  This study proposes an interpretable neural network-based non-proportional
odds model (N$^3$POM) for ordinal regression. N$^3$POM is different from
conventional approaches to ordinal regression with non-proportional models in
several ways: (1) N$^3$POM is defined for both continuous and discrete
responses, whereas standard methods typically treat the ordered continuous
variables as if they are discrete, (2) instead of estimating response-dependent
finite-dimensional coefficients of linear models from discrete responses as is
done in conventional approaches, we train a non-linear neural network to serve
as a coefficient function. Thanks to the neural network, N$^3$POM offers
flexibility while preserving the interpretability of conventional ordinal
regression. We establish a sufficient condition under which the predicted
conditional cumulative probability locally satisfies the monotonicity
constraint over a user-specified region in the covariate space. Additionally,
we provide a monotonicity-preserving stochastic (MPS) algorithm for effectively
training the neural network. We apply N$^3$POM to several real-world datasets.
