Rare Safety Events And Zero-Event Trials

Why Zero-Event Predictions Need Attention

For a rare adverse event, a trial can observe no events even when the event probability is not zero. That makes a predicted zero count clinically important: it helps reveal whether a proposed prior and planned sample size jointly imply that “no events observed” should be routine or surprising. This check does not establish that a safety risk is absent. It describes what the prior model predicts before trial outcomes are observed.

The calculation depends on both the event probability and the number of participants observed. A fixed-probability calculation gives the chance of zero events at one specified probability. A Bayesian prior predictive calculation averages that chance over the prior distribution for the probability. These are not generally the same calculation, even if the fixed probability is set equal to the prior mean.

Model and Assumptions

For a simple calculation, suppose each participant contributes one binary outcome: the specified event either occurs or does not occur during the defined window. Conditional on a common event probability \(p\), assume the participant outcomes are independent and have the same probability \(p\). Then \(K\mid p\) has a binomial distribution. To represent uncertainty about the event probability before seeing trial data, use a beta prior with positive shape parameters \(a\) and \(b\).

This model requires more than a small event rate. The endpoint and observation window must be defined consistently for every participant; event ascertainment must be sufficiently reliable; and the common-probability, conditional-independence assumption must be reasonable for the intended calculation. For example, this setup treats each participant as contributing one event indicator, not a recurrent-event count. Differences in follow-up or substantial dependence between participants may require a different model. The calculation is a prior predictive check under stated assumptions, not a guarantee about the trial.

The beta prior has mean \(a/(a+b)\). Conditional on a particular \(p\), the probability of no events is \((1-p)^n\). Integrating this conditional probability over the prior gives the prior predictive probability of zero events:

Here \(B(\cdot,\cdot)\) is the beta function, and \(\pi(p)\) is the beta prior density. More generally, integrating the binomial probability for any count \(k\) gives the beta-binomial predictive distribution:

The predictive mean count is \(E[K]=n\,a/(a+b)\), but a mean does not determine how much probability lies at zero. The probability of at least one event is \(P(K\geq 1)=1-P(K=0)\). Both quantities are useful: the mean describes the average count over prior predictive repetitions, while the zero-event probability directly describes how often the model predicts no observed events.

Worked Prior Predictive Calculations

How Prior Uncertainty Affects the Zero Probability

For \(n\geq 2\), the function \(f(p)=(1-p)^n\) is convex on \(0\leq p\leq 1\), since \(f''(p)=n(n-1)(1-p)^{n-2}\geq 0\). Jensen’s inequality therefore gives:

The left side is the prior predictive probability of zero events; the right side is the fixed-probability calculation at the prior mean. Thus, for this model and \(n\geq2\), substituting the prior mean generally understates the prior predictive chance of a zero count. The inequality is strict when the prior has nonzero uncertainty and \(n\geq2\). When \(n=1\), the probability of zero is linear in \(p\), so the two calculations are equal.

This result helps explain why uncertainty in a rare-event probability can make zero-event trials more probable than a calculation at the average probability suggests. It does not say that every prior with greater variance has a larger zero probability than every other prior: the full distribution matters, and comparisons should use the actual prior predictive calculation.

Diagnostics, Alternatives, and Limitations

A prior predictive check is most useful when it is tied to a specific safety question. Before interpreting a zero probability, verify the endpoint definition, denominator, follow-up window, and event ascertainment. Then check whether the prior predictive probability of zero events, and the expected count, are plausible to the clinical team for that trial size. If simulations are used, draw \(p\) from the prior and then draw \(K\mid p\); fixing \(p\) at its mean skips the uncertainty the prior predictive check is meant to include.

The binomial calculation is a useful baseline, not a universal safety-event model. It may be unsuitable if participants have materially different exposure times, event risks vary in ways not represented by the shared \(p\), outcomes are dependent, or participants can have multiple events and the target is a recurrent-event count. In those settings, the probability of zero events should be calculated from a model that represents the relevant follow-up and dependence structure. A zero count can also arise from limited follow-up or incomplete ascertainment, neither of which is resolved by choosing a different prior for \(p\).

A fixed-\(p\) calculation is simpler and appropriate when the question is explicitly conditional on one specified event probability. A Poisson approximation may be convenient for a binomial count when \(p\) is small and \(n\) is sufficiently large, with mean \(np\); at fixed \(p\), it approximates the zero probability by \(\exp(-np)\). It is an approximation, and substituting a prior mean into it still does not integrate over prior uncertainty. The beta-binomial expression gives the exact prior predictive calculation for the beta-binomial model. Simulation can check the calculation or handle extensions, but the simulation must reproduce the intended model and prior.

A common mistake is to say, “The prior mean implies 0.25 events, so a zero-event trial is unlikely,” or to calculate zero probability only at the prior mean and call it prior predictive. An expected count of 0.25 does not mean that 0.25 events will occur in a trial, and the plug-in calculation does not average across plausible probabilities. Calculate \(P(K=0)\) under the full prior predictive distribution, then interpret it in the context of the event definition, sample size, and assumptions.