A General Machine Learning Framework for Survival Analysis

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# A General Machine Learning <br> Framework for Survival Analysis
## ECML PKDD 2020
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### <strong>Andreas Bender (<i class='fa fa-twitter'><span class="citation">@adibender</span></i>)</strong>, <br>David Rügamer, Fabian Scheipl, Bernd Bischl
### <br>Department of Statistics, LMU Munich

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### The framework is *general* in the sense that

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1. It supports different Survival Tasks
  - right-censoring, left-truncation
  - time-varying effects, time-varying features
  - competing risks, multi-state models

2. Does not require specialized Software, can be applied across programming languages and using any algorithm that supports optimization of the Poisson Likelihood

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???

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<h1 style="color:#005500;">Survival Task as Poisson Task</h1>

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Consider setting with right-censored data:
  - we observe `$(t_i, \delta_i), i = 1,\ldots, n$`, where
    - `$t_i = \min(T_i, C_i)$`; `$T_i \sim F \perp C_i \sim G; T_i,C_i > 0$`
    - `$\delta_i = I(T_i \leq C_i) \in \{0,1\}$`

To approximate
`$$\lambda(t; \bfx_i) = \exp(g(\bfx_i(t), t)) \stackrel{PH}{=}\lambda_0(t)\exp(\bfx_i'\bsbeta)$$`

- split the follow-up in `$J$` intervals `$(\kappa_{j-1}, \kappa_j], j = 1,\ldots, J$`

- assume piece-wise constant hazards:
`$$\begin{align}
\lambda(t| \bfx_i(t))
  & \equiv \exp(g(\bfx_{ij}, t_j)):=\lambda_{ij},\ \ \forall t \in (\kappa_{j-1}, \kappa_j],\\
\end{align}$$`

- Estimation using Piece-wise Exponential Model (e.g. [Friedman (1982)](http://www.jstor.org/stable/2240502)) <br>
( `$\ra$` Poisson regression with transformed data)

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