Bioequivalence and Bioavailability Forum • sample size in bioequivalence studies

daryazyatina
☆

Ukraine,
2017-08-03 11:34
(2784 d 14:49 ago)

Posting: # 17646
Views: 21,810

sample size in bioequivalence studies [Power / Sample Size]

Hi, guys!

Today I have a new question related to the the sample size formula for average bioequivalence.

Before now I сalculated the sample size for ABE by formula:
[image]

(described in the book Shein-Chung Chow, Jun Shao and Hansheng Wang "Sample size calculations in clinical research" Copyright © 2003 by Marcel Dekker (Chapter 10. Bioequivalence Testing) http://www.crcnetbase.com/doi/book/10.1201/9780203911341)

I decided to try to calculate the sample size in R with package "PowerTOST" and function sampleN.TOST()

Comparing the results of the calculation in R and calculation with formula Shein-Chung Chow, Jun Shao and Hansheng Wang, I realized that they are different. I looked at the articles and books that this function refers to, and understand that different formulas are used.

On this basis, I had a question. What formula should I use to calculate the sample size for the ABE using two one-sided tests?

BE-proff ● 2017-08-03 11:55 (2784 d 14:29 ago) @ daryazyatina Posting: # 17647 Views: 19,997	sample size in bioequivalence studies Post reply
	Hi darya, Results must be different because PowerTOST uses iterative method for calculation. AFAIK, PowerTOST is used worldwide and no claims from regulators have been received.

daryazyatina
☆

Ukraine,
2017-08-03 12:08
(2784 d 14:15 ago)

@ BE-proff
Posting: # 17648
Views: 19,927

sample size in bioequivalence studies

Post reply

Hi, BE-proff. Thank you for your answer.

❝ Results must be different because PowerTOST uses iterative method for calculation.

Have you looked at the formula what I used? There is also used an iterative method.

❝ AFAIK, PowerTOST is used worldwide and no claims from regulators have been received.

It's good. But I asked another question.
I used formula from book:

and also any claims from regulators have not been received.

ElMaestro
★★★

Denmark,
2017-08-03 12:17
(2784 d 14:07 ago)

@ daryazyatina
Posting: # 17649
Views: 19,983

sample size in bioequivalence studies

Post reply

Hi Daryazyatina,

❝ I decided to try to calculate the sample size in R with package "PowerTOST" and function sampleN.TOST()

Good choice :-D

❝ Comparing the results of the calculation in R and calculation with formula Shein-Chung Chow, Jun Shao and Hansheng Wang, I realized that they are different. I looked at the articles and books that this function refers to, and understand that different formulas are used.

Can you tell what your design is and which values you want to plug in for the calculation? If the difference is 2 or 4 subjects, then so be it, subtle differences in approximation may account for that. If the difference is 46 or something then I'd wonder, too. I am sure there is an explanation and that your confidence in the powerTOST package can easily be restored. Apart from that you are of course right if you intended to hint that the author of the power.TOST family of R functions is a dubious character :lol:

—
Pass or fail!
ElMaestro

daryazyatina
☆

Ukraine,
2017-08-03 12:58
(2784 d 13:26 ago)

@ ElMaestro
Posting: # 17650
Views: 20,042

sample size in bioequivalence studies

Post reply

Hi ElMaestro,

❝ Good choice :-D

Thank you

❝ Can you tell what your design is and which values you want to plug in for the calculation? If the difference is 2 or 4 subjects, then so be it, subtle differences in approximation may account for that. If the difference is 46 or something then I'd wonder, too. I am sure there is an explanation and that your confidence in the powerTOST package can easily be restored.

For comparison, I use all the standard properties. Design 2x2, confidence intervals 0.8 - 1.25, power 0.8, alpha 0.05.
The only thing about what I'm not sure is CV. Because in formula that I used this is intra-subject variability, but in PowerTOST() this is coefficient of variation as ratio. In calculations in both cases I used СV - 0.3.

And this is results from sampleN.TOST()
sampleN.TOST(logscale = TRUE, CV = 0.3, details = TRUE)

+++++++++++ Equivalence test - TOST +++++++++++

            Sample size estimation

-----------------------------------------------

Study design:  2x2 crossover 

log-transformed data (multiplicative model)



alpha = 0.05, target power = 0.8

BE margins = 0.8 ... 1.25 

True ratio = 0.95,  CV = 0.3



Sample size (total)

 n     power

40   0.815845

The sample size by the formula I used:

n=28

❝ Apart from that you are of course right if you intended to hint that the author of the power.TOST family of R functions is a dubious character.

