A Tutorial on Bayesian Multi-Study Factor Analysis With Applications in Nutrition and Genomics

High-dimensional data are crucial in biomedical research. Integrating such data from multiple studies is a critical process that relies on the choice of advanced statistical models, enhancing statistical power, reproducibility, and scientific insight compared to analyzing each study separately. Factor analysis (FA) is a core dimensionality reduction technique that models observed data through a small set of latent factors. Bayesian extensions of FA have recently emerged as powerful tools for multi-study integration, enabling researchers to disentangle shared biological signals from study-specific variability. In this tutorial, we provide a practical and comparative guide to seven advanced Bayesian integrative factor models: Perturbed Factor Analysis (PFA), Bayesian Factor Regression with non-local spike-and-slab priors (MOM-SS), Subspace Factor Analysis (SUFA), Bayesian Multi-study Factor Analysis (BMSFA), a variational-inference implementation of BMSFA (CAVI), Bayesian Latent Analysis through Spectral Training (BLAST), and Bayesian Combinatorial Multi-study Factor Analysis (Tetris). To contextualize these methods, we also include two benchmark approaches: Standard FA applied to pooled data (Stack FA), and FA applied separately to each study (Ind FA). We evaluate all methods through extensive simulations, assessing computational efficiency, accuracy in estimation of loadings, and the number of factors. To bridge theory and practice, we present a full analytical workflow-with detailed R code-demonstrating how to apply these models to real-world datasets in nutrition and genomics. This tutorial is designed to guide applied researchers through the landscape of Bayesian integrative factor analysis, offering insights and tools for extracting interpretable, robust patterns from complex multi-source data. Simulation code, R package "bmfaToolkits" and a user-friendly guidebook can be found at https://github.com/Mavis-Liang/Bayesian_integrative_FA_tutorial.