Estimating Relative Risk When Observing Zero Events—Frequentist Inference and Bayesian Credibility Intervals

Relative risk (RR) is a preferred measure for investigating associations in clinical and epidemiological studies with dichotomous outcomes. However, if the outcome of interest is rare, it frequently occurs that no events are observed in one of the comparison groups. In this case, many of the standard methods used to obtain confidence intervals (CIs) for the RRs are not feasible, even in studies with strong statistical evidence of an association. Different strategies for solving this challenge have been suggested in the literature. This paper, which uses both mathematical arguments and statistical simulations, aims to present, compare, and discuss the different statistical approaches to obtain CIs for RRs in the case of no events in one of the comparison groups. Moreover, we compare these frequentist methods with Bayesian approaches to determine credibility intervals (CrIs) for the RRs. Our results indicate that most of the suggested approaches can be used to obtain CIs (or CrIs) for RRs in the case of no events, although one-sided intervals obtained by methods based on deliberate, probabilistic considerations should be preferred over ad hoc methods. In addition, we demonstrate that Bayesian approaches can be used to obtain CrIs in these situations. Thus, it is possible to obtain statistical inference for the RR, even in studies with no events in one of the comparison groups, and CIs for the RRs should always be provided. However, it is important to note that the obtained intervals are sensitive to the method chosen in the case of small sample sizes.