Anu
★    

India,
2013-02-01 08:01
(4469 d 18:53 ago)

Posting: # 9938
Views: 12,093
 

 Sample size for RSABE [Power / Sample Size]

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Hi All,

Greetings.

Kindly help me out with, how can we calculate sample size in case of RSABE according to US FDA Guidelines?
 
d_labes
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Berlin, Germany,
2013-02-01 10:46
(4469 d 16:08 ago)

@ Anu
Posting: # 9940
Views: 11,290
 

 Sample size for RSABE

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Dear Anu,

there is no simple (or complicated) analytical way to calculate power or estimate the sample size for RSABE due to the complex conditions and requirements for the described methods (FDA or EMA) :no:.

The only way known at moment is via simulations.
Have a look into the paper:
Laszlo Tothfalusi and Laszlo Endrenyi
"Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs"
J Pharm Pharmaceut Sci (www.cspsCanada.org) 15(1) 73 - 84, 2011

Can be accessed online.

See also this thread here in the forum.

Hope this helps.
—
Regards,

Detlew
 
Anu
★    

India,
2013-02-07 07:58
(4463 d 18:56 ago)

@ d_labes
Posting: # 9982
Views: 10,544
 

 Sample size for RSABE

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Hi Detlew,

Thankyou so much for your reply.
I have used the R-code as well that you have provided. In my data (CV=40%, Ratio=1.16), through R-code sample size came 39 and with ref to the Table A3, Sample-size is between (47-104).
 
Helmut
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2013-02-07 13:32
(4463 d 13:22 ago)

@ Anu
Posting: # 9985
Views: 10,590
 

 No analytic solution!

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Hi Anu,

❝ I have used the R-code as well that you have provided. In my data (CV=40%, Ratio=1.16), through R-code sample size came 39 and with ref to the Table A3, Sample-size is between (47-104).


As Detlew already pointed out there is no analytical solution for RSABE due to the GMR-restriction (+ the 50% CV cap for EMA). Either you use the tables or you set up your own simulations.

require(PowerTOST)
sigma0  <- 0.25 # FDA's
CVwr    <- 0.4
sigmawr <- CV2se(CVwr)
L       <- exp(-log(1.25)/sigma0*sigmawr)
U       <- exp(+log(1.25)/sigma0*sigmawr)
if(CVwr <= 0.3){L <- 0.8; U <-1.25} # no scaling
for(theta0 in seq(1.15, 1.20, by=0.05)){
  sampleN.TOST(CV=CVwr, theta0=theta0, theta1=L, theta2=U, design="2x3x3")
}

+++++++++++ Equivalence test - TOST +++++++++++
            Sample size estimation
-----------------------------------------------
Study design:  partial replicate (2x3x3)
log-transformed data (multiplicative model)

alpha = 0.05, target power = 0.8
BE margins        = 0.7090232 ... 1.410391
Null (true) ratio = 1.15,  CV = 0.4

Sample size (total)
 n     power
36   0.822516

+++++++++++ Equivalence test - TOST +++++++++++
            Sample size estimation
-----------------------------------------------
Study design:  partial replicate (2x3x3)
log-transformed data (multiplicative model)

alpha = 0.05, target power = 0.8
BE margins        = 0.7090232 ... 1.410391
Null (true) ratio = 1.2,  CV = 0.4

Sample size (total)
 n     power
54   0.803675

Table A3 (FDA method, 3-period, 80% power) for CV 40% gives 47 (GMR 1.15) and 104 (GMR 1.20) subjects. Sample sizes are substantially larger (especially for GMR 1.20) due to the [0.80–1.25] restriction.

Good news: For GMR 1.16 your 39 would be correct …
Bad news:  … but only by ignoring the restriction.
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d_labes
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Berlin, Germany,
2013-02-08 12:57
(4462 d 13:57 ago)

@ Helmut
Posting: # 9992
Views: 10,602
 

 Coming soon ...

Post reply
Dear Helmut, dear Anu,

❝ As Detlew already pointed out there is no analytical solution for RSABE due to the GMR-restriction (+ the 50% CV cap for EMA). Either you use the tables or you set up your own simulations.


