Kro
☆    

France,
2009-05-19 12:56
(5428 d 02:41 ago)

Posting: # 3699
Views: 19,760
 

 Sample size for PK linearity [Power / Sample Size]

Hi,

I would like to estimate a sample size for a PK linearity study.
The study design plans to test 4 different doses (10, 20, 30 and 40).
I need to test the bioequivalence for each dose, compared to the reference dose which is 20. In addition I need to calculate the linear correlation slope for the four doses, a deviation of 20% between doses being acceptable.

What is the required methodology and what is the formula to calculate the sample size?

Regards,

Kro
Helmut
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Vienna, Austria,
2009-05-19 15:19
(5428 d 00:19 ago)

@ Kro
Posting: # 3702
Views: 18,520
 

 Sample size for PK linearity

Dear Kro!

❝ I would like to estimate a sample size for a PK linearity study.


Keep terminology high: since you don’t know the relationship beforehand, you should call it a PK proportionality study. Dose linearity is a subcategory.
  1. Dose linearity
    1. A straight line (PK response v.s. dose) through the origin = no significant (p>0.05) difference of the intercept from zero = 95% confidence interval of the intercept includes zero.
    2. A straight line, but with nonzero intercept.
  2. Dose proportionality
    Any relationship other than linear; generally a power model.
In other words if PK linearity is shown, PK proportionality is also given – but not the other way ’round.

❝ The study design plans to test 4 different doses (10, 20, 30 and 40).

❝ I need to test the bioequivalence for each dose, compared to the reference dose which is 20.


In the most simple case run a 4-way cross-over, normalize PK-responses (10, 30, 40 doses) to the 20 mg dose, and evaluate BE at 80–125%. A Bonferroni-correction of the alpha-level was advised by the German BfArM and was acceptable in a recent MRP in 15 EU member states. Your would perform three comparisons; therefore α/3 = 0.05/3 or 96.6% confidence intervals instead of 90% CIs. The overall patient’s risk is kept at <0.05, since 1 – (1 – 0.05/3)³ = 0.0492. See also this thread and Martin’s suggestions of another approach.

❝ In addition I need to calculate the linear correlation slope for the four doses, a deviation of 20% between doses being acceptable.


I’m not sure what you mean by ‘a deviation of 20% between doses being acceptable’. If you run model I (1 and/or 2) you get a common slope which can have any value (depending on the units of dose and PK response). So a 20% deviation from what?

❝ What is the required methodology and what is the formula to calculate the sample size?


If you opt for the Bonferroni-approach, run a sample size estimation as usual based on the expected T/R-ratio (generally 0.95), CVintra, and α 0.05/3 (instead of 0.05).
Example (T/R 0.95, power 80%), total cross-over sample sizes):
                 α
CV%      0.05000  0.01667
10.0        8       10 
12.5       10       14
15.0       12       16
17.5       16       22
20.0       20       28
22.5       24       34
25.0       28       40
27.5       34       48
30.0       40       56

Sample sizes are larger in order to compensate for simultaneous testing.

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Kro
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France,
2009-05-19 15:56
(5427 d 23:41 ago)

@ Helmut
Posting: # 3703
Views: 17,936
 

 Sample size for PK linearity

Dear Helmut,

Thanks for your quick answer.
My understanding is that your sample size calculation is based on the BE tests and not on the proportionallity test.
What happens if the principal criteria is the proportionaly test and how to calculate the sample size in this case?

Kindest regards,

Kro
Helmut
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Vienna, Austria,
2009-05-19 17:09
(5427 d 22:28 ago)

@ Kro
Posting: # 3706
Views: 18,085
 

 Sample size for PK linearity

Dear Kro!

❝ My understanding is that your sample size calculation is based on the BE tests and not on the proportionallity test.


Right.

❝ What happens if the principal criteria is the proportionaly test


You didn’t answer my question on the ‘20% deviation’, so I still do not know which test you will perform. Most people apply a power model to assess nonlinear PK.

