A Unified Framework for Modeling Feedback and Endogeneity in Longitudinal Binary Outcomes Using Bayesian Methods

Longitudinal studies with binary outcomes often involve time-dependent covariates that evolve alongside outcomes, introducing feedback effects and endogeneity. Standard approaches, such as Generalized Estimating Equations (GEE) and Generalized Linear Mixed Models (GLMM), typically assume covariate exogeneity, leading to biased and inconsistent estimates when feedback is present. We propose a unified three-step hierarchical Bayesian framework to address these challenges. First, the Generalized Method of Moments (GMM) identifies valid instruments to correct for endogeneity. Second, a Bayesian hierarchical logistic regression model estimates outcome probabilities while accounting for within-subject correlation. Third, a reversal model explicitly captures feedback by modeling time-dependent covariates as outcomes influenced by prior responses. Through simulations, we demonstrate that the proposed framework substantially reduces bias, lowers root mean squared error (RMSE), and improves uncertainty quantification compared to traditional methods, especially under moderate to strong feedback. We illustrate the practical utility of the approach using a synthetic longitudinal diabetes dataset, where feedback between prior glucose levels and self-monitoring behavior meaningfully alters inference. This integrated framework provides a flexible and interpretable solution for longitudinal binary data analysis, particularly in clinical, behavioral, and public health research.