No need to set up your own simulations. Just let PowerTOST do the job :cool::

sampleN.scABEL(CV=0.4, theta0=1.16, design="2x3x3", regulator="FDA")

+++++++++++ scaled (widened) ABEL +++++++++++
            Sample size estimation
---------------------------------------------
Study design:  2x3x3
log-transformed data (multiplicative model)

alpha = 0.05, target power = 0.8
CVw(T) = 0.4; CVw(R) = 0.4
Null (true) ratio = 1.16
PE constraints    = 0.8 ... 1.25
Regulatory body: FDA
- CVswitch =  0.3, no cap on ABEL
- Regulatory constant = 0.8925742

Sample size search
 n     power
36   0.717989
39   0.744064
42   0.768658
45   0.789295
48   0.807810

And the best of all: computational time for each step approx. 1-2 sec for 1E+06 sims, believe it or not :yes:.

But ...
Cave 1: Only functions with PowerTOST V1.1-00. Distribution via CRAN may take some time.

Cave 2: PowerTOST is doing the sample size estimation only for balanced designs since the break down of the total subject number in case of unbalanced sequence groups is not unique. Moreover the formulas used are only for balanced designs. That may give different sample sizes compared to the two Laszlos.

Cave3: In case of regulator="FDA" the sample size is only approximate since the BE decision method via widened BE limits is not exactly what is expected by the FDA. But the two Laszlos state that the scABEL method should be 'operational' equivalent to the FDA method. Thus the sample size should be comparable.
—
Regards,

Detlew
 
Helmut
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2013-02-08 13:19
(4462 d 13:35 ago)

@ d_labes
Posting: # 9993
Views: 10,663
 

 Wow!

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Dear Detlew!

❝ No need to set up your own simulations. Just let PowerTOST do the job :cool::

❝ And the best of all: computational time for each step approx. 1-2 sec for 1E+06 sims, believe it or not :yes:.


Do you want to marry me? The community will love you for that! Opens even the door for two-stage simulations where the intermediate power/sample size was the actual show-stopper.

❝ In case of regulator="FDA" the sample size is only approximate since the BE decision method via widened BE limits is not exactly what is expected by the FDA. But the two Laszlos state that the scABEL method should be 'operational' equivalent to the FDA method. Thus the sample size should be comparable.


Agree. Remember this one? Somewhere in electronic nirvana I have presentations by Donald Schuirmann showing scaled limits ‘for demonstrational purposes’ as well.
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Anu
★    

India,
2013-02-09 10:20
(4461 d 16:34 ago)

@ Helmut
Posting: # 9994
Views: 10,483
 

 Wow!

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Hi Helmut & Detlew,

Many thanks for your valueable replies.

Please help me with one thing that while using the code
sampleN.scABEL(CV=0.4, theta0=1.16, design="2x3x3", regulator="FDA") inspite of PowerTOST & mvtnorm, do we need to install someother package as well.
The following error came while execution:
Error: could not find function "sampleN.scABEL"

Thanks & Regards
 
Helmut
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2013-02-09 10:35
(4461 d 16:19 ago)

@ Anu
Posting: # 9995
Views: 10,533
 

 read…

Post reply
Hi Anu,

see Detlew’s first warning above. You need V1.1-00 of PowerTOST which is not available at CRAN yet.

Edit 2013-02-10: Go ahead – it’s there. ;-)
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d_labes
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Berlin, Germany,
2013-02-09 19:11
(4461 d 07:43 ago)

@ Helmut
Posting: # 9996
Views: 10,486
 

 Bigamist

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Dear Helmut!

❝ Do you want to marry me? The community will love you for that!


They would call me "Bigamist" and send me to prison in many countries. And I think my spouse wouldn't be very amused if I convert to a Mussulmán in my old days :-D.

But anyhow: Thanks for the flowers.

❝ Opens even the door for two-stage simulations where the intermediate power/sample size was the actual show-stopper.


