[Paper Review] DeepPAMM: Deep Piecewise Exponential Additive Mixed Models for Complex Hazard Structures in Survival Analysis

Philipp Kopper, Simon Wiegrebe+3 more|arXiv (Cornell University)|Feb 12, 2022

Statistical Methods and Inference4 citations

TL;DR

DeepPAMM is a deep learning framework that unifies piecewise exponential additive mixed models with neural networks to model complex hazard structures in survival analysis. It enables interpretable, high-dimensional feature learning while handling censored, truncated, competing risks, and correlated data, achieving competitive predictive performance with statistical rigor and flexibility for multimodal and time-varying features.

ABSTRACT

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.

Motivation & Objective

• To address the limitations of traditional survival models in handling high-dimensional, unstructured, or multimodal data while preserving interpretability.
• To extend piecewise exponential additive mixed models (PAMM) with deep neural networks for modeling complex, non-linear feature interactions and time-varying effects.
• To support diverse survival data types, including right-censored, left-truncated, competing risks, and recurrent events, within a unified statistical framework.
• To enable the modeling of correlated or repeated observations through random effects while maintaining predictive accuracy.
• To provide a flexible yet interpretable deep learning framework for survival analysis that retains the theoretical foundation of PAMMs.

Proposed method

• DeepPAMM uses a semi-structured additive predictor combining parametric and non-parametric components: a linear predictor with penalized splines for smooth time effects and random effects for clustering.
• It integrates a deep neural network as a non-parametric component to model high-dimensional feature interactions and unstructured data, such as images or tabular features.
• The model estimates the log-hazard function as a sum of structured effects (e.g., cyclic splines for time-of-day, tensor product splines for spatial coordinates) and an unstructured deep neural network component.
• Random effects (frailty terms) are incorporated to account for correlation in grouped or repeated observations, such as patients from different hospitals or fire response units.
• The framework supports multiple survival data types, including competing risks and multi-state processes, via a unified hazard parameterization.
• Model fitting is performed using existing software stacks (e.g., deepregression), enabling efficient optimization with regularization to prevent overfitting.

Experimental results

Research questions

• RQ1 Can a deep learning framework maintain interpretability while modeling complex, high-dimensional feature interactions in survival analysis?
• RQ2 How does the inclusion of a deep neural network component improve predictive performance over classical PAMM models in the presence of unstructured data?
• RQ3 To what extent can DeepPAMM model non-proportional hazards, time-varying effects, and competing risks while preserving statistical interpretability?
• RQ4 Does the addition of an unstructured deep component significantly enhance predictive performance beyond structured effects in real-world survival data?
• RQ5 Can DeepPAMM effectively handle multimodal data (e.g., tabular and spatial data) in a single coherent model?

Key findings

• DeepPAMM achieves competitive predictive performance, with a 16.7 IBS (Integrated Brier Score) at the median (Q50) on the London fire brigade dataset, outperforming the baseline KM and PAMM models.
• The addition of the unstructured deep component slightly improves predictive performance, but the structured effects alone are sufficient, as indicated by overlapping confidence intervals and low standard deviations.
• The model successfully captures non-linear, cyclic effects of time-of-day on emergency response times, with higher hazards (shorter response times) during daytime and lower hazards at night.
• Spatial effects show higher hazards in city peripheries, consistent with prior findings, demonstrating the model's ability to recover interpretable spatial patterns.
• DeepPAMM maintains interpretability through smooth, structured splines (e.g., cyclic and bivariate splines), while allowing complex interactions via the deep neural network component.
• The framework is capable of modeling complex survival data types, including competing risks, left-truncation, and correlated data, with a single unified model architecture.