A Penalized Likelihood-Based Approach For Multivariate Categorical Response Regression

In the era of big data,data types are diverse,and more and more fields need to analyze the effects of explanatory variables on multivariate response variables.For the multivariate categorical response problem,in binary terms,most scholars map the multivariate categorical response to the univariate categorical response (4?,and subseuuently do polynomial logistic regression on the univariate categorical response,and their proposed methods can predict well and have interpretability in polynomial logistic regression.However,when the data dimensionality is high,there is also dependence between response variables,and these methods above do not explain (4? is constructed using two different response variables,because these methods ignore the association between two responses.Therefore,in 2021,Molstad and Rothman developed a new penalized likelihood framework to fit regression models for multiple categorical responses,which takes into account the association between two responses and obtains a low prediction misspecification rate with interpretability.In this paper,we propose a regression method for multivariate categorical responses that takes into account the penalized likelihood of the margins,based on Molstad and Rothman,by adding a penalty for the marginal effects of the two responses.The method proposed in this paper not only estimates which predictors are uncorrelated and which predictors simultaneously affect the interaction effects of the binary response variables,but also estimates which predictors affect only the marginal effects of the binary response variables and,on this basis,estimates which response specifically is affected by the predictors’ marginal effects.In this paper,we apply an accelerated proximal gradient descent algorithm to compute our estimates and also further extend our method to the case of arbitrary multivariate categorical responses.Simulation studies show that our proposed method is feasible,and in this paper we compare our method with Molstad and Rothman’s method under different models,and the results show that our method is comparable to their method in terms of prediction misspecification in different models,and in particular,our method has a lower marginal misspecification rate,while being able to estimate more accurately which predictors affect only the In particular,our method has a lower marginal misspecification rate and is able to estimate more accurately which predictors affect only the marginal effects of the binary response variables and,on this basis,to estimate the marginal effects of the predictors on which responses specifically.Further,the proposed method is also applied to breast cancer data analysis,and the results show that the joint misspecification and marginal misspecification rates of the proposed method for predicting binary response variables are lower than those of existing methods.