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Meta-analysis is a powerful tool to synthesize findings from multiple studies. The normal–normal random-effects model is widely used to account for between-study heterogeneity. However, meta-analyses of sparse data, which may arise when the event rate is low for binary or count outcomes, pose a challenge to the normal–normal random-effects model in the accuracy and stability in inference since the normal approximation in the within-study model 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 model with an exact model. Publication bias is one of the most serious threats in meta-analysis. Several quantitative sensitivity analysis methods for evaluating the 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 useful guidance to address publication bias in the meta-analysis of sparse data. [ABSTRACT FROM AUTHOR]