I'm not quite sure if you are right here. In simulating 2-stage adaptive designs the power/sample size steps may contribute considerably to the computation times. Thus they may stay the show-stopper even if we can push on the calculations of stage 1/stage 2 BE by the ideas behind power.scABEL() or behind the code inthis post.
But lets see ...

2-stage design combined with RSABE will have another drawback: Where is the point for simulations which represents alpha :confused:? Simply GMR=1.25 (or whatever BE limit applies) surely not.
Eventually power.scABEL() will help to explore.
—
Regards,

Detlew
 
Helmut
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2013-02-10 19:35
(4460 d 07:19 ago)

@ d_labes
Posting: # 9997
Views: 10,815
 

 Some testing (Part I)

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Dear Detlew!

❝ They would call me "Bigamist" and send me to prison in many countries. And I think my spouse wouldn't be very amused if I convert to a Mussulmán in my old days :-D.


I know. Was in the tradition of this post. :-D

❝ 2-stage design combined with RSABE will have another drawback: Where is the point for simulations which represents alpha :confused:? Simply GMR=1.25 (or whatever BE limit applies) surely not.


Duno. Of course the limits are flexible, but the GMR-restriction cuts in anyway. Have to think about it.

In the following results of some testing:

The sample size converges pretty fast if the ratio is not extreme.

[image]

In the common range nsims=1e+05 should be sufficient.

In some cases the sample size is lower than Lászlós’. For CV 0.5, ratio 0.85, 90% power, 3-period (EMA) I got n=150 even with 100 mio sim’s whereas in Table A1 I find n=158. Likely their number of simulations (10,000) was too low.

EMA, 3-period, 80% power
        0.85    0.90   0.95  1.00  1.05  1.10   1.15     1.20
      PT   TE  PT  TE PT TE PT TE PT TE PT TE  PT  TE  PT   TE
0.30 195  194  54  53 27 27 24 22 27 26 45 45 105 104 429 >201
0.35 129  127  51  51 30 29 27 25 30 29 45 45  84  84 204 >201
0.40  93   90  45  44 30 29 27 27 30 30 42 42  66  68 138  139
0.45  78   77  42  40 30 29 27 27 30 29 39 37  57  57 126  124
0.50  75   75  39  40 30 30 27 28 30 30 39 37  54  53 144  133
0.55  81   81  42  42 33 32 30 30 33 32 39 40  57  56 171  172
0.60  90   88  48  46 36 36 33 33 36 36 45 44  63  63 198 >201
0.65 102   99  54  53 42 40 39 37 39 40 51 50  72  71 225 >201
0.70 117  109  60  58 45 45 42 41 45 45 57 56  78  80 258 >201
0.75 129  136  66  67 51 50 48 46 51 50 63 62  90  89 285 >201
0.80 144  144  75  72 57 54 51 51 54 55 69 68  99  97 315 >201

EMA, 3-period, 90% power
        0.85    0.90   0.95  1.00  1.05  1.10   1.15     1.20
      PT   TE  PT  TE PT TE PT TE PT TE PT TE  PT  TE  PT   TE
0.30 276 >201  75  74 36 36 27 28 36 36 63 62 147 147 609 >201
0.35 180  181  72  70 39 39 33 32 39 39 63 63 117 117 288 >201
0.40 135  130  63  61 39 38 33 33 39 39 57 57  93  94 231 >201
0.45 129  132  57  55 39 37 33 33 39 38 51 51  84  85 273 >201
0.50 150  158  54  55 39 39 36 34 39 38 51 51  84  84 330 >201
0.55 180  178  57  59 42 41 39 37 42 41 54 53  96  97 393 >201
0.60 207  199  66  64 45 45 42 41 45 46 60 60 111 112 456 >201
0.65 237 >201  72  72 51 51 48 46 51 51 69 67 126 125 519 >201
0.70 270 >201  81  82 60 57 51 52 57 57 75 76 144 141 588 >201
0.75 300 >201  93  93 66 66 57 58 63 64 84 85 159 161 663 >201
0.80 333 >201 102 100 72 70 66 63 72 71 93 93 177 176 723 >201