E(Y|x) = α·x β, where α>0 and β≠0, weighted regression (w = 1/x)


There’s no consensus how to judge the degree of nonlinear PK. Chow & Liu (2009) suggest to assess the 95% confidence interval (L, U) of the exponent in the power model b (based on earlier work of Smith, 1986):1
  • 0.75 < L < 1 < U < 1.25
    no departure from dose linearity
  • 1 < L < U < 1.25 or 0.75 < L < U < 1
    slight departure from dose linearity, but no practical significance from dose linearity
  • L > 1.25 or U < 0.75
    reject hypothesis of dose linearity
Don’t ask me what ‘no practical significance from dose linearity’ means…

The power model is an empirical one with no support from PK theory. Another option would be to test a linear model v.s. a quadratic one. See also another often-quoted reference (Gough et al. 1995).2

A survey on the application of the power model performed at David Bourne’s PKPD-list in 2007 gave following results:

Of the 11 responders (6 pharmaceutical companies, 3 CROs and 2 independent consultants), 9 estimated an exponent from the power model with confidence intervals. Of these 9 responders, 5 claimed dose proportionality if the exponent included unity, 3 have used the approach recommended by Smith and 1 concluded dose proportionality if the exponent was close to unity. Thus, testing of the hypothesis that beta=1 was the most common approach; however, this was not proposed by Gough et al.,2 and Senn3,4 emphasised that “It will presumably not be adequate simply to test the null hypothesis that beta is equal to one”.

Senn’s quote continues with “In many applications we shall wish to be able to show that beta is sufficiently close to 1, perhaps by demonstrating that the confidence limits lie within some suitable range.”

❝ and how to calculate the sample size in this case?


Nasty. Monte Carlo Simulations? :confused:
Chow, Shao and Wang5 give an example of calculating the minimum effective dose. The method is a little bit esoteric, but probably can be modified by a hard-core statistician (not me!).

  1. T Smith
    Statistical methods - dose proportionality
    Technical Report, Ayerst Laboratories, New York (1986)
  2. Gough K, Byrom B, Ellis S, Lacey L, McKellar J, Hutchison M and O Keene
    Assessment of Dose Proportionality: Report From the Statisticians in the Pharmaceutical Industry/Pharmacokinetics UK Joint Working Party
    Drug Information Journal 29(3), 1039-1048 (1995)
  3. S Senn
    Statistical Issues in Drug Development
    John Wiley & Sons, Chichester, pp 300-302 (1997, reprint with corrections 2004)
  4. S Senn
    Statistical Issues in Drug Development
    John Wiley & Sons, Chichester, pp 345-347 (2nd ed., 2007)
  5. Chow S-C, Shao J and H Wang
    Sample Size Calculations In Clinical Research
    Marcel Dekker, New York, pp 296-301 (2003)

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ElMaestro
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Denmark,
2009-05-26 19:26
(5420 d 20:11 ago)

@ Helmut
Posting: # 3767
Views: 17,769
 

 Sample size for PK linearity

Dear HS,

α>0 and β≠0, weighted regression (w = 1/x)


Could you explain where that weight comes from? I would intuitively say it should be non-weighted, as nonlinearity can easily be e.g. a phenomenon only visible/measurable in the upper dose region. In this case I would think this weight would work against the chance of finding it.

EM.
Helmut
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2009-05-26 21:52
(5420 d 17:45 ago)

@ ElMaestro
Posting: # 3771
Views: 17,826
 

 Weighting scheme

Dear ElMaestro!

❝ ❝ α>0 and β≠0, weighted regression (w = 1/x)


❝ Could you explain where that weight comes from?


Chow & Liu (all editions), Chapter ‘Dose Proportionality Study’:

Let Y be AUC or Cmax and X be the dose level. Because, often, the standard deviation of Y increases as the dose increases, the primary assumption of dose proportionality is that the standard deviation of Y is proportional to X; that is,

Var(Y) = X2σ2,

where σ2 usually consists of inter- and intra-subject variability.

After describing the different models, they continue

It can be seen that model 1 can be used to evaluate dose proportionality by […] using a weighted linear regression with weights equal to X -1 based on the original data (X, Y).
[…] Model 2 […] can be tested using a weighted linear regression with weights equal to X -1 and with the original data (X, Y).
[…] similar to model 2, model 3 can be tested using a weighted linear regression with log-transformed data (logX, logY).


