  Correlated binary response data with covariates are ubiquitous in
longitudinal or spatial studies. Among the existing statistical models the most
well-known one for this type of data is the multivariate probit model, which
uses a Gaussian link to model dependence at the latent level. However, a
symmetric link may not be appropriate if the data are highly imbalanced. Here,
we propose a multivariate skew-elliptical link model for correlated binary
responses, which includes the multivariate probit model as a special case.
Furthermore, we perform Bayesian inference for this new model and prove that
the regression coefficients have a closed-form unified skew-elliptical
posterior. The new methodology is illustrated by application to COVID-19
pandemic data from three different counties of the state of California, USA. By
jointly modeling extreme spikes in weekly new cases, our results show that the
spatial dependence cannot be neglected. Furthermore, the results also show that
the skewed latent structure of our proposed model improves the flexibility of
the multivariate probit model and provides better fit to our highly imbalanced
dataset.
