Bioequivalence and Bioavailability Forum

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 Senn^3,4 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?
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!).

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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 (2^nd 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)