  In this paper, we deduce a new multivariate regression model designed to fit
correlated binary data. The multivariate distribution is derived from a
Bernoulli mixed model with a nonnormal random intercept on the marginal
approach. The random effect distribution is assumed to be the generalized
log-gamma (GLG) distribution by considering a particular parameter setting. The
complement log-log function is specified to lead to strong conjugacy between
the response variable and random effect. The new discrete multivariate
distribution, named MBerGLG distribution, has location and dispersion
parameters. The MBerGLG distribution leads to the MBerGLG regression (MBerGLGR)
model, providing an alternative approach to fitting both unbalanced and
balanced correlated response binary data. Monte Carlo simulation studies show
that its maximum likelihood estimators are unbiased, efficient, and consistent
asymptotically. The randomized quantile residuals are performed to identify
possible departures from the proposal model and the data and detect atypical
subjects. Finally, two applications are presented in the data analysis section.
