A Generalization of the Spatial Binary Model to the Longitudinal Spatial Setup

When spatial data are repeatedly collected over a short period of time, they exhibit two-way correlations. More specifically at a given point of time the neighboring responses from a spatial family become pair-wise spatially correlated, and over time each of these familial responses become longitudinally correlated, leading to a two-way multivariate spatial longitudinal correlation model. Recently, this type of two-way multivariate correlation model was studied by Sutradhar (2021, Sankhya A, 83, 206-244) for linear longitudinal spatial data. However, as the binary correlations can not be obtained directly from linear correlations, the construction of spatial longitudinal model for binary data requires special efforts mainly for combining the two structures, spatial and longitudinal, those are themselves complex for binary data. In this paper we resolve this challenge by using first a conditional (on spatial family-based random effects) binary dynamic logits (CBDL) model for longitudinal binary data, and then integrating out the family-based random effects over their distributional properties those used originally to generate spatial correlations. For inferences for the proposed model, we develop the so-called generalized quasi-likelihood (GQL) and method of moments approaches. Asymptotic properties such as consistency of the estimators of all parameters and asymptotic normality for the GQL estimators of main regression parameters are discussed in details.