❝ I would intuitively say it should be non-weighted, as nonlinearity can easily be e.g. a phenomenon only visible/measurable in the upper dose region. In this case I would think this weight would work against the chance of finding it.


Hhm – maybe. Weighting comes from the quoted proportionality of the standard deviation of Y to X. I just checked it with one of my data sets (6 dose levels from 0.5–12, lower than dose proportional: b=0.59) and got an AIC of 286 for the unweighted model and an AIC of 263 for w=1/X. Residuals of the unweighted model showed the classical funnel shape. So I think it's not unreasonable.

Martin pointed me to a recent review*. Again, w=1/X is claimed for the linear model. They don’t give a weighting scheme for the power model, but derive a sample size estimation!
Interesting stuff.

  • Hummel J, McKendrick S, Brindley C, and R French
    Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion
    Pharmaceut Statist 8(1), 38–49 (2009)
    DOI: 10.1002/pst.326

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mjons
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Sweden,
2011-12-08 19:03
(4494 d 19:35 ago)

@ Helmut
Posting: # 7768
Views: 16,625
 

 Sample size for PK linearity

Hi,

Being new to dose proportionality studies, I am struggling with preparing a protocol for such a study. (Please excuse any stupid questions…) Since it should be a fairly common study to perform, it is strange there seems to be no consensus on how to do it properly. To just test if b≠1 seems illogical since this would favour an underpowered study.

A CI approach and acceptance criteria seems to be a more logical approach to me if you want to show that b is sufficiently close to 1. I found another paper by Smith (2000) and liked the power model approach suggested there, since you get a lot of (seemingly?) useful data out of this:
  • Possibility to claim dose proportionality (rather than just concluding that dose proportionality could not be rejected).
  • Even if failed (not demonstrated over the whole range) you get this rho-parameter with which you could potentially conclude a smaller interval that is dose proportional. (Or indicate dose proportionality over an even wider range...)
My questions with regards to this are:
  • Did I understand this correctly?
  • What are the pros and cons with this approach compared to other approaches? (What did I miss...?, Which method would you recommend?)
  • Should you just use the standard BE interval, 0.8000 to 1.2500 to calculate the acceptance interval (eq. 4) or could you expand it further?
  • Would a 90% CI for b be appropriate to test against the acceptance interval?
Best regards,
/Martin
Lucas
★    

Brazil,
2014-09-11 23:41
(3486 d 15:57 ago)

@ Helmut
Posting: # 13488
Views: 13,969
 

 Sample size for PK linearity

Hello Helmut!

If I'm going to use the Power Model, I think I should use this reference. Can you suggest me which software does that calculation?

Thanks in advance.

Lucas
d_labes
★★★

Berlin, Germany,
2014-09-12 17:01
(3485 d 22:36 ago)

@ Lucas
Posting: # 13499
Views: 14,061
 

 Sample size for PK linearity - power model

❝ Hello Helmut!


Not interested in an answer of others?

❝ If I'm going to use the Power Model, I think I should use this reference. Can you suggest me which software does that calculation?


I'm only aware of a solution in SAS given in the Patterson/Jones book for the case of crossover studies:
data a;
  * total number of subjects (needs to be a multiple of number
    of sequences, seq);
  * p is the number of periods;

  n=12; seq=3; p=3;
  * significance level;
  a=0.05;
  * true dose proportionality;
  beta=1;
  * sigmaW = within-subject standard deviation;
  sigmaW=0.25;
  * css is the corrected sum of squares of doses;
  * s assumes period effects are orthogonal to dose;

  css=CSS(log(1),log(2),log(8));
  s=sigmaW/sqrt(n*css);
  * error degrees of freedom for cross-over with n subjects in total
    assigned equally to seq sequences;

  n2=(n*p)-(n+p-1)-1;
  * t = acceptance limit;
  theta=1.25;
  r=8/1;
  t=log(theta)/log(r);
run;
data b; set a;
  * calculate power;
  t1=tinv(1-a,n2); t2=-t1;
  nc1=(sqrt(n))*((beta-(1-t))/s);
  nc2=(sqrt(n))*((beta-(1+t))/s);
  df=n2;
  prob1=probt(t1,df,nc1);
  prob2=probt(t2,df,nc2);
  answer=prob2-prob1;
  power=answer*100;
run;
proc print data=b; run;


I'm not quite sure if this all is correct.
Only power is calculated. Use this with different n's until you get your desired power. But for crossover designs you get high power with small numbers already. The above code with n=12 gives power=1!


