An interpretable neural network-based non-proportional odds model for ordinal regression with continuous response

This paper proposes an interpretable neural network-based non-proportional odds model (N$^3$POM) for ordinal regression, where the response variable can take not only discrete but also continuous values, and the regression coefficients vary depending on the predicting ordinal response. In contrast to conventional approaches estimating the linear coefficients of regression directly from the discrete response, we train a non-linear neural network that outputs the linear coefficients by taking the response as its input. By virtue of the neural network, N$^3$POM may have flexibility while preserving the interpretability of the conventional ordinal regression. We show a sufficient condition so that the predicted conditional cumulative probability~(CCP) satisfies the monotonicity constraint locally over a user-specified region in the covariate space; we also provide a monotonicity-preserving stochastic (MPS) algorithm for training the neural network adequately.