Ordinal Neural Network Transformation Models: Deep and interpretable regression models for ordinal outcomes.

Outcomes with a natural order, such as quality of life scores or movie ratings, commonly occur in prediction tasks. The available input data are often a mixture of complex inputs like images and tabular predictors. Deep Learning (DL) methods have shown outstanding performances on perceptual tasks. Yet, most DL applications treat ordered outcomes as unordered classes and lack interpretability of individual predictors. In contrast, traditional ordinal regression models are specific for ordered outcomes and enable to interpret predictor effects but are limited to tabular input data. Here, we present the highly modular class of ordinal neural network transformation models (ONTRAMs) which can include both tabular and complex data using multiple neural networks. All neural networks are jointly trained to optimize the likelihood, which is parametrized to take the outcome's natural order into account. We recapitulate statistical ordinal regression models and discuss how they can be understood as transformation models. Transformation models use a parametric transformation function and a simple distribution, the former of which determines the flexibility and interpretability of the individual model components. We demonstrate how to set up interpretable ONTRAMs with tabular and/or image data. We show that the most flexible ONTRAMs achieve on-par performance with existing DL approaches while outperforming them in training speed. We highlight that ONTRAMs with image and tabular predictors yield correct effect estimates while keeping the high prediction performance of DL methods. We showcase how to interpret individual components of ONTRAMs and discuss the case where the included tabular predictors are correlated with the image data. In this work, we demonstrate how to join the benefits of DL and statistical regression methods to create efficient and interpretable models for ordinal outcomes.