we see a difference: n=30 (RSABE) and n=34 (ABE). Why? ABE assumes the CV as “carved in stone”, whereas in real life the CV might be larger → apply scaling → fewer sub­jects needed to achieve the target power.

❝ {I must try the freeware package you recommend}.


Go ahead!

❝ I do have a Study Size program from Sweden, but have not used it for this project as yet. It seems to be an excellent program. Update using Study Size for the above sample size calculation again I get 67 subjects.


Yes, but for unscaled ABE in a 2×2 design. Study Size does not work for RSABE as well. I you apply the “rule of thumb” n/2 and perform the study in 34 subjects, it will be overpowered:
power.RSABE(theta0=0.93, CV=0.3005, n=34, design="2x2x4")
[1] 0.93941

Explain to your boss why you plan to run the study in 34 subjects if 30 will give you already a power of 91.5%.

❝ There is a reference I have found…


Hey, that’s the reference I gave you in my last post. ;-)

❝ When I consult this paper I see for the fully replicate design with 4 periods that for CV of 30% and power 0.9 with GMR ratio of 0.9 that 38 subjects are estimated.


You have to look it up in Table A4, second part. For CV 30% they report n=44 for GMR 0.90 and n=23 (imbalanced: round up to 24) for GMR 0.95.

❝ So I am thinking ~ 40 subjects is needed.


Nope. 30.


Homework:

sampleN.RSABE(CV=0.30, theta0=0.90, targetpower=0.9, design="2x2x4")
sampleN.RSABE(CV=0.30, theta0=0.95, targetpower=0.9, design="2x2x4")

Do the results match Table A4?



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AngusMcLean
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USA,
2014-06-25 22:05
(3564 d 02:20 ago)

@ Helmut
Posting: # 13146
Views: 15,681
 

 sampleN.RSABE()

Helmut: My apologies I am having difficult getting access to the site and posting. It is a battle and I am distracted by it and making many errors. I have now switched to Firefox to see if that will work.

Yes; the paper is the same paper you recommended, but I have it as 2012

J Pharm Pharmaceut Sci (www.cspsCanada.org) 15(1) 73 - 84, 2012

Laszlo Tothfalusi1 and Laszlo Endrenyi2

I am focusing on Table 4 second part (FDA approach) and I see that at 90% power for GMR=1.1 {as recommended by authors} the sample size needed is 38 subjects for a CV of 30%.

I note that your program in R estimates sample size for RSABE. I will see if I can download and see what it provides.

I will switch back to Explorer and see if I can get it to work.


Angus
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2014-06-25 23:29
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@ AngusMcLean
Posting: # 13147
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 sampleN.RSABE()

Hi Angus,

❝ […] I am having difficult getting access to the site and posting. It is a battle and I am distracted by it and making many errors. I have now switched to Firefox to see if that will work.


Sorry about that. I checked my server’s log-files and it seems that some script-kiddies are trying to hack into the database. Low chances, but side-effects due to high server load.
My suggestion: Regularly copy the entire text-area to the clipboard (Ctrl-A Ctrl-C). If you get disconnected you could insert the stuff into a new reply.

❝ I am focusing on Table 4 second part (FDA approach)…


I guess you mean Table A4 on p84.

❝ …and I see that at 90% power for GMR=1.1 {as recommended by authors} the sample size needed is 38 subjects for a CV of 30%.


Contrary to 2×2 crossovers (5% deviation) in HVDs/HVDPs PEs “jump around” between studies. There­fore, the Lászlós recommend a 10% deviation. If you have no idea about the direction, always use the lower (<1) one. The upper one will be covered as well. Example 10% deviation. Assuming 90% we will get the same power for 1/0.9=1.1111% (110% is covered as well). This doesn’t work the other way ’round. If you start with 110%, you will get the same power for 1/1.1=0.9090. 90% is not covered!

But you expect the ratio at 0.93, right? Therefore, according to Table A4 the sample size will be between 23 (→24!) for 0.95 and 44 for 0.90. PowerTOST suggests 30 for 0.93.

❝ I note that your program in R estimates sample size for RSABE.


It also allows sample size estimation for cases where CVWR  CVWT. This reduces the sample size if you know that the reference is lousy and the test shows lower variability (an effect commonly seen in studies of PPIs). From a 2×2 cross­over you only get CVintra (pooled from CVWR and CVWT). However, in the backyard you can play around with assump­tions. Let’s say got CVintra 30% in a 2×2 cross­over and assume T/R-ratios of intra-subject variances to be 1:1, 3:4, and 1:2. Try this code:
CVs <- CVp2CV(0.3, ratio=c(1, 3/4, 1/2))
CVs

You will get decomposed per-treatment CVs:
          CVwT      CVwR
[1,] 0.3000000 0.3000000
[2,] 0.2768811 0.3217173
[3,] 0.2431753 0.3489488

Now you can feed the rows to sampleN.RSABE in order to assess their impact on sample size. Example for CVintra 30%, target power 90%, T/R-ratio 90%, equal and dif­fe­rent CVs, 2×2×4 RSABE:
for (j in 1:3) {
  sampleN.RSABE(CV=CVs[j, ], theta0=0.90,
  targetpower=0.9, design="2x2x4", details=F)
}


───────────────────────────
CVWT %   CVWR %   n  % power
───────────────────────────
30.00   30.00   44   90.02 
27.69   32.17   38   90.90 
24.32   34.89   30   90.30 
───────────────────────────

This is one of the reasons why it makes sense to perform already the pilot study in a fully replicated design. It may pay off in a smaller pivotal.

