The framework is general in the sense that

Consider setting with right-censored data:

• we observe $(t_i,{\delta }_i),i=1,\dots ,n$, where
  • $t_i=min(T_i,C_i)$; $T_i\sim F\perp C_i\sim G;T_i,C_i>0$
  • ${\delta }_i=I(T_i\le C_i)\in \{0,1\}$

To approximate $$\lambda (t;x_i)=\exp \left(g\right(x_i\left(t\right),t\left)\right)\stackrel{PH}{=}{\lambda }_0\left(t\right)\exp \left(x_i^{\prime }\beta \right)$$

• split the follow-up in $J$ intervals $({\kappa }_{j-1},{\kappa }_j],j=1,\dots ,J$

• assume piece-wise constant hazards: $$\begin{array}{cc}\lambda \left(t|x_i\right(t\left)\right)& \equiv \exp \left(g\right(x_{ij},t_j\left)\right):={\lambda }_{ij},\forall t\in ({\kappa }_{j-1},{\kappa }_j],\end{array}$$

• Estimation using Piece-wise Exponential Model (e.g. Friedman (1982))
  ( $\to$ Poisson regression with transformed data)

$\to$ transform to PED using ${\kappa }_0=0,{\kappa }_1=1,{\kappa }_2=1.5,{\kappa }_3=3$

• define: ${\delta }_{ij}=\{\begin{array}{cc}1& t_i\in ({\kappa }_{j-1},{\kappa }_j]\wedge {\delta }_i=1\\ 0& \text{else}\end{array}$, $t_{ij}=\{\begin{array}{cc}{t}_i-{\kappa }_{j-1}& {\delta }_{ij}=1\\ {\kappa }_j-{\kappa }_{j-1}& \text{else}\end{array}$, $t_j:={\kappa }_j$

Consider 3 subjects in competing risks setting with event types $k\in \{1,2\}$

• $i=1$: $(t_1=1.3,{\delta }_1=2)$
• $i=2$: $(t_2=0.5,{\delta }_2=0)$
• $i=3$: $(t_3=2.7,{\delta }_3=1)$

$\to$ estimate $\lambda (t|x,k)=\exp \left(f\right(x\left(t\right),t,k\left)\right),k\in \{1,2\}$

Time-varying effects Shared vs. cause-specific effects (in CR)

Comparison with ORSF (single-event, right-censoring)
Evaluation w.r.t. Integrated Brier Score

Comparison with DeepHit (single-event and competing risks, right-censoring)
Evaluation w.r.t. weighted Brier Score

Choice of interval split points

• Number and placement of interval split points could potentially be a tuning parameter

• In our experience setting split points at observed event times results in good performance $\to$ many split points where many events observed

• For large data sets select subset of unique event times for split points

References

Bender, A, D. Rügamer, F. Scheipl, et al. (2020). "A General Machine Learning Framework for Survival Analysis". In: arXiv:2006.15442 [cs, stat]. arXiv: 2006.15442.

Chen, T. and C. Guestrin (2016). "XGBoost: A Scalable Tree Boosting System". In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD '16, pp. 785-794. DOI: 10.1145/2939672.2939785. arXiv: 1603.02754.

Friedman, M. (1982). "Piecewise Exponential Models for Survival Data with Covariates". In: The Annals of Statistics 10.1, pp. 101-113. ISSN: 00905364. URL: http://www.jstor.org/stable/2240502.

Jaeger, B. C, D. L. Long, D. M. Long, et al. (2019). "Oblique random survival forests". In: The Annals of Applied Statistics 13.3, pp. 1847-1883. ISSN: 1932-6157, 1941-7330. DOI: 10.1214/19-AOAS1261.

Lee, C., W. R. Zame, J. Yoon, et al. (2018). "DeepHit: A Deep Learning Approach to Survival Analysis With Competing Risks". In: Thirty-Second AAAI Conference on Artificial Intelligence. Thirty-Second AAAI Conference on Artificial Intelligence. URL: https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16160.