# Bayesian Factor Analysis for Inference on Interactions

## TL;DR

This paper introduces a Bayesian latent factor model, FIN, for inferring complex interactions among correlated chemical exposures affecting health, with theoretical guarantees and practical evaluation.

## Contribution

It develops a novel Bayesian factor analysis framework that captures interactions and main effects in correlated predictors and responses, extending to higher order interactions.

## Key findings

- The model achieves accurate inference in simulations.
- Application to NHANES data reveals meaningful chemical interactions.
- Posterior consistency is established under certain conditions.

## Abstract

This article is motivated by the problem of inference on interactions among chemical exposures impacting human health outcomes. Chemicals often co-occur in the environment or in synthetic mixtures and as a result exposure levels can be highly correlated. We propose a latent factor joint model, which includes shared factors in both the predictor and response components while assuming conditional independence. By including a quadratic regression in the latent variables in the response component, we induce flexible dimension reduction in characterizing main effects and interactions. We propose a Bayesian approach to inference under this Factor analysis for INteractions (FIN) framework. Through appropriate modifications of the factor modeling structure, FIN can accommodate higher order interactions and multivariate outcomes. We provide theory on posterior consistency and the impact of misspecifying the number of factors. We evaluate the performance using a simulation study and data from the National Health and Nutrition Examination Survey (NHANES). Code is available on GitHub.