  We consider modeling, inference, and computation for analyzing multivariate
binary data. We propose a new model that consists of a low dimensional latent
variable component and a sparse graphical component. Our study is motivated by
analysis of item response data in cognitive assessment and has applications to
many disciplines where item response data are collected. Standard approaches to
item response data in cognitive assessment adopt the multidimensional item
response theory (IRT) models. However, human cognition is typically a
complicated process and thus may not be adequately described by just a few
factors. Consequently, a low-dimensional latent factor model, such as the
multidimensional IRT models, is often insufficient to capture the structure of
the data. The proposed model adds a sparse graphical component that captures
the remaining ad hoc dependence. It reduces to a multidimensional IRT model
when the graphical component becomes degenerate. Model selection and parameter
estimation are carried out simultaneously through construction of a
pseudo-likelihood function and properly chosen penalty terms. The convexity of
the pseudo-likelihood function allows us to develop an efficient algorithm,
while the penalty terms generate a low-dimensional latent component and a
sparse graphical structure. Desirable theoretical properties are established
under suitable regularity conditions. The method is applied to the revised
Eysenck's personality questionnaire, revealing its usefulness in item analysis.
Simulation results are reported that show the new method works well in
practical situations.
