A hierarchical model for binary data with dependence between the design and outcome success probabilities

Statistical theory requires that analysis of study outcomes be conducted conditional on the design process. Ignoring this process may result in severely biased estimates, leading to false inferences, especially when the outcome variable is associated with design variables. We propose in this paper a class of hierarchical models to investigate the dependence between the design process and the study outcomes of primary interest. We discuss a fully parametric and a semi-parametric formulation of the hypothesized model and propose the EM algorithm to obtain maximum likelihood estimates. Our numerical results show that the semi-parametric approach outperforms the fully parametric model with respect to some key features of the model. The methodology is used to gain insight into the mechanism that generates breast cancer literacy outcomes in a study conducted among medically underserved females in Michigan.