The role of spatially varying loadings in dynamic spatial factor models for modeling the opioid syndemic.

Understanding the interactions and spatio-temporal variations of public health outcomes is crucial for gaining insight into interrelated epidemics across different locations and time periods. Dynamic spatial factor models provide a flexible framework for capturing shared variability among multiple outcomes through a latent factor and its corresponding loadings. A common assumption in these models is that factor loadings are spatially constant, implying uniform relationships between outcomes across the study region. However, this assumption may overlook important regional differences in how outcomes relate to the underlying latent factor. In this study, we derive the covariance structure of the outcome vector to highlight how spatially varying versus constant loadings influence the overall correlation structure. We find that when loadings vary across space, the spatial covariance of the outcomes is shaped by both the spatial covariance of the loadings and the latent factors. In contrast, when loadings are spatially constant, the spatial covariance of the outcomes is determined primarily by the latent factors, leading to uniform variation across the spatial domain. To assess these differences in practice, we apply a Bayesian hierarchical spatial dynamic factor model to analyze the opioid syndemic in North Carolina. Our results suggest that incorporating spatially varying loadings provides a more detailed, county-specific understanding of the epidemic. This added flexibility enables a localized interpretation of opioid-related interactions and offers insights that could inform targeted public health interventions.