BTW:
If you have the paper you linked to I would happy to get a copy. Then it may eventually possible to incorporate it into R-package PowerTOST.


S Patterson and B Jones
Bioequivalence and Statistics in Clinical Pharmacology
Chapman & Hall/CRC, Boca Raton (2006)
page 239

Regards,

Detlew
d_labes
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Berlin, Germany,
2014-10-01 11:09
(3467 d 04:28 ago)

@ d_labes
Posting: # 13627
Views: 15,459
 

 Power dose proportionality - power model - Correction

Dear all,

have posted here the above SAS code from the book Patterson/Jones for calculating power for dose proportionality studies evaluated via power model.

As already stated this code gives extraordinary high power for low n's already. The authors state "These are typically very powerful designs for the assessment of dose proportionality, and interested readers will find that power for the above design approaches 100%. Although as few as six normal healthy volunteers will serve to provide a very robust dose-proportionality assessment in most settings, it is recommended that cross-over studies supporting a regulatory file include at least 10 to 12 subjects to ensure application of the central-limit theorem is appropriate."

My gut feeling was that the power obtained is too high and that there must be a bug.
This feeling was supported by the following:
  • I would expect that the power equals the power obtained via the ordinary formulas for a 2x2x2 crossover, assuming true ratio=1, if we calculate for 2 doses only.
    But this is not the case.
    Example sigmaW=0.25 (CV=25.39567%), N=12 and doses 1, 4:
    power = 100% with the SAS code
    power(2x2x2) with true ratio 1 = 33.0069%
  • In the paper of Hummel et al.1 formulas via the normal distribution (large sample approximation I think) are given for the power of the parallel group design. I could not reproduce the values obtained via these formulas if I adapt the above SAS code to parallel group design (different df and different css). Again the values are much higher with the SAS code.
  • The paper of Sethuraman et al.2 gives examples for sample sizes also not compatible with the power calculated via the cited SAS code.
A closer look revealed that the non-centrality parameters have to changed to:
  nc1=((beta-(1-t))/s);
  nc2=((beta-(1+t))/s);

i.e. without the term sqrt(n) and voila, all above issues vanish.

Morals of the story: Even the pope may err ;-).

BTW: New version of R-package PowerTOST3 is out now. From the NEWS: "... Contains further experimental functions for power calculations / samplesize estimation for dose proportionality studies using the Power model".
Experimental in the sense that I have pending a discussion with the authors of above SAS code. But meanwhile I'm quite sure that the implementation in PowerTOST is correct.

BTW2: PowerTOST contains more new goodies to be discovered :cool:.


1Hummel et al.
"Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion"
Pharm. Stat. Vol. 8(1):38-49 (2007)

2Sethuraman VS, Leonov S, Squassante L, Mitchell TR, Hale MD
"Sample size calculation for the Power Model for dose proportionality studies"
Pharm. Stat. Vol. 6(1):35-41 (2007)

3Detlew Labes and Helmut Schuetz (2014).
PowerTOST: Power and Sample size based on two one-sided t-tests (TOST) for (bio)equivalence studies.
R package version 1.2-01.

Regards,

Detlew
d_labes
★★★

Berlin, Germany,
2014-09-13 17:05
(3484 d 22:32 ago)

@ Lucas
Posting: # 13502
Views: 13,794
 

 Sample size for PK linearity

Dear Lucas,

Forget the sentence "I'm not quite sure if this all is correct.". It's only my lack of understanding.

Regards,

Detlew
Lucas
★    

Brazil,
2014-10-13 22:36
(3454 d 17:02 ago)

@ d_labes
Posting: # 13694
Views: 13,386
 

 Sample size for PK linearity

Hello Detlew! I hope I find you well.

