Testing bioequivalence statistically using FDA guidance

Question

I need to statistically test whether a drug is bioequivalent to another drug by using the pharmacokinetic parameters AUC (area under the curve of a plasma concentration curve) and Cmax (maximum concentration). According to the FDA one should calculate the 90% confidence interval (CI) of the proportion of test drug and reference (test/reference) for both parameters. In my case the data are unpaired, meaning that a few persons took drug X and a few persons took drug Y and the mean AUC and Cmax were calculated for both groups. So this means that if the group that took drug X had a mean AUC of 2.7 and the group that took drug Y had a mean AUC of 5.5, the 90% confidence interval would be for the proportion $\frac{2.7}{5.5}$. Now my question is, how can I calculate this 90% confidence interval? According to the FDA one should first log transform the data and calculate the 90% CI for the difference of the logarithm of both parameters, so the 90% CI of differences for: ln AUC$_1$ - ln AUC$_2$ and ln Cmax$_1$ - ln Cmax$_2$. And then transform the lower and upper bounds back by taking $e$ to the power of the boundaries to get the lower and upper bounds of the proportion since $ln(x) - ln( y) = ln (x/y)$. The only example I got was with paired data so I assumed it worked for unpaired data too. However when I tried this with my unpaired data the confidence intervals I got were way off way since I got CIs boundaries with way higher values than the maximum observed values. I first calculated group means for both groups and then took the Ln of both. Then subtracted the log transformed parameters like I referenced before (ln AUC$_1$ - ln AUC$_2$). And then calculated the pooled standard deviation. And lastly the 90% CI for the difference of the log transformed parameters using the formula $(Ln(AUC_1)-Ln(AUC_2)) ± t_{0.10} * s_p * \sqrt{1/n_1 +1/n_2}$ Sorry for the somewhat long and vague question. Thanks in advance to anyone who can help me with this though. If further explaination needed I can elaborate or if the data are needed I could post them .

Answer 1

You said: I first calculated group means for both groups and then took the Ln of both. Then subtracted the log transformed parameters like I referenced before (ln AUC1 - ln AUC2).

You should at least take the logarithms and then take the average logarithms. The point being that if AUC is lognormal distributed then it is the average logarithm that is normally distributed, and the average of raw AUC's will not measure the location of the data. It is difficult for me to follow what you did, not so much because you did not explain it, but because you did so many steps that it is a bit confusing. Showing data might make it easier to understand, but, at least try what I suggest here and see if that is enough, please.