❝ I will see if I can download and see what it provides.


Some hints about installation in this post.

❝ I will switch back to Explorer and see if I can get it to work.


I’m afraid the problems are on my side of the pond. :no:

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AngusMcLean
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2014-06-27 19:37
(3562 d 04:48 ago)

@ Helmut
Posting: # 13159
Views: 15,611
 

 sampleN.RSABE()

Helmut: Thank you for your help and suggestion. It seems that Firefox works for me.

For this work there is no marketed reference formulation; it is a unique MR formulation of an existing drug. We are doing a site change and making pivotal batches and we wish to compare the new site to the original site material, which will be the reference.
Logically we should perform or have performed a Pilot study to get good parameter estimates prior to pivotal. ……”Schadenfreude”

Why I said 0.93 is that I think is a reasonable target for us to aim for a ratio of (0.93-1.07) or perhaps (0.92-1.08). I will use the lower one in future for calculations.

I will try your program; certainly it does look great. There is a mirror near here at N.I.H. in Bethesda,

Angus
d_labes
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2014-06-30 11:47
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@ Helmut
Posting: # 13168
Views: 15,836
 

 power2.TOST()

Dear Helmut!

❝ We could calculate power for n=67 as well:

power.TOST(CV=0.3005, theta0=0.93, n=67, design="2x2")

❝ [1] 0.9011207


That is not recommended. Just to cite the help page of power.TOST:
"The formulas used assume balanced studies, i.e. equal number of subjects in the (sequence) groups."

Use instead:
power2.TOST(CV=0.3005, theta0=0.93, n=c(34,33), design="2x2")
[1] 0.9010638


Depending on the imbalance you may get even power below 90%:
power2.TOST(CV=0.3005, theta0=0.93, n=c(35,32), design="2x2")
[1] 0.9006074
power2.TOST(CV=0.3005, theta0=0.93, n=c(36,31), design="2x2")
[1] 0.8996887
power2.TOST(CV=0.3005, theta0=0.93, n=c(37,30), design="2x2")
[1] 0.8982959


May be I should improve the documentation with a more direct reference to power2.TOST() and a warning.

Regards,

Detlew
Helmut
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2014-06-30 14:02
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@ d_labes
Posting: # 13170
Views: 15,667
 

 power2.TOST()

Dear Detlew,

❝ […] "The formulas used assume balanced studies, i.e. equal number of subjects in the (sequence) groups."

❝ May be I should improve the documentation with a more direct reference to power2.TOST() and a warning.


Yes, please! Though I knew power2.TOST() – introduced in Dec 2011 – in the heat of battle I forgot using it. ;-)

BTW, FARTSSIE17 reports for n=67 a power of 0.9011206.

power2.TOST(CV=0.3005, theta0=0.93, n=c(34,33), design="2x2", method="nct")
[1] 0.9010638


Close, but not identical.

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ElMaestro
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2014-06-30 14:31
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@ Helmut
Posting: # 13171
Views: 15,595
 

 power2.TOST()

Hi Hötzi,

❝ BTW, FARTSSIE17 reports for n=67 a power of 0.9011206.


power2.TOST(CV=0.3005, theta0=0.93, n=c(34,33), design="2x2", method="nct")

[1] 0.9010638


❝ Close, but not identical.

power2.TOST(CV=0.3005, theta0=0.93, n=c(33.5,33.5), design="2x2", method="nct")
[1] 0.9011207

Pass or fail!
ElMaestro
Helmut
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2014-06-30 14:43
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@ ElMaestro
Posting: # 13172
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 Elementary, my dear Watson!

Hi ElMaestro,

❝ ❝ BTW, FARTSSIE17 reports for n=67 a power of 0.9011206.

power2.TOST(CV=0.3005, theta0=0.93, n=c(33.5,33.5), design="2x2", method="nct")

[1] 0.9011207


Great detective work. Amazing Kinetica-style!

  Never trust in any piece of software you haven’t written yourself
(and even then you should be cautious…)
    Jaime R.

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ElMaestro
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2014-06-30 15:07
(3559 d 09:18 ago)

@ Helmut
Posting: # 13173
Views: 15,446
 

 Elementary, my dear Watson!

Hi Hötzi,

❝ Great detective work. Amazing Kinetica-style!


I had a phunny pheeling you might comment something like that :-D

From the study report page 673 section 45.274.48Z:
"To achieve 90% power this study aimed at 33.5 completing healthy adult children with chronic persistent Alzheimer in each sequence for a total of 67. Due to a dropout rate of 1/26 (=3.85%) observed in the pilot trial 34.84 subjects were enrolled in each sequence out of a total of 23.87+41i subjects screened. Our in-house statistician decided to take early retirement prior to signing the SAP."

Pass or fail!
ElMaestro
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