  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.
