  We propose a penalized likelihood method to fit the bivariate categorical
response regression model. Our method allows practitioners to estimate which
predictors are irrelevant, which predictors only affect the marginal
distributions of the bivariate response, and which predictors affect both the
marginal distributions and log odds ratios. To compute our estimator, we
propose an efficient first order algorithm which we extend to settings where
some subjects have only one response variable measured, i.e., the
semi-supervised setting. We derive an asymptotic error bound which illustrates
the performance of our estimator in high-dimensional settings. Generalizations
to the multivariate categorical response regression model are proposed.
Finally, simulation studies and an application in pan-cancer risk prediction
demonstrate the usefulness of our method in terms of interpretability and
prediction accuracy. An R package implementing the proposed method is available
for download at github.com/ajmolstad/BvCategorical.
