The two one-sided tests (TOST) procedure for average bioequivalence is operationally equivalent to an ordinary \(100 (1 - 2 \alpha)\%\) or an expanded \(100 (1 - \alpha)\%\) confidence interval.

OrdCI <- c(est - sd * qt(1 - alpha, n - 1) / sqrt(n),
           est + sd * qt(1 - alpha, n - 1) / sqrt(n))
ExpCI <- c(min(0, est - sd * qt(1 - alpha, n - 1) / sqrt(n)),
           max(0, est + sd * qt(1 - alpha, n - 1) / sqrt(n)))

We simulate 10,000 normal datasets with \(\sigma=0.1\), and \(n=20\) and calculate both the 90% ordinary and the 95% expanded CI using a simulation function Sim(mu, sd, n, alpha, nsim) (not shown here).

For \(\mu=log(1.25)\) we get the expected coverage probabilities (CP) of 90% and 95%; the implied tests, however, have a “power” of 5% for both CIs. The expanded CI is substantially wider on average, but this involves (by construction) no power disadvantage whatsoever.

Sim(mu=log(1.25), sd=0.1, n=20, alpha=0.05, nsim=10000)
##          ordinary expanded
## Coverage    0.902    0.952
## Power       0.048    0.048
## Width       0.076    0.261

For \(\mu=0\) the CP of the expanded CI is 100% because it always contains zero (again by construction). The discrepancy in average width is now much smaller because few CIs are actually being “expanded”.

Sim(mu=0, sd=0.1, n=20, alpha=0.05, nsim=10000)
##          ordinary expanded
## Coverage    0.901    1.000
## Power       1.000    1.000
## Width       0.076    0.077