Multivariate binary data are widely collected in many disciplines, including in finance, psychometrics, and ecology. Often, many binary responses are driven by only a subset of predictors, and groups of binary responses may exhibit similar effects to a predictor. Motivated by a survey containing presence-absence records of 22 demersal fish species recorded across the U.S. Northeast shelf, we propose a novel method for simultaneous coefficient clustering and variable selection in multivariate binary data using a penalized Ising regression model. The Ising regression model formulates an explicit joint distribution for a binary response vector via main and pairwise interaction coefficients, where the former is modeled as a function of covariates and the latter captures conditional dependence relationships between responses. To cluster coefficients within each covariate, and encourage sparsity across both covariates and pairwise interaction coefficients, we propose to augment the Ising regression model with adaptive fused lasso and adaptive lasso penalties. Such a structured penalty to encourage simultaneous sparsity and groupings is aligned with goals of achieving sparse species-covariate relationships, and homogeneity of environmental responses across species, in the motivating demersal fish survey. Through a reparametrization, we show that the proposed estimator can be efficiently obtained by fitting a single, adaptive lasso logistic regression model. Simulation studies and an application to the demersal fish survey demonstrate competitive performance of the proposed method relative to several existing Ising regression models for multivariate binary data, and leads to an interpretable and parsimonious set of response-covariate relationships.
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