EMA, 4-period, 80% power
        0.85  0.90  0.95  1.00  1.05  1.10  1.15    1.20
      PT  TE PT TE PT TE PT TE PT TE PT TE PT TE  PT   TE
0.30 128 127 36 35 18 19 14 15 18 18 30 30 68 68 280 >201
0.35  88  88 34 34 20 20 18 18 20 20 30 31 58 57 140  140
0.40  66  64 32 31 20 20 18 18 20 20 28 28 46 47  98   98
0.45  56  57 28 29 20 21 18 19 20 21 26 27 40 41  88   90
0.50  54  54 28 28 22 22 20 20 22 21 26 27 38 38  96  100
0.55  56  55 30 30 22 23 22 21 22 23 28 28 38 40 114  116
0.60  60  60 32 32 24 25 22 23 24 25 30 31 42 44 132  124
0.65  70  74 36 37 28 28 26 26 28 28 34 33 48 49 150  155
0.70  78  78 40 40 30 31 28 28 30 31 38 38 54 55 170  167
0.75  88  85 44 45 34 34 32 32 34 34 42 42 60 61 192  186
0.80  98  95 50 50 38 38 34 35 36 37 46 46 66 66 212 >201

EMA, 4-period, 90% power
        0.85   0.90  0.95  1.00  1.05  1.10   1.15    1.20
      PT   TE PT TE PT TE PT TE PT TE PT TE  PT  TE  PT   TE
0.30 182  180 50 49 24 25 18 19 24 24 42 42  96  95 400 >201
0.35 124  123 48 48 26 27 22 22 26 27 42 43  80  80 202 >201
0.40  94   93 44 42 26 26 24 23 26 26 38 39  66  66 158  165
0.45  90   90 40 40 26 27 24 24 26 27 36 37  60  59 184  181
0.50 100  102 38 39 28 27 24 25 26 27 36 36  60  60 222 >201
0.55 118  123 40 41 28 29 26 26 28 29 38 38  64  63 262 >201
0.60 138  139 44 45 32 32 28 29 32 31 40 41  74  71 304 >201
0.65 158  159 48 51 34 36 32 32 34 35 46 46  84  81 348 >201
0.70 180  172 54 55 40 40 36 36 38 40 50 52  96  97 394 >201
0.75 200  195 62 62 44 43 40 39 44 44 56 58 106 106 442 >201
0.80 222 >201 68 69 48 49 44 45 48 49 62 62 118 113 488 >201

PT: PowerTOST (higher due to balanced sequences, higher, lower), TE: Tóthfalusi/Endrényi
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Helmut
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2013-02-11 02:00
(4460 d 00:54 ago)

@ d_labes
Posting: # 9998
Views: 10,613
 

 Some testing (Part II)

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Continuing…

FDA, 3-period, 80% power
       0.85   0.90  0.95  1.00  1.05  1.10  1.15    1.20
      PT  TE PT TE PT TE PT TE PT TE PT TE PT TE  PT   TE
0.30 144 145 45 45 24 24 21 21 24 24 39 39 81 82 306 >201
0.35  69  74 36 37 24 24 21 22 24 25 33 34 51 54 105  109
0.40  60  60 33 33 24 24 21 22 24 24 30 31 42 47  99  104
0.45  60  59 30 31 24 23 21 22 24 24 27 29 42 43 120  116
0.50  66  66 30 30 24 24 21 22 24 23 27 28 42 41 144  133
0.55  78  80 30 30 24 24 24 22 24 24 27 28 45 44 171  172
0.60  90  88 30 31 24 24 24 23 24 24 30 30 48 50 198 >201
0.65 102  98 33 32 27 25 24 24 24 25 30 31 57 53 228 >201
0.70 117 106 36 35 27 26 24 25 27 26 33 31 63 62 258 >201
0.75 132 136 39 38 27 27 27 26 27 27 36 34 69 70 288 >201
0.80 144 144 42 40 30 29 27 27 30 29 39 37 78 76 318 >201

