Sensitivity analysis for publication bias in meta-analysis of sparse data based on exact likelihood

Meta-analysis is a powerful tool to synthesize findings from multiple studies. The normal-normal random-effects model is widely used accounting for between-study heterogeneity. However, meta-analysis of sparse data, which may arise when the event rate is low for binary or count outcomes, poses a challenge to the normal-normal random-effects model in the accuracy and stability in inference since the normal approximation in the within-study likelihood may not be good. To reduce bias arising from data sparsity, the generalized linear mixed model can be used by replacing the approximate normal within-study likelihood with an exact likelihood. Publication bias is one of the most serious threats in meta-analysis. Several objective sensitivity analysis methods for evaluating potential impacts of selective publication are available for the normal-normal random-effects model. We propose a sensitivity analysis method by extending the likelihood-based sensitivity analysis with the $t$-statistic selection function of Copas to several generalized linear mixed-effects models. Through applications of our proposed method to several real-world meta-analyses and simulation studies, the proposed method was proven to outperform the likelihood-based sensitivity analysis based on the normal-normal model. The proposed method would give a useful guidance to address publication bias in meta-analysis of sparse data.