SVM competing risks

book/P1C6_survtsk.qmd

@@ -254,7 +254,7 @@ These are composed as either:
 1. $\hatbh(t)\exp(\hat{\eta})$ if proportional hazards is assumed; or
 2. $\hatbh(\exp(-\hat{\eta})t)\exp(-\hat{\eta})$ for accelerated failure time models.
 
-## Beyond single-event
+## Beyond single-event {#sec-survtsk-beyond}
 
 In @sec-eha, competing risks and multi-state models were introduced with a focus on estimating probability distributions.
 

book/P3C15_svm.qmd

@@ -132,7 +132,8 @@ $$
 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 +181,8 @@ 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
 
 $$
@@ -208,6 +207,7 @@ $$
 
 Where $\mu_i$ are again Lagrange multipliers.
 
+There do not appear to be any adaptations to the ranking SSVM for other censoring or truncation types.
 
 ### Hybrid SSVMs
 
@@ -241,6 +241,13 @@ $$
 
 where $\mu_i, \mu_i^*, \mu_i'$ are Lagrange multipliers and $K$ is a chosen kernel function, which may have further hyper-parameters to select or tune.
 
+### Competing risks
+
+As of the time of publication, no SSVMs for competing risks appear to have been published [@Kantidakis2023; @Monterrubio-Gómez2024; @Djangang2025].
+As discussed in @sec-survtsk-beyond, there is not a straightforward concept of time-to-event competing risks predictions so survival time SSVMs are unlikely to be extended to competing risks.
+For the ranking SSVMs, 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.
+Moreover, the risks predicted by SSVMs correspond to an abstract relative risk and not an interpretable hazard function, meaning there is no method to transform these predictions to CIFs or other common competing risks statistics.
+
 ## Conclusion
 
 :::: {.callout-warning icon=false}