DavidManteigas
★

Portugal,
2017-08-03 14:40
(2784 d 11:43 ago)

@ daryazyatina
Posting: # 17651
Views: 20,180

sample size in bioequivalence studies

Post reply

Hi daryazyatina,

For some reason, I can't see the image with the formula that you put on the first post.

❝ For comparison, I use all the standard properties. Design 2x2, confidence intervals 0.8 - 1.25, power 0.8, alpha 0.05.

❝ The only thing about what I'm not sure is CV. Because in formula that I used this is intra-subject variability, but in PowerTOST() this is coefficient of variation as ratio. In calculations in both cases I used СV - 0.3.

Within subject standard deviation and within subject CV are different parameters. Nevertheless, I think that this is not the only reason for such a big difference.

There is a sentence in one of the articles that is quoted on SampleNTOST formula that may clarify this issue:

"This formula is less conservative than Formula (5), but it may result in a lower actual power than the required. For example, when α = 0.05, σ = 0.3, Δ = 0.2, θ = 0.01 and a required power = 0.80, the sample size from Formula (6) [Formula from Chow] is 17 per sequence, but the actual power obtained by this sample size is only 0.69."

So by reading this, I am not sure if Chow formula might be appropriate to calculate sample size for BABE trials. I have just quickly read the article, so I may not be doing a proper analysis. Perhaps dlabes might clarify this, since he is the master that we all should thank for the amazing PowerTOST package :-D

daryazyatina
☆

Ukraine,
2017-08-03 15:46
(2784 d 10:37 ago)

@ DavidManteigas
Posting: # 17652
Views: 19,876

sample size in bioequivalence studies

Post reply

Hi, DavidManteigas

❝ For some reason, I can't see the image with the formula that you put on the first post.

Look again at the picture with formula. At first it was not visible, now you can see it.

❝ Within subject standard deviation and within subject CV are different parameters. Nevertheless, I think that this is not the only reason for such a big difference.

I did not write anything about the standard deviation.

❝ "This formula is less conservative than Formula (5), but it may result in a lower actual power than the required. For example, when α = 0.05, σ = 0.3, Δ = 0.2, θ = 0.01 and a required power = 0.80, the sample size from Formula (6) [Formula from Chow] is 17 per sequence, but the actual power obtained by this sample size is only 0.69."

If you looked at the formula I use, you would see in the article that this is another formula Chow. And that's why I do not understand what formula should be used.

❝ So by reading this, I am not sure if Chow formula might be appropriate to calculate sample size for BABE trials. I have just quickly read the article, so I may not be doing a proper analysis. Perhaps dlabes might clarify this, since he is the master that we all should thank for the amazing PowerTOST package :-D

I want to understand how to do it right, just this. ;-)

Helmut
★★★

Vienna, Austria,
2017-08-03 17:37
(2784 d 08:47 ago)

@ DavidManteigas
Posting: # 17653
Views: 20,196

Don’t use the formula by Chow, Shao, Wang!

Post reply

Hi David & Darya,

❝ For some reason, I can't see the image with the formula that you put on the first post.

I uploaded a copy. Should be visible by now.

❝ ❝ The only thing about what I'm not sure is CV. Because in formula that I used this is intra-subject variability, but in PowerTOST() this is coefficient of variation as ratio. In calculations in both cases I used СV - 0.3.

❝

❝ Within subject standard deviation and within subject CV are different parameters. Nevertheless, I think that this is not the only reason for such a big difference.

Chow used the standard deviation, whilst in Power.TOST the CV (fraction, not in %!) is used. However, no big difference since \(CV_w = \sqrt{e^{s_{w}^{2}} - 1}\) and the other way ’round \(s_w = \sqrt{\log{(CV_{w}^{2}) + 1}}\). In Power.TOST for convenience you can use se2CV(foo) for the former and CV2se(foo) for the latter.

❝ There is a sentence in one of the articles that is quoted on SampleNTOST formula that may clarify this issue: […]

❝ So by reading this, I am not sure if Chow formula might be appropriate to calculate sample size for BABE trials.

