用神经网络替代线性项，让混合效应模型更灵活地捕捉复杂关系。
Neural Generalized Mixed-Effects Models
- 用神经网络取代传统线性函数，提升模型对非线性关系的拟合能力。
- 在合成数据上优于经典GLMM，真实数据集表现超越已有方法。
- 适合处理具有分组结构的复杂数据，如教育、医疗等场景。
广义线性混合效应模型（GLMM）广泛用于分析分组和层级数据。在GLMM中，每个响应变量服从指数族分布，其自然参数由可观测协变量与潜变量组特异性随机效应的线性组合决定。由于对随机效应进行精确边际化通常不可行，模型参数通过最大化近似边际似然估计。本文将线性函数替换为神经网络，提出神经广义混合效应模型（NGMM），显著增强模型对协变量与响应间复杂关系的捕捉能力。为拟合NGMM，我们设计了一种高效可微优化算法，最大化近似边际似然。理论证明，目标函数的近似误差随用户设定参数呈高斯尾部衰减。在合成数据上，当协变量-响应关系为非线性时，NGMM优于标准GLMM；在真实数据集上，性能超过先前方法。最后，我们利用大规模学生能力数据集，展示NGMM向更复杂潜变量模型扩展的可行性。
原文摘要 · Abstract (English)
Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.
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