Piece-wise exponential (Additive Mixed) Modeling Tools

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# Piece-wise exponential (Additive Mixed) Modeling Tools
## ISCB41, 2020
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### <br><br><strong>Andreas Bender (<i class='fa fa-twitter'><span class="citation">@adibender</span></i>)</strong>, <br> Fabian Scheipl, David Rügamer, Philipp Kopper, Bernd Bischl, Helmut Küchenhoff
### <br><br>Department of Statistics, LMU Munich

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<div class="my-footer"><span><i class="fa fa-twitter">@adibender</i>
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???

`$$\usepackage{amsmath,amssymb,bm}
\newcommand{\ra}{\rightarrow}
\newcommand{\bs}[1]{\boldsymbol{#1}}
\newcommand{\tn}[1]{\textnormal{#1}}
\newcommand{\mbf}[1]{\mathbf{#1}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\bfx}{\mathbf{x}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfB}{\mathbf{B}}
\newcommand{\bff}{\mathbf{f}}
\newcommand{\bsbeta}{\boldsymbol{\beta}}$$`

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

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.font120[

1. it supports different Survival Tasks
  - right-censoring, left-truncation
  - time-varying effects, time-varying features
  - cumulative effects (weighted cumulative exposure, distributed lag models)
  - 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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(source: [Bender, et al. (2020)](https://arxiv.org/abs/2006.15442))

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.font200[Survival Analysis as Poisson Regression]

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<div class="my-footer"><span><i class="fa fa-twitter">@adibender</i>
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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 (PEM; e.g.: [Laird, et al. (1981)](http://www.jstor.org/stable/2287816); [Friedman (1982)](http://www.jstor.org/stable/2240502); [Carstensen, et al. (2011)](https://www.jstatsoft.org/index.php/jss/article/view/v038i06))
  - Piece-wise exponential Additive Mixed Models (PAMM, e.g.: [Cai, et al. (2002)](http://dx.doi.org/10.1198/106186002862); [Kauermann (2005)](http://www.sciencedirect.com/science/article/pii/S0167947304001240); [Argyropoulos, et al. (2015)](http://dx.doi.org/10.1371/journal.pone.0123784); [Bender, et al. (2018)](https://doi.org/10.1177/1471082X17748083))

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