Latent factor models that integrate data from multiple sources/studies or modalities have garnered considerable attention across various disciplines. However, existing methods predominantly focus either on multi-study integration or multi-modality integration, rendering them insufficient for analyzing the diverse modalities measured across multiple studies. To address this limitation and cater to practical needs, we introduce a high-dimensional generalized factor model that seamlessly integrates multi-modality data from multiple studies, while also accommodating additional covariates. We conduct a thorough investigation of the identifiability conditions to enhance the model's interpretability. To tackle the complexity of high-dimensional nonlinear integration caused by 4 large latent random matrices, we utilize a variational lower bound to approximate the observed log-likelihood by employing a variational posterior distribution. By profiling the variational parameters, we establish the asymptotical properties of estimators for model parameters using M-estimation theory. Furthermore, we devise a computationally efficient variational expectation maximization (EM) algorithm to execute the estimation process and a criterion to determine the optimal number of both study-shared and study-specific factors. Extensive simulation studies and a real-world application show that the proposed method significantly outperforms existing methods in terms of estimation accuracy and computational efficiency.
Liu et al. (Thu,) studied this question.