A reproducible, diagnostic-guided hybrid Laplace workflow for generalized linear mixed models.

Laplace approximation is widely used for generalized linear mixed models because it is fast and compatible with automatic-differentiation software, yet its accuracy can deteriorate when clusters are weakly informative, outcomes are sparse, or conditional random-effect distributions are skewed, heavy-tailed, or poorly conditioned. This article presents a reproducible diagnostic-guided hybrid Laplace workflow. The method begins with an ordinary Laplace fit, assigns blockwise approximation-risk scores, and selectively upgrades only flagged random-effect blocks using higher-fidelity corrections such as adaptive Gauss-Hermite quadrature, importance sampling, or skew-aware correction. This selectivity retains ordinary Laplace contributions for unflagged blocks but should not be interpreted as a demonstrated wall-clock speed-up: in the scalar benchmarks reported here, diagnostic overhead makes the hybrid slower than global 15-point adaptive Gauss-Hermite quadrature. A computational advantage is expected only in regimes where the cost of high-fidelity correction materially exceeds the cost of diagnostic screening and remains to be established there. In a rare-event Bernoulli random-intercept validation, the diagnostic score ranked absolute Laplace log-integral error well (Spearman ρ=0.84). Correcting the highest-risk third of blocks reduced the mean absolute error on flagged blocks from