❝ Not interested in an answer of others?


Not at all, sorry for that. HS was the one explaining to the OP how to do it, but any response would be great. I had to leave work for a few days and only now have seen your response. Thank you very much for it, very sorry for the late response of mine.

❝ If you have the paper you linked to I would happy to get a copy. Then it may eventually possible to incorporate it into R-package PowerTOST.


Do you still want it? Seems like u've put your hands in it to make the last post, but if not I can e-mail it to you.

We have created a dinamic Excel spreadsheet that calculates the sample size, based on the publication of Sethuraman et al. If you want to see it I could sent it to you also. It is nothing compared to your PowerTOST, since our programming skills here are basic.

❝ Morals of the story: Even the pope may err ;-).


That is where we separate "the children from the adults", or I could say "the button-clickers analysts from the actual analysts/statisticians". Critic sense! :ok:

❝ BTW: New version of R-package PowerTOST3 is out now. From the NEWS: "... Contains further experimental functions for power calculations / samplesize estimation for dose proportionality studies using the Power model".


That's excellent! I'm downloading it now to see.

Again, thank u very much for the help, and excuse my late response!

Best regards.

Lucas
d_labes
★★★

Berlin, Germany,
2014-10-14 11:26
(3454 d 04:12 ago)

@ Lucas
Posting: # 13697
Views: 13,332
 

 Sample size for PK linearity

Dear Lucas,

❝ Hello Detlew! I hope I find you well.


Thanks, one can't enough lament :cool:.

❝ Do you still want it? Seems like u've put your hands in it to make the last post, but if not I can e-mail it to you.

❝ We have created a dinamic Excel spreadsheet that calculates the sample size, based on the publication of Sethuraman et al. If you want to see it I could sent it to you also. ...


Meanwhile I have the paper. But of course interested in your EXCEL sheet.

Do you have any idea what they do in the paper for incomplete block designs? Is it worth to deal with such designs?

Regards,

Detlew
Lucas
★    

Brazil,
2014-10-14 17:24
(3453 d 22:13 ago)

@ d_labes
Posting: # 13703
Views: 13,320
 

 Sample size for PK linearity

Hi Detlew!

❝ Meanwhile I have the paper. But of course interested in your EXCEL sheet.


I sent you a message to get your e-mail adress. Do not expect anything fancy, it is only to avoid using pencil and paper. :-D

❝ Do you have any idea what they do in the paper for incomplete block designs? Is it worth to deal with such designs?


I've studied Dose proportionality studies and also Incomplete Block designs but yet did not had to conduct any of those, so I may not be the best person to instruct you. But what I know is the basic knowledge that incomplete block designs would be very good to DP studies when one is interested in assessing dose proportionality with more than 4 or 5 doses, since it's very difficult to conduct a crossover study with more than 4 periods, and a parallel group design does not seem like (IMHO) not a good idea at all. The author only explain how to calculate the sample size for those designs, but does not say if it is worth dealing with it. We decided to conduct our first DP study with a 4-way crossover, only testing 4 doses.

Hope I've been of some help.
martin
★★  

Austria,
2009-05-25 10:46
(5422 d 04:51 ago)

@ Helmut
Posting: # 3745
Views: 18,688
 

 dose proportionality vs. dose linearity

Dear HS !

I think you mixed up dose-linearity and dose-proportionality.

Dose-proportionality: doubling of the dose within this range leads to doubling of Cmax or AUC (e.g. Cawello, 2003; page 149).

Using the power-law model AUC = a × Doseb, dose-proportionality holds when b = 1 (straight line through origin) and dose-proportionality implies dose-linearity but not vice versa.

best regards

Martin
Helmut
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Vienna, Austria,
2009-05-25 16:59
(5421 d 22:39 ago)

@ martin
Posting: # 3748
Views: 18,771
 

 dose proportionality vs. dose linearity

Dear Martin!

❝ I think you mixed up dose-linearity and dose-proportionality.


Oops! You are exactly right – it’s exactly they other way ’round. Sorry for the confusion caused. :-(

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