Philipp Kopper

Raphael Sonabend, John Zobolas, Riccardo Be Bin et al.

Scoring rules promote rational and honest decision-making, which is important for model evaluation and becoming increasingly important for automated procedures such as `AutoML'. In this paper we survey common squared and logarithmic scoring rules for survival analysis, with a focus on their theoretical and empirical properness. We introduce a marginal definition of properness and show that both the Integrated Survival Brier Score (ISBS) and the Right-Censored Log-Likelihood (RCLL) are theoretically improper under this definition. We also investigate a new class of losses that may inform future survival scoring rules. Simulation experiments reveal that both the ISBS and RCLL behave as proper scoring rules in practice. The RCLL showed no violations across all settings, while ISBS exhibited only minor, negligible violations at extremely small sample sizes, suggesting one can trust results from historical experiments. As such we advocate for both the RCLL and ISBS in external validation of models, including in automated procedures. However, we note practical challenges in estimating these losses including estimation of censoring distributions and densities; as such further research is required to advance development of robust and honest evaluation in survival analysis.

Simon Wiegrebe, Philipp Kopper, Raphael Sonabend et al.

The influx of deep learning (DL) techniques into the field of survival analysis in recent years has led to substantial methodological progress; for instance, learning from unstructured or high-dimensional data such as images, text or omics data. In this work, we conduct a comprehensive systematic review of DL-based methods for time-to-event analysis, characterizing them according to both survival- and DL-related attributes. In summary, the reviewed methods often address only a small subset of tasks relevant to time-to-event data - e.g., single-risk right-censored data - and neglect to incorporate more complex settings. Our findings are summarized in an editable, open-source, interactive table: https://survival-org.github.io/DL4Survival. As this research area is advancing rapidly, we encourage community contribution in order to keep this database up to date.

Lucas Kook, Chris Kolb, Philipp Schiele et al.

Neural network representations of simple models, such as linear regression, are being studied increasingly to better understand the underlying principles of deep learning algorithms. However, neural representations of distributional regression models, such as the Cox model, have received little attention so far. We close this gap by proposing a framework for distributional regression using inverse flow transformations (DRIFT), which includes neural representations of the aforementioned models. We empirically demonstrate that the neural representations of models in DRIFT can serve as a substitute for their classical statistical counterparts in several applications involving continuous, ordered, time-series, and survival outcomes. We confirm that models in DRIFT empirically match the performance of several statistical methods in terms of estimation of partial effects, prediction, and aleatoric uncertainty quantification. DRIFT covers both interpretable statistical models and flexible neural networks opening up new avenues in both statistical modeling and deep learning.

Philipp Kopper, David Rügamer, Raphael Sonabend et al.

Scoring rules are an established way of comparing predictive performances across model classes. In the context of survival analysis, they require adaptation in order to accommodate censoring. This work investigates using scoring rules for model training rather than evaluation. Doing so, we establish a general framework for training survival models that is model agnostic and can learn event time distributions parametrically or non-parametrically. In addition, our framework is not restricted to any specific scoring rule. While we focus on neural network-based implementations, we also provide proof-of-concept implementations using gradient boosting, generalized additive models, and trees. Empirical comparisons on synthetic and real-world data indicate that scoring rules can be successfully incorporated into model training and yield competitive predictive performance with established time-to-event models.

Philipp Kopper, Simon Wiegrebe, Bernd Bischl et al.

Survival analysis (SA) is an active field of research that is concerned with time-to-event outcomes and is prevalent in many domains, particularly biomedical applications. Despite its importance, SA remains challenging due to small-scale data sets and complex outcome distributions, concealed by truncation and censoring processes. The piecewise exponential additive mixed model (PAMM) is a model class addressing many of these challenges, yet PAMMs are not applicable in high-dimensional feature settings or in the case of unstructured or multimodal data. We unify existing approaches by proposing DeepPAMM, a versatile deep learning framework that is well-founded from a statistical point of view, yet with enough flexibility for modeling complex hazard structures. We illustrate that DeepPAMM is competitive with other machine learning approaches with respect to predictive performance while maintaining interpretability through benchmark experiments and an extended case study.

David Rügamer, Chris Kolb, Cornelius Fritz et al.

In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library \pkg{TensorFlow} for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as \pkg{mgcv}. The packages' modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models.

Philipp Kopper, Sebastian Pölsterl, Christian Wachinger et al.

We propose a versatile framework for survival analysis that combines advanced concepts from statistics with deep learning. The presented framework is based on piecewise exponential models and thereby supports various survival tasks, such as competing risks and multi-state modeling, and further allows for estimation of time-varying effects and time-varying features. To also include multiple data sources and higher-order interaction effects into the model, we embed the model class in a neural network and thereby enable the simultaneous estimation of both inherently interpretable structured regression inputs as well as deep neural network components which can potentially process additional unstructured data sources. A proof of concept is provided by using the framework to predict Alzheimer's disease progression based on tabular and 3D point cloud data and applying it to synthetic data.