We present generalized additive latent and mixed models (GALAMMs) for analysis of clustered data with responses and latent variables depending smoothly on observed variables. A scalable maximum likelihood estimation algorithm is proposed, utilizing the Laplace approximation, sparse matrix computation, and automatic differentiation. Mixed response types, heteroscedasticity, and crossed random effects are naturally incorporated into the framework. The models developed were motivated by applications in cognitive neuroscience, and two case studies are presented. First, we show how GALAMMs can jointly model the complex lifespan trajectories of episodic memory, working memory, and speed/executive function, measured by the California Verbal Learning Test (CVLT), digit span tests, and Stroop tests, respectively. Next, we study the effect of socioeconomic status on brain structure, using data on education and income together with hippocampal volumes estimated by magnetic resonance imaging. By combining semiparametric estimation with latent variable modeling, GALAMMs allow a more realistic representation of how brain and cognition vary across the lifespan, while simultaneously estimating latent traits from measured items. Simulation experiments suggest that model estimates are accurate even with moderate sample sizes.

翻译：我们提出广义可加潜在混合模型（GALAMMs），用于分析响应变量与潜在变量随观测变量平滑变化的聚类数据。本研究提出一种可扩展的极大似然估计算法，该算法利用拉普拉斯近似、稀疏矩阵计算和自动微分技术。混合响应类型、异方差性及交叉随机效应均被自然地纳入该框架。所开发模型的灵感源于认知神经科学的应用场景，并呈现两个案例研究。首先，我们展示了GALAMMs如何联合建模情景记忆、工作记忆与速度/执行功能的复杂生命跨期轨迹——这些指标分别通过加州言语学习测试（CVLT）、数字广度测试和斯特鲁普测试测量。其次，利用教育与收入数据及磁共振成像测得的海马体积，我们研究了社会经济地位对大脑结构的影响。通过将半参数估计与潜在变量建模相结合，GALAMMs能够更真实地呈现大脑和认知在生命跨期中的变化规律，同时从测量条目中同步估计潜在特质。模拟实验表明，即使样本量适中，模型估计仍保持较高精度。