I think it’s crap. The formula Darya posted (10.2.6) of p.259 of the book gives the impression that n is the total sample size. The text continues with:

Since the above equations do not have an explicit solution, for convenience, for a 2 × 2 crossover design, the total sample size needed to achieve a power of 80% or 90% at 5% level of significance with various combinations of ε and δ is given in Table 10.2.1.

However, in the example which follows on p.260 we read:

By referring to Table 10.2.1, a total of 24 subjects per sequence is needed in order to achieve an 80% power at the 5% level of significance.

(my emphases)

❝ […] Perhaps dlabes might clarify this, since he is the master that we all should thank for the amazing PowerTOST package :-D

Detlew is on vacation. Some clarifications:

Zhang¹
(5) where n = sample size per sequence
[image]
(9) with a correction term c
[image]

Hauschke et al.²
[image]

Chow & Liu (5.4.10)³
[image]

Using the formulas of Zhang or Chow & Liu, you get the sample size / sequence. To obtain the total, multiply by 2, which is already done it the right-hand side of Hauschke’s formulas.

Comparison with the references:
Table 2, untransformed data, 90% Power, Δ 0.2, σ 0.3, ∕θ 0.05 (p.537¹): 37 / sequence

library(PowerTOST) sampleN.TOST(CV=se2CV(0.3), theta0=0.05, theta1=-0.2, theta2=0.2, logscale=FALSE, targetpower=0.90, print=FALSE)[["Sample size"]]/2 [1] 37 [image]

Table 5.4.1, untransformed data, 80% Power, Δ 0.2μ_R, α 0.05 (p.158³): 52

library(PowerTOST) sampleN.TOST(CV=0.30, theta0=0.05, theta1=-0.2, theta2=0.2, logscale=FALSE, targetpower=0.80, print=FALSE)[["Sample size"]] [1] 52 [image]

Table 5.1, log-transformed data, 80% Power, (θ₁, 1∕θ₁) = (0.80, 1.25), θ 0.95, α 0.05 (p.113²): 40

library(PowerTOST) sampleN.TOST(CV=0.30, theta0=0.95, theta1=0.8, theta2=1.25, targetpower=0.80, print=FALSE)[["Sample size"]] [1] 40 [image]

Stop using the formula given by Chow, Shao, Wang! Sample sizes are way too small – which compromises power. If you used it in the past for 80% power – if all assumptions (CV, θ₀) turned out to be “correct” – substantially more than 20% of studies should have failed. If not, you were lucky!

Zhang P. A Simple Formula for Sample Size Calculation in Equivalence Studies. J Biopharm Stat. 2003;13(3):529-38. doi:10.1081/BIP-120022772
Hauschke D, Steinijans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007. p.116. doi:10.1002/9780470094778
Chow S-C Liu J-p. Design and Analysis of Bioavailability and Bioequivalence Studies.
Boca Raton: Chapman & Hall/CRC Press; 3^rd ed. 2009. p.157. doi:10.1201/9781420011678

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daryazyatina
☆

Ukraine,
2017-08-03 17:52
(2784 d 08:32 ago)

@ Helmut
Posting: # 17654
Views: 19,869

Don’t use the formula by Chow, Shao, Wang!

Post reply

Hi, Helmut

❝ Stop using the formula given by Chow, Shao, Wang! Sample sizes are way too small – which compromises power. If you used it in the past for 80% power – if all assumptions (CV, θ₀) turned out to be “correct” – substantially more than 20% of studies should have failed. If not, you were lucky!

Thank you for such a comprehensive answer.
It's good that now we will know about this.

DavidManteigas ★ Portugal, 2017-08-03 18:12 (2784 d 08:11 ago) @ Helmut Posting: # 17657 Views: 19,831	Don’t use the formula by Chow, Shao, Wang! Post reply
	Thank you for the clarification Helmut, very helpful as always. In the example you've mentioned on page 260 there is also another mistake, since z0.10 is 1.28 and not 0.84, so according to the reduced formula, 28 subjects per sequence would be necessary and not the reported 21. Still, significantly lower than the sample obtained with sampleNTOST.

Helmut
★★★

Vienna, Austria,
2017-08-03 18:33
(2784 d 07:51 ago)

@ DavidManteigas
Posting: # 17658
Views: 20,056

Don’t use the formula by Chow, Shao, Wang!

Post reply

Hi David,

❝ […] on page 260 there is also another mistake, since z0.10 is 1.28 and not 0.84, so according to the reduced formula, 28 subjects per sequence would be necessary and not the reported 21.

