Bioequivalence and Bioavailability Forum

Dear Ben!

❝ On slide 31 from Helmut's lecture it's written that exact methods rely on AS 243 (or AS 184). I thought the exact method stands for solving the integral defined by Owens Q function (see also this post); but AS 243 (and AS 184) are algorithms to compute cumulative probabilities of the noncentral t-distribution (aren't they?) and hence are only applicable in the case where the approximation via noncentral t-distribution is being used. So one cannot talk about using the exact method and AS 243 in one sentence.

Well, I could. I must confess that my wording might be confusing. I used ‘exact’ in a sense of numerically approximating Owen’s solution.

❝ Or am I wrong here?

Not at all. Though Owen gave the solution in his 1965 paper – the differences between the two definite integrals cannot be solved explicitly. Hence numeric methods have to be applied (see the nice numbers in log-Γ section of the FORTRAN90-source of AS 234).

❝ Also, the nQuery v7 user manual (Appendix 7-5, page 153) says that an algorithm due to Owen is used in order to calculate the power, therefore on slide 33 I don't understand why the algorithm from nQuery is "AS 184"

Blast! Where did I get this from? If I recall it right once there was a paper on Statsol’s website. Gone yet. I can also no confirm which version is implemented in FARTSSIE… No version is given in the VBA-code. My knowlege of FORTRAN is better than of VBA; maybe I hack myself through. There are some differences in the implementation, but I cannot say yet whether the VBA-code is more close to 184 or 234. Where did I get all these algo-numbers from? Have to chimney-sweep my future slides.

❝ (well, if the algorithm from Owen is exactly AS 184, then it's ok...).

Owen’s method is no algo. Unfortunately nothing is stated in the manual (neither in v7 nor in v5).

❝ Coming from another point of view it's getting clear why it cannot be the algorithm of Owen (like in the row above: Diletti et al (1991)): the sample sizes for example in case of CV=0.075 from Diletti et al (1991) and nQuery Advisor 7 do not match. So what about the user manual...?

Note that nQuery always give the sample size in integers per sequence. Therefore in a 2×2 cross-over the output is 3 which gives a total of 6. Diletti et al. in their Table 1 give also 6, but in the heading the additional statement “Calculated odd sample sizes have been rounded up and are given in italics.” – which was the case for power 80%, PE 0.95, CV 7.5%. The respective rows in my table (slide 33) give the unrounded (odd) numbers, if applicable.

❝ Another thing is: AS 184 is older than AS 243, but is it worse?

Good question. Next question.

❝ For example if the true ratio=1, CV=0.075 and n=4 (see also this post) the exact method from PowerTOST gives a power of 0.7290143. nQuery 7 (see the table in the post just mentioned) gives 71.559% whereas FARTSSIE 1.6 (with AS 243) gives 66.674%. The result from nQuery is closer to the "exact" result, although it uses an older algorithm.

Interesting! But as I said above right now I cannot confirm which algo R. Lenth implemented into FARTSSIE. David’s statement ‘Dr. Russel Lenth generously provided the library subroutines to calculate non-central distributions (NCt)’ is not telling. See also the end of this post. Maybe VBA runs into trouble at T/R=1; have to dig out Dieter’s paper (can’t promise – piles are high).