FDA, 3-period, 90% power
       0.85   0.90  0.95  1.00  1.05  1.10    1.15    1.20
      PT  TE  PT TE PT TE PT TE PT TE PT TE  PT  TE  PT   TE
0.30 228 >201 66 65 33 33 27 26 33 32 57 55 123 122 495 >201
0.35 102  106 48 51 33 32 27 28 33 32 45 47  72  77 177  186
0.40 102   99 45 45 30 31 27 28 30 31 39 43  63  68 222 >201
0.45 126  128 42 43 30 30 27 28 30 30 39 40  69  69 273 >201
0.50 150  158 45 45 30 31 27 28 30 30 39 40  81  79 333 >201
0.55 177  178 48 50 33 31 30 28 30 31 45 42  96  96 393 >201
0.60 207  199 57 54 33 33 30 30 33 34 48 50 111 112 456 >201
0.65 237 >201 63 61 36 35 33 32 36 36 54 53 126 125 522 >201
0.70 270 >201 72 68 39 39 36 34 39 37 63 61 144 141 594 >201
0.75 300 >201 81 80 45 43 39 37 42 41 69 68 159 161 660 >201
0.80 333 >201 90 83 48 48 42 41 48 47 75 75 177 176 732 >201

FDA, 4-period, 80% power
      0.85  0.90  0.95  1.00  1.05  1.10  1.15    1.20
     PT TE PT TE PT TE PT TE PT TE PT TE PT TE  PT   TE
0.30 96 96 30 30 18 17 14 15 16 17 26 27 54 55 206  200
0.35 50 54 26 26 18 18 16 16 18 18 24 24 36 39  76   79
0.40 42 43 22 24 18 18 16 16 16 17 22 22 32 33  68   72
0.45 42 44 22 23 16 18 16 16 16 17 20 21 30 32  80   82
0.50 46 45 22 22 18 17 16 17 16 17 20 21 30 31  96   99
0.55 52 52 22 22 18 18 16 17 18 17 20 21 32 31 114  116
0.60 60 58 22 23 18 18 16 17 18 18 20 21 34 34 132  124
0.65 68 74 24 24 18 19 18 18 18 18 22 22 38 36 152  155
0.70 78 75 24 24 18 19 18 18 18 19 22 23 42 44 170  167
0.75 88 81 26 26 20 20 18 19 20 20 24 24 46 47 190  186
0.80 98 95 28 29 20 21 20 20 20 20 26 25 52 51 210 >201

FDA, 4-period, 90% power
        0.85  0.90  0.95  1.00  1.05  1.10    1.15    1.20
      PT   TE PT TE PT TE PT TE PT TE PT TE  PT  TE  PT   TE
0.30 152  152 44 44 22 23 18 18 22 22 38 38  82  81 332 >201
0.35  76   80 36 38 22 23 20 20 22 23 32 34  52  55 126  128
0.40  70   70 32 32 22 22 20 20 22 22 28 30  46  48 148  158
0.45  84   84 30 32 22 22 20 20 22 22 28 30  48  49 184  181
0.50 100  102 32 32 22 23 20 20 22 22 28 30  54  54 220 >201
0.55 118  123 34 34 22 23 20 21 22 22 30 31  64  61 262 >201
0.60 138  139 38 38 24 24 22 22 24 24 34 33  74  71 304 >201
0.65 160  159 42 44 26 26 22 23 24 25 36 35  84  81 348 >201
0.70 180  172 48 46 28 26 24 24 26 27 42 43  96  97 394 >201
0.75 200  195 54 53 30 29 26 26 30 29 46 48 106 106 442 >201
0.80 222 >201 60 60 32 33 28 28 32 31 52 51 118 113 488 >201

I will ask them whether they have used 0.89 like in their paper from 2009 instead of log(1.25)/0.25 ≈ 0.8925742…

I like the option to specify unequal CVs of test and reference as CV=c(T,R). Example for EMA, 3-period, T/R 0.95, 80% power (pooled CVintra 0.3 in all cases):
  CVWT     CVWR   n
 0.2500  0.3458  24
 0.3000  0.3000  27
 0.3458  0.2500  30
It will pay off to have a ‘better’ formulation.
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d_labes
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Berlin, Germany,
2013-02-11 13:03
(4459 d 13:51 ago)

@ Helmut
Posting: # 10001
Views: 10,532
 

 Power testing

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Dear Helmut!