Yep, I noticed that as well. :-D

BTW, z_0.05 is 1.64 and not 1.96. In all its “beauty”:

delta <- log(1.25) epsilon <- 0.05 sigma1.1 <- 0.40 z.05 <- qnorm(0.05, lower.tail=FALSE) z.10 <- qnorm(0.10, lower.tail=FALSE) (z.05+z.10)^2*sigma1.1^2/(2*(delta-epsilon)^2) [1] 22.85316

Of course, PowerTOST contains the large sample approximation based on z as well (by an internal and hence, undocumented) function.

library(PowerTOST) CV <- seq(0.15, 0.8, 0.025) n <- data.frame(CV=CV, exact=rep(NA, length(CV)), nc.t=rep(NA, length(CV)), shifted.t=rep(NA, length(CV)), normal=rep(NA, length(CV)), row.names=NULL) for (j in seq_along(CV)) { # exact method (Owen's Q) n[j, 2] <- sampleN.TOST(CV=CV[j], theta0=0.95, theta1=0.8, theta2=1.25, targetpower=0.8, method="exact", print=FALSE)[["Sample size"]] # approximation by the noncentral t-distribution n[j, 3] <- sampleN.TOST(CV=CV[j], theta0=0.95, theta1=0.8, theta2=1.25, targetpower=0.8, method="nct", print=FALSE)[["Sample size"]] # approximation by the shifted central t-distribution n[j, 4] <- sampleN.TOST(CV=CV[j], theta0=0.95, theta1=0.8, theta2=1.25, targetpower=0.8, method="shifted", print=FALSE)[["Sample size"]] # (large sample) approximation by the normal distribution n[j, 5] <- PowerTOST:::.sampleN0(se=CV2se(CV[j]), diffm=log(0.95), ltheta1=log(0.8), ltheta2=log(1.25), targetpower=0.8) } print(n, row.names=FALSE)

Gives:

CV exact nc.t shifted.t normal 0.150 12 12 12 10 0.175 16 16 16 12 0.200 20 20 20 16 0.225 24 24 24 20 0.250 28 28 28 26 0.275 34 34 34 30 0.300 40 40 40 36 0.325 46 46 46 42 0.350 52 52 52 48 0.375 58 58 60 56 0.400 66 66 66 62 0.425 74 74 74 70 0.450 82 82 82 78 0.475 90 90 90 86 0.500 98 98 98 94 0.525 106 106 108 102 0.550 116 116 116 110 0.575 126 126 126 120 0.600 134 134 134 128 0.625 144 144 144 138 0.650 154 154 154 148 0.675 164 164 164 158 0.700 174 174 174 166 0.725 184 184 184 176 0.750 194 194 194 186 0.775 204 204 204 196 0.800 214 214 216 208

The approximation by the noncentral t-distribution does an excellent job. That’s why we’ve set it as the default in package Power2Stage for speed reasons (~40 times faster than Owen’s Q).
The shifted central t is also good. Only in a few cases higher sample sizes. Conservative, no worries.
The large sample size approximation sucks. Always smaller sample sizes than with the other methods, power compromised – unless one dares to submit a study evaluated by z. ;-)

Quoting from a conversation with an eminent regulator of the Iberian Peninsula:

“Frankly between z and t methods the difference is ridiculous when variability is not large and later a few subjects is added to compensate drop-outs. I do not see any problem in using z-method. I use it because it is very straightforward in Excel and there is no need to have special software.”

Well, cough…

Brain-dead. 38.9 ℃ and rising, no AC in my office…
[image]

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d_labes
★★★

Berlin, Germany,
2017-08-16 18:01
(2771 d 08:22 ago)

@ Helmut
Posting: # 17694
Views: 19,324

Don’t use the Book by Chow, Shao, Wang!

Post reply

Hi all!

In addition to all what was said up to now:
Don’t use the Book by Chow, Shao, Wang!

It's full of errors, full of a terminology not compatible to other publications so that one has no chance to compare.
(F.i. what are epsilon, delta, sigma1.1 compared to our use of theta0, upper or lower BE limit, intra-subject CV. No one knows! To cite our ol'Sailor "garbage in, garbage out, It's that simple.")

