Expanded Safety (ES) trials are phase IV trials designed to estimate the frequency of uncommon adverse events that may have been undetected in earlier studies. These studies may be nonrandomized.

Typically, we assume that the study population is large, the probability of an adverse event is small (because it did not crop up in prior trials), and all participants in the cohort of size m are followed for approximately the same length of time. Under these assumptions, we can model the probability of exactly d events occurring based on a Poisson probability function, i.e.,

\( Pr \left[ D = d \right] = ( \xi m)^d exp(- \xi m)/d! \)

where \( \xi \) is the adverse event rate.

The cohort should be large enough to have a high probability of observing at least one event when the event rate is \( \xi \). Thus, we want

\( \beta = Pr \left[D \ge 1 \right] = 1 - Pr \left[ D = 0 \right] = 1 - exp(-\xi m) \)

to be relatively large. With respect to the cohort size, this means that m should be selected such that

\( m = -log_e(1 - \beta)/ \xi \)