We consider random effects meta-analysis where the outcome variable is the occurrence of some event of interest. The data structures handled are where one has one or more groups in each study, and in each group either the number of subjects with and without the event, or the number of events and the total duration of follow-up is available. Traditionally, the meta-analysis follows the summary measures approach based on the estimates of the outcome measure(s) and the corresponding standard error(s). This approach assumes an approximate normal within-study likelihood and treats the standard errors as known. This approach has several potential disadvantages, such as not accounting for the standard errors being estimated, not accounting for correlation between the estimate and the standard error, the use of an (arbitrary) continuity correction in case of zero events, and the normal approximation being bad in studies with few events. We show that these problems can be overcome in most cases occurring in practice by replacing the approximate normal within-study likelihood by the appropriate exact likelihood. This leads to a generalized linear mixed model that can be fitted in standard statistical software. For instance, in the case of odds ratio meta-analysis, one can use the non-central hypergeometric distribution likelihood leading to mixed-effects conditional logistic regression. For incidence rate ratio meta-analysis, it leads to random effects logistic regression with an offset variable. We also present bivariate and multivariate extensions. We present a number of examples, especially with rare events, among which an example of network meta-analysis.
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