This paper adapts sparse factor models for exploring covariation in multivariate binary data, with an application to measuring latent factors in U.S. Congressional roll-call voting patterns. We focus on the advantages of using formal probability models for inference in this context, drawing parallels with the seminal findings of Poole and Rosenthal (1991). Our methodological innovation is to introduce a sparsity prior on a latent covariance matrix that descibes common factors in binary and ordinal outcomes. We apply the method to analyze sixty years of roll-call votes from the United States Senate, focusing primarily on the interpretation of posterior summaries that arise from the model.

We also explore two advantages of our approach over traditional factor analysis. First, patterns of sparsity in the factor-loadings matrix often have natural subject-matter interpretations. For the roll-call vote data, the sparsity prior enables one to conduct a formal hypothesis test about whether a given vote can be explained exclusively by partisanship. Moreover, the factor scores provide a novel way of ranking Senators by the partisanship of their voting patterns. Second, by introducing sparsity into existing factor-analytic probit models, we effect a favorable bias-variance tradeoff in estimating the latent covariance matrix. Our model can thus be used in situations where the number of variables is very large relative to the number of observations.

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