non-central t is not the question [🇷 for BE/BA]

posted by d_labes  – Berlin, Germany, 2009-10-05 14:58 (5288 d 00:57 ago) – Posting: # 4296
Views: 30,144

Dear ElMaestro,

don't waste your time in attempting to improve Al Gore's rhythms.
The inventors of them for the non-central t distri had already made every effort for its accuracy and reliability, I think.

Your time is better invested for NLYW (blonde downstairs) or for the Seven seas :-D .

The formula given in Julious is definitely approximate, albeit a good one for the usual power range ≥70% as the sample size calculations with that formula show.

The exact algo for the power of the equivalence test (two one sided tests aka TOST) in a 2x2 cross-over is ([1],[2]):

Power=Prob(t1≥t1-alpha,(n-2) and t2≤-t1-alpha,(n-2)|bioequivalence)


The t1 and t2 values are the t-variates of the two one-sided t-tests of bioequivalence.

(t1, t2) have according to Owen a bivariate non-central t-distribution and the power can be calculated as the difference of two definite integrals (Owen's Q functions dependent on df=degrees of freedom and with four arguments)

Power=Qdf(-t1-alpha,(n-2),d2,0,R)-Qdf(t1-alpha,(n-2),d1,0,R)


See [1] and [2] for that and for the formulas for d1, d2 and R.

So you need not a better algo for the non-central t but rather an algo for Owen's Q-functions :-P .
My SAS beast has these functions implemented (but not documented!).

❝ In a previous thread I asked what the power in a 2,2,2-BE trial is at N=6, CV=65% and T/R=95%.


And my sasophylistic answer to the your ElMaestrophylistic question "what the power in a 2,2,2-BE trial is at N=38, CV=65% and T/R=95%." given here, power=4.5%, is based on these Q-functions.
Implementing the Julious formulas (according to Helmut's R-code) in "The power to know" results in power=-0.027799235, same as Fartssie!

[1] Diletti, E., Hauschke, D., and Steinijans, V.W. (1991),
"Sample Size Determination for Bioequivalence Assessment by Means of Confidence Intervals"
International Journal of Clinical Pharmacology, Therapy and Toxicology, Vol. 29, 1 -8.
[2] Hauschke, Steinijans, Pigeot
"Bioequivalence studies in drug development"
Wiley 2007, pages 120-121


Regards,

Detlew

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