Meta-analysis of diagnostic test accuracy (DTA) studies is commonly based on bivariate generalized linear mixed models (GLMMs) that assume normally distributed study-specific random effects. In practice, this distributional assumption is rarely examined, particularly in individual participant data meta-analysis (IPDMA), where empirical Bayes (EB) estimates exhibit heterogeneous precision across studies. Applying standard multivariate normality tests to such estimates can therefore lead to misleading inference. We propose a weighted extension of the Henze-Zirkler (HZ) test to assess bivariate normality of random effects, accounting for study-specific precision via inverse-variance weighting. To obtain valid finite-sample inference, we develop a parametric bootstrap calibration procedure. Extensive simulation studies showed that the classical HZ test applied to EB estimates can exhibit substantial Type I error inflation, whereas the proposed weighted procedures maintain nominal error rates while retaining power to detect departures from normality. The proposed method provides a practical, statistically principled diagnostic tool for assessing random-effects distributional assumptions in hierarchical models, with particular relevance to IPDMA of DTA studies. Although motivated by IPDMA, the proposed approach also extends to aggregate-data diagnostic test accuracy meta-analysis based on study-level 2-by-2 tables.
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