Bioequivalence and Bioavailability Forum

Dear all,

during development of my famous code for exact power and sample size for BE studies, especially for incorporating replicate designs, I came across of approximate formulas. The use of non-central t-distribution as approximation is extensively discussed in this thread.

But in [1] or [2] the following formula is used (only additive model shown here)
power=Fdf( (delta-theta)/(CV*SQRT(bk/n)) - t(alpha,df) )
     -Fdf( t(alpha,df) - (delta+theta)/(CV*SQRT(bk/n)) )

where delta is the bioequivalence limit, theta=(µ_T-µ_R)/µ_R, bk the design constant and t(alpha,df) the critical value of the t-distribution. F_df() is the cumulative distribution function of the central! t-distribution with df degrees of freedom.

Astonishing enough the authors phrase it as "exact" (page 757 of [1]:"Hence, the exact formula for determination of n required to achieve 1-ß power ...).
Does anybody know where this comes from?
It seems an approximation of the non-central t-distribution (also an approximation as we know meanwhile).

This formula using central 'shifted' t-distribution (and the variant for the multiplicative model, log-transformed values) is implemented in PASS 2008.

The 'approximation of an approximation' works very well as a comparison with the exact results show. But only little numeric experimentation done yet with target power=0.8.
For a one to one correspondence note the different degrees of freedom in my implementation (without carry over) and the different design constant in case of the "2x2x4" replicate design (also without carry over).

In experimenting with this approximate formula and the exact results I noticed that the Tables (pages 295 ff of [2]) for sample size in the multiplicative model can only reproduced if the values of CV are used as sigma_W, also it is termed "Coefficient of variation (%)" (take their value as sigma_W and calculate CV=sqrt(exp(sigma_W²)-1) and then call my R function sampleN.TOST()).
This is also described in the help files of PASS2008.

[1] K.-W. Chen, S.-C. Chow, G. Li
"A note on sample size determination for bioequivalence studies with higher-order cross-over designs"
J. Pharmacokin. Biopharm. Vol. 25 (No.6), 753-765 (1997)
[2] S.-C. Chow, J.-P. Liu
"Design and analysis of bioavailability and bioequivalence studies"
Third edition, Chapman & Hall (2009), page 289 ff