  Understanding the dependence structure between response variables is an
important component in the analysis of correlated multivariate data. This
article focuses on modeling dependence structures in multivariate binary data,
motivated by a study aiming to understand how patterns in different U.S.
senators' votes are determined by similarities (or lack thereof) in their
attributes, e.g., political parties and social network profiles. To address
such a research question, we propose a new Ising similarity regression model
which regresses pairwise interaction coefficients in the Ising model against a
set of similarity measures available/constructed from covariates. Model
selection approaches are further developed through regularizing the
pseudo-likelihood function with an adaptive lasso penalty to enable the
selection of relevant similarity measures. We establish estimation and
selection consistency of the proposed estimator under a general setting where
the number of similarity measures and responses tend to infinity. Simulation
study demonstrates the strong finite sample performance of the proposed
estimator in terms of parameter estimation and similarity selection. Applying
the Ising similarity regression model to a dataset of roll call voting records
of 100 U.S. senators, we are able to quantify how similarities in senators'
parties, businessman occupations and social network profiles drive their voting
associations.
