The identity of two meta-analytic likelihoods and the ignorability of double-zero studies

In meta-analysis, the conventional two-stage approach computes an effect estimate for each study in the first stage and proceeds with the analysis of effect estimates in the second stage. For counts of events as outcome, the risk ratio is often the effect measure of choice. However, if the meta-analysis includes many studies with no events the conventional method breaks down. As an alternative one-stage approach, a Poisson regression model and a conditional binomial model can be considered where no event studies do not cause problems. The conditional binomial model excludes double-zero studies, whereas this is seemingly not the case for the Poisson regression approach. However, we show here that both models lead to the same likelihood inference and double-zero studies (in contrast to single-zero studies) do not contribute in either case to the likelihood.