  Network meta-analysis (NMA) of rare events has attracted little attention in
the literature. Until recently, networks of interventions with rare events were
analyzed using the inverse-variance NMA approach. However, when events are rare
the normal approximation made by this model can be poor and effect estimates
are potentially biased. Other methods for the synthesis of such data are the
recent extension of the Mantel-Haenszel approach to NMA or the use of the
non-central hypergeometric distribution. In this article, we suggest a new
common-effect NMA approach that can be applied even in networks of
interventions with extremely low or even zero number of events without
requiring study exclusion or arbitrary imputations. Our method is based on the
implementation of the penalized likelihood function proposed by Firth for bias
reduction of the maximum likelihood estimate to the logistic expression of the
NMA model. A limitation of our method is that heterogeneity cannot be taken
into account as an additive parameter as in most meta-analytical models.
However, we account for heterogeneity by incorporating a multiplicative
overdispersion term using a two-stage approach. We show through simulations
that our method performs consistently well across all tested scenarios and most
often results in smaller bias than other available methods. We also illustrate
the use of our method through two clinical examples. We conclude that our
"penalized likelihood NMA" approach is promising for the analysis of binary
outcomes with rare events especially for networks with very few studies per
comparison and very low control group risks.
