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The Concordance Index decomposition: A measure for a deeper understanding of survival prediction models

Abdallah Alabdallah, Mattias Ohlsson, Sepideh Pashami, Thorsteinn Rögnvaldsson

TL;DR

This work introduces a decomposition of the Concordance Index into a weighted harmonic combination of $CI_{ee}$ and $CI_{ec}$ with weight $\alpha$, enabling finer assessment of a survival model's ranking of event-event versus event-censored pairs. It also presents SurVED, a continuous-time Survival Variational Encoder-Decoder, and demonstrates its competitive performance against state-of-the-art methods across four public datasets with varying censoring. The decomposition reveals where models differ in ranking strength—whether in $CI_{ee}$ or $CI_{ec}$—which is often hidden in the aggregate $CI$ metric. Overall, the study shows that deep learning models can better leverage observed events to maintain stable performance under censoring, and the decomposition offers a principled guide for developing improved survival models.

Abstract

The Concordance Index (C-index) is a commonly used metric in Survival Analysis for evaluating the performance of a prediction model. In this paper, we propose a decomposition of the C-index into a weighted harmonic mean of two quantities: one for ranking observed events versus other observed events, and the other for ranking observed events versus censored cases. This decomposition enables a finer-grained analysis of the relative strengths and weaknesses between different survival prediction methods. The usefulness of this decomposition is demonstrated through benchmark comparisons against classical models and state-of-the-art methods, together with the new variational generative neural-network-based method (SurVED) proposed in this paper. The performance of the models is assessed using four publicly available datasets with varying levels of censoring. Using the C-index decomposition and synthetic censoring, the analysis shows that deep learning models utilize the observed events more effectively than other models. This allows them to keep a stable C-index in different censoring levels. In contrast to such deep learning methods, classical machine learning models deteriorate when the censoring level decreases due to their inability to improve on ranking the events versus other events.

The Concordance Index decomposition: A measure for a deeper understanding of survival prediction models

TL;DR

This work introduces a decomposition of the Concordance Index into a weighted harmonic combination of and with weight , enabling finer assessment of a survival model's ranking of event-event versus event-censored pairs. It also presents SurVED, a continuous-time Survival Variational Encoder-Decoder, and demonstrates its competitive performance against state-of-the-art methods across four public datasets with varying censoring. The decomposition reveals where models differ in ranking strength—whether in or —which is often hidden in the aggregate metric. Overall, the study shows that deep learning models can better leverage observed events to maintain stable performance under censoring, and the decomposition offers a principled guide for developing improved survival models.

Abstract

The Concordance Index (C-index) is a commonly used metric in Survival Analysis for evaluating the performance of a prediction model. In this paper, we propose a decomposition of the C-index into a weighted harmonic mean of two quantities: one for ranking observed events versus other observed events, and the other for ranking observed events versus censored cases. This decomposition enables a finer-grained analysis of the relative strengths and weaknesses between different survival prediction methods. The usefulness of this decomposition is demonstrated through benchmark comparisons against classical models and state-of-the-art methods, together with the new variational generative neural-network-based method (SurVED) proposed in this paper. The performance of the models is assessed using four publicly available datasets with varying levels of censoring. Using the C-index decomposition and synthetic censoring, the analysis shows that deep learning models utilize the observed events more effectively than other models. This allows them to keep a stable C-index in different censoring levels. In contrast to such deep learning methods, classical machine learning models deteriorate when the censoring level decreases due to their inability to improve on ranking the events versus other events.
Paper Structure (13 sections, 11 equations, 4 figures, 5 tables)

This paper contains 13 sections, 11 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Method
  3. The Concordance Index Decomposition
  4. SurVED: Survival Variational Encoder-Decoder
  5. The Objective Function
  6. Description of Datasets
  7. Experimental Settings
  8. Results and Discussion
  9. Comparison on the four data sets
  10. The Effect of Censoring and Size
  11. Conclusion
  12. Declarations of interest
  13. Acknowledgement

Figures (4)

  • Figure 1: The results of $CI$, $\alpha$, $CI_{ee}$, $CI_{ec}$, and $|\alpha\text{-Deviation}|$ in eq. (\ref{['eq:CI_ee_ec']}) of the SurVED, DATE, and VSI models on the four datasets (Censoring level decreases from the highest censoring (NWTCO) to the lowest censoring (SUPPORT)).
  • Figure 2: The Win/Lose/Draw comparison based on $CI$, $CI_{ee}$, $CI_{ec}$, and $\alpha$-Deviation in eq. (\ref{['eq:CI_ee_ec']}) of the compared models on the four datasets.
  • Figure 3: The change of $CI$ as the size of the dataset and the ratio of events change. The x-axis shows the sizes of the datasets and percentages of the events (for the SUPPORT dataset) in the three experiments.
  • Figure 4: The change of $CI$, $CI_{ee}$, $CI_{ec}$, and $\alpha$ in eq. (\ref{['eq:CI_ee_ec']}) as the ratio of events changes. The x-axis shows different percentages of events (for the SUPPORT dataset).