From 7e977a808c782dfb172e0c2c2bf4cead4f075428 Mon Sep 17 00:00:00 2001 From: Raphael Sonabend Date: Fri, 29 Aug 2025 15:47:18 +0100 Subject: [PATCH 1/6] add notes about other censoring Signed-off-by: Raphael Sonabend --- book/P3C15_svm.qmd | 9 ++++++--- book/library.bib | 36 ++++++++++++++++++++++++++++++++++++ 2 files changed, 42 insertions(+), 3 deletions(-) diff --git a/book/P3C15_svm.qmd b/book/P3C15_svm.qmd index 4e57420c..e6bbd648 100644 --- a/book/P3C15_svm.qmd +++ b/book/P3C15_svm.qmd @@ -14,6 +14,9 @@ Support vector machines are a popular class of models in regression and classifi Survival support vector machines (SSVMs) predict continuous responses that can be used as ranking predictions with some formulations that provide survival time interpretations. This chapter starts with SVMs in the regression setting before moving to adaptions for survival analysis. +As of the time of publication, no SSVMs for competing risks appear to have been published [@Kantidakis2023; @Monterrubio-Gómez2024; @Djangang2025]. +Theoretically, one could use any of the methods to estimate per-cause risk by considering each risk separately and censoring observations that experience a different risk, however this has not been validated in the literature. + ## SVMs for Regression In simple linear regression, the aim is to estimate the line $y = \alpha + x\beta_1$ by estimating the $\alpha,\beta_1$ coefficients. @@ -132,7 +135,8 @@ $$ If no observations are censored then the optimization becomes the regression optimization in (@eq-svm-opt). Note that in SSVMs, the $\epsilon$ parameters are typically removed to better accommodate censoring and to help prevent the same penalization of over- and under-predictions. In contrast to this formulation, one *could* introduce more $\epsilon$ and $C$ parameters to separate between under- and over-predictions and to separate right- and left-censoring, however this leads to eight tunable hyperparameters, which is inefficient and may increase overfitting [@pkgsurvivalsvm; @Land2011]. -The algorithm can be simplified to right-censoring only by removing the second constraint completely for anyone censored: + +If only right-censoring is present in the data then the algorithm can be simplified by removing the second constraint completely for anyone censored: $$ \begin{aligned} @@ -180,10 +184,8 @@ This is visualized in @fig-svm-surv-redux where six observations are sorted by o Starting from right to left, the first pair is made by matching the observation to the first uncensored outcome to the left, this continues to the end. In order for all observations to be used in the optimisation, the algorithm sets the first outcome to be uncensored hence observation $2$ being compared to observation $1$. - ![Van Belle SVM nearest neighbors reduction. Sorted observations are paired with the nearest uncensored outcome 'to the left'. Red squares are uncensored observations and blue circles are censored. The observation with the smallest outcome time is always treated as uncensored.](Figures/svm/comparable.png){#fig-svm-surv-redux fig-alt="x-axis says 'observation', y-axis says 'outcome time'. There are six observations that increase linearly from bottom-left to top-right. The order is: uncensored, censored, uncensored, censored, uncensored, uncensored. Arrows show observation 6 matched with 4, 5 matched with 5, 4 matched with 2, 3 matched with 2, 2 matched with 1." width=500} - Using this reduction, the algorithm becomes $$ @@ -208,6 +210,7 @@ $$ Where $\mu_i$ are again Lagrange multipliers. +There do not appear to any adaptations to the ranking SSVM for other censoring or truncation types. ### Hybrid SSVMs diff --git a/book/library.bib b/book/library.bib index e5f727cf..2a86cab0 100644 --- a/book/library.bib +++ b/book/library.bib @@ -7523,3 +7523,39 @@ @inproceedings{Chapfuwa2018 volume = {80}, } +@article{Monterrubio-Gómez2024, +author = {Monterrubio-Gómez, Karla and Constantine-Cooke, Nathan and Vallejos, Catalina A.}, +title = {A review on statistical and machine learning competing risks methods}, +journal = {Biometrical Journal}, +volume = {66}, +number = {2}, +pages = {2300060}, +keywords = {competing risks, risk prediction, survival analysis, time-to-event data}, +doi = {https://doi.org/10.1002/bimj.202300060}, +url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/bimj.202300060}, +eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/bimj.202300060}, +year = {2024} +} + +@misc{Djangang2025, + title={Comparative Review of Modern Competing Risk Methods in High-dimensional Settings}, + author={Paul M. Djangang and Summer S. Han and Nilotpal Sanyal}, + year={2025}, + eprint={2503.12824}, + archivePrefix={arXiv}, + primaryClass={stat.ME}, + url={https://arxiv.org/abs/2503.12824}, +} + +@article{Kantidakis2023, + author = {Georgios Kantidakis and Hein Putter and Saskia Litière and Marta Fiocco}, + doi = {10.1186/s12874-023-01866-z}, + issn = {1471-2288}, + issue = {1}, + journal = {BMC Medical Research Methodology}, + pages = {51}, + title = {Statistical models versus machine learning for competing risks: development and validation of prognostic models}, + volume = {23}, + url = {https://doi.org/10.1186/s12874-023-01866-z}, + year = {2023} +} From f6c609513227cb48b27c3c768215d3c5563bc9f7 Mon Sep 17 00:00:00 2001 From: RaphaelS1 Date: Fri, 29 Aug 2025 14:49:50 +0000 Subject: [PATCH 2/6] Push pdf [skip ci] --- .../Machine-Learning-in-Survival-Analysis.pdf | Bin 6109629 -> 6408287 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/book/_book/Machine-Learning-in-Survival-Analysis.pdf 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