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2017-08-03 18:04
(2784 d 08:19 ago)

@ daryazyatina
Posting: # 17655
Views: 19,924

compromised power

Post reply

Hi Darya,

❝ The sample size by the formula I used:

❝ n=28

To keep everything equal you should convert the standard deviation to the CV. Hence,

library(PowerTOST) sampleN.TOST(CV=se2CV(0.3)) +++++++++++ Equivalence test - TOST +++++++++++ Sample size estimation ----------------------------------------------- Study design: 2x2 crossover log-transformed data (multiplicative model) alpha = 0.05, target power = 0.8 BE margins = 0.8 ... 1.25 True ratio = 0.95, CV = 0.3068783 Sample size (total) n power 42 0.818541

Of course higher than the 28 by the dubious formula. As David mentioned above (quoting Zhang’s paper) power is compromised. Let’s check how much:

power.TOST(CV=se2CV(0.3), n=28) [1] 0.6250166

Oops!

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daryazyatina ☆ Ukraine, 2017-08-04 10:15 (2783 d 16:09 ago) @ Helmut Posting: # 17663 Views: 19,713	compromised power Post reply
	Hi Helmut, ❝ To keep everything equal you should convert the standard deviation to the CV. Hence, ❝ `se2CV(0.3)` You are wrote about converting the standard deviation to СV, but use the formula to convert the standard error to СV. I thought that the standard deviation and standard error are different parameters. Maybe I'm wrong.

Helmut
★★★

Vienna, Austria,
2017-08-04 14:41
(2783 d 11:42 ago)

@ daryazyatina
Posting: # 17664
Views: 19,824

terminology

Post reply

Hi Darya,

❝ You are wrote about converting the standard deviation to СV, but use the formula to convert the standard error to СV. I thought that the standard deviation and standard error are different parameters.

Like in this thread it is a question of terminology; see this thread for details.
In PowerTOST type ?se2CV to get the formulas for conversion. If you want a second opinion:¹

[image]

Nitpick: Chaos everywhere. We have only s_w – which is the estimate of the unknown σ_w…
BTW, when it comes to reference-scaling, both the EMA and the FDA correctly use s_wR. Since scaling is based on the unknown parameters, we might observe an inflation of the type I error.²

Patterson S, Jones B. Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: Chapman & Hall / CRC; 2^nd ed. 2016: p.47.
Labes D, Schütz H. Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its Control. Pharm Res. 2016;33(11):2805–14. doi:10.1007/s11095-016-2006-1. full text view-only.

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Dif-tor heh smusma 🖖🏼 Довге життя Україна!
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daryazyatina
☆

Ukraine,
2017-08-16 12:12
(2771 d 14:12 ago)

@ Helmut
Posting: # 17689
Views: 19,555

terminology

Post reply

Hi Helmut

❝ Like in this thread it is a question of terminology; see this thread for details.

I have some questions about this.
Some articles describe the difference between a standard deviation and a standard error. How to understand such articles? for example

The terms “standard error” and “standard deviation” are often confused. The contrast between these two terms reflects the important distinction between data description and inference, one that all researchers should appreciate.¹

Douglas G Altman, J Martin Bland. Standard deviations and standard errors.
doi:10.1136/bmj.331.7521.903
David L Streiner, PhD. Maintaining Standards: Differences between the Standard Deviation and Standard Error, and When to Use Each.
http://ww1.cpa-apc.org/Publications/Archives/PDF/1996/Oct/strein2.pdf

Helmut
★★★

Vienna, Austria,
2017-08-16 20:54
(2771 d 05:30 ago)

@ daryazyatina
Posting: # 17695
Views: 19,447

terminology

Post reply

Hi Darya,

❝ Some articles describe the difference between a standard deviation and a standard error.

THX for the second one! Was great fun to read.

❝ How to understand such articles?

Duno. Perfect description. Read them again. ;-)

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Helmut Schütz

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balinskyi
☆

Ukraine,
2018-06-03 14:58
(2480 d 11:26 ago)

@ daryazyatina
Posting: # 18844
Views: 15,248

sample size in bioequivalence studies

Post reply

Hello everyone,

I should note, that this formula was published in Ukrainian Pharmacopoeia 1st edition (obsolete since 2016) with reference to Liu and Chow 2008 book. Now, while planning a study, the sponsor has applied the formula (referring to Pharmacopoeia) and consequently argue, that my estimate for the sample size (calculated with PowerTOST) is unnecessarily high. So in my search for an answer I have found this thread. Thank you everyone for the answers!