## CI for Transformed data Unbalanced study [General Sta­tis­tics]

Dear USFDA_EMEA!

I guess that’s not your real name

» I am looking for help on estimation of 90% confidence interval for log transformed data of AUC (0-inf) for a bioequivalence study. This study involved two-way crossover, two-period, two-sequence design with a total of 33 subjects completing the study. Each sequence had Period I TR=16 and RT=17, and in Period II TR=17 and RT=16; thus, total of 33 completed the study.

OK, the number of subjects in each sequence is n1 and n2, respectively, where n = n1 + n2.

n1 = 17 n2 = 16 n  = 33

The assignment of sequence 1 to RT follows the literature, but is only a convention (if you want you may call TR sequence 1 as long as you keep this convention through all calculations…)

» In order to estimate the 90% confidence interval, can some one help me with the formula and a worked out example for the ease of understanding. OR Guide me to a site where this can be found.

Let’s see. Which formula are you using right now?
If you have something like 1/n in it, for unbalanced data you have to replace it by 1/n1 + 1/n2.

» I believe least-squares mean estimate for each formulation needs to be used to form the difference, together with the standard error. However, in lack of exact formula i am seeking the help of experts out there.

Yes, you are on the right track!
Let's call Xt the LSM of the test, and Xr the LSM of the reference (log-scale). sigma-w is the within- (or intra-) subject standard deviation (sqrt(MSE) from your ANOVA), t(1-alpha,n1+n2-2) is the 0.95 quantile of the central t-distribution with n1+n2-2 degrees of freedom. Then the upper/lower 90% confidence limits (log scale) are given by
 Xt - Xr ± t(1-alpha,n1+n2-2) × sqrt(MSE) × sqrt[ 1/2 × (1/n1 + (1/n2) ]

Just a reminder: don’t try evaluations in M\$-Excel; the t-quantile (TINV) is implemented in a rather queer way: in order to get the correct value for t(1-alpha,n1+n2-2), you would have to use TINV(2 × alpha, n1+n2-2)!
See
http://www.practicalstats.com/xlsstats/excelstats.html
http://www.mis.coventry.ac.uk/~nhunt/pottel.pdf

I would heartly recommend two textbooks:
1. S-C Chow and J-p Liu
Design and Analysis of Bioavailability and Bioequivalence Studies
Marcel Dekker, New York, 2nd Ed. (2000)
2. D Hauschke, VW Steinijans and I Pigeot
Bioequivalence Studies in Drug Development
Wiley, Chichester (2007)

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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