Thank you so much for this comprehensive testing :clap:!
Saves me a lot of work.

Can you give me a statement on the validity of power.scABEL() and sampleN.scABEL()?

❝ I will ask them whether they have used 0.89 like in their paper from 2009 instead of log(1.25)/0.25 ≈ 0.8925742…


Maybe some of the differences in case of regulator="FDA" are due to the widened ABEL approach instead of the approach via the linearized SABE criterion and its 95% upper confidence limit <0.
Again: Coming soon ... or later ... :cool:

But if you ask them: Ask in this respect if they have used "The unknown x" used in the SAS code of the progesterone guidance in their implementation of the linearized SABE criterion. I haven't found any hint in their papers on HVD/HVDP regarding such a 'correction' for the upper CI part of the point estimator.
—
Regards,

Detlew
 
Helmut
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2013-02-11 15:40
(4459 d 11:14 ago)

@ d_labes
Posting: # 10002
Views: 10,537
 

 the unknown x

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Dear Detlew!

❝ Can you give me a statement on the validity of power.scABEL() and sampleN.scABEL()?


Cross-validating two simulations? :-D

❝ Maybe some of the differences in case of regulator="FDA" are due to the widened ABEL approach instead of the approach via the linearized SABE criterion and its 95% upper confidence limit <0.


Yes, may be another cause.

❝ Again: Coming soon ... or later ... :cool:


Relax.

❝ But if you ask them: Ask in this respect if they have used "The unknown x" used in the SAS code of the progesterone guidance in their implementation of the linearized SABE criterion. I haven't found any hint in their papers on HVD/HVDP regarding such a 'correction' for the upper CI part of the point estimator.


Right guess, IMHO. Remember this thread? I’m short in time right now; if you want I can give you their e-mail addresses and you do it yourself.
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d_labes
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Berlin, Germany,
2013-02-15 09:47
(4455 d 17:07 ago)

@ Helmut
Posting: # 10027
Views: 10,401
 

 Unequal CVs

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Dear Helmut,

a follow up to your post:

❝ I like the option to specify unequal CVs of test and reference as CV=c(T,R). Example for EMA, 3-period, T/R 0.95, 80% power (pooled CVintra 0.3 in all cases):

❝   CVWT     CVWR   n

❝  0.2500  0.3458  24
❝  .....

❝ It will pay off to have a ‘better’ formulation.


This is what one would expect. I least I think I had read something like that in SABE papers.

[nitpicking] But I doubt that your chosen CVs give a pooled CV of 0.3.
As a one-liner using the PowerTOST helper functions I got:
mse2CV(mean(CV2mse(c(0.25,0.3458))))
[1] 0.3011071.
[/nitpicking]

What did the one-liner do? Calculate the intra-subject variances for T and R, calculate the mean of both and back-calculate from that mean the CV :cool:.

Or did I miss somefink Welch here?
—
Regards,

Detlew
 
Helmut
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2013-02-15 14:01
(4455 d 12:53 ago)

@ d_labes
Posting: # 10032
Views: 10,478
 

 Typo

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Dear Detlew,

❝ As a one-liner using the PowerTOST helper functions I got:

❝ mse2CV(mean(CV2mse(c(0.25,0.3458))))

❝ [1] 0.3011071.

❝

❝ What did the one-liner do? Calculate the intra-subject variances for T and R, calculate the mean of both and back-calculate from that mean the CV :cool:.


Sure. ;-)

❝ Or did I miss somefink Welch here?


Shame on me; was a typo – should read 0.3438…
mse2CV(mean(CV2mse(c(0.25,0.3438))))
[1] 0.2999892
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