Bridging conditional and marginal inference for spatially referenced binary data

Spatially referenced binary data are common in epidemiology and public health. Owing to its elegant log-odds interpretation of the regression coefficients, a natural model for these data is logistic regression. To account for missing confounding variables that might exhibit a spatial pattern (say, socioeconomic, biological, or environmental conditions), it is customary to include a Gaussian spatial random effect. Conditioned on the spatial random effect, the coefficients may be interpreted as log odds ratios. However, marginally over the random effects, the coefficients no longer preserve the log-odds interpretation, and the estimates are hard to interpret and generalize to other spatial regions. To resolve this issue, we propose a new spatial random effect distribution through a copula framework which ensures that the regression coefficients maintain the log-odds interpretation both conditional on and marginally over the spatial random effects. We present simulations to assess the robustness of our approach to various random effects, and apply it to an interesting dataset assessing periodontal health of Gullah-speaking African Americans. The proposed methodology is flexible enough to handle areal or geo-statistical datasets, and hierarchical models with multiple random intercepts.