  Publication bias (PB) is one of the serious issues in meta-analysis. Many
existing methods dealing with PB are based on the normal-normal (NN)
random-effects model assuming normal models in both the within-study and the
between-study levels. For rare-event meta-analysis where the data contain rare
occurrences of event, the standard NN random-effects model may perform poorly.
Instead, the generalized linear mixed effects model (GLMM) using the exact
within-study model is recommended. However, no method has been proposed for
dealing with PB in rare-event meta-analysis using the GLMM. In this paper, we
propose sensitivity analysis methods for evaluating the impact of PB on the
GLMM based on the famous Copas-Heckman-type selection model. The proposed
methods can be easily implemented with the standard software coring the
nonlinear mixed-effects model. We use a real-world example to show how the
usefulness of the proposed methods in evaluating the potential impact of PB in
meta-analysis of the log-transformed odds ratio based on the GLMM using the
non-central hypergeometric or binomial distribution as the within-study model.
An extension of the proposed method is also introduced for evaluating PB in
meta-analysis of proportion based on the GLMM with the binomial within-study
model.
