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Semi-Parametric/Cox Proportional Hazard models

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Cox Proportional Hazard model

Hazard function's formula

When it comes to predicting the survival function for a specific unit, the Cox Proportional Hazard Model (CoxPH) is usually the go-to model. The CoxPH model is a semi-parametric model that focuses on modeling the hazard function $h(t, x_i)$, by assuming that its time component $\lambda_0(t)$ and feature component $\eta(\vec{x_i})$ are proportional such that: $$\begin{equation*} h(t, \vec{x_i}) = \lambda_0(t)\eta(\vec{x_i}) \end{equation*}$$ with:

• $\lambda_0(t)$, is the baseline function, which is usually not specified.

• $\eta(\vec{x_i})$, is the risk function usually expressed via a linear representation such that $\eta(\vec{x_i}) = \exp \left( \sum_{j=1}^p x^i_j\omega_j \right)$. $\omega_j$ are the coefficients to determine

Building the model

The model can be built by calculating the Efron's partial likelihood to take ties into account. The partial likelihood $L(\omega)$ can be written such that: $\displaystyle L(\omega )=\prod _{j}{\frac {\prod _{i\in H_{j}}\theta _{i}}{\prod _{s =0}^{m_j-1}[\sum _{i:T_{i}\geq t_{j}}\theta _{i}-{\frac {s}{m_j}}\sum _{i\in H_{j}}\theta _{i}]}}.$

• the log partial likelihood is $\displaystyle l (\omega )=\sum _{j}\left(\sum _{i\in H_{j}}X_{i}\cdot \omega -\sum _{s =0}^{m_j-1}\log(\phi _{j,s ,m_j})\right).$

• the gradient is $\displaystyle \vec{\nabla l(\omega )}=\sum _{j}\left(\sum _{i\in H_{j}}X_{i}-\sum _{s =0}^{m_j-1}{\frac {z_{j,s ,m_j}}{\phi _{j,s ,m_j}}}\right).$

• the Hessian matrix is $\displaystyle \nabla^2 l(\omega )=-\sum _{j}\sum _{s =0}^{m_j-1}\left({\frac {{Z}_{j,s ,m_j}}{\phi _{j,s ,m_j}}}-{\frac {z_{j,s ,m}z_{j,s ,m_j}^{T}}{\phi _{j,s ,m_j}^{2}}}\right).$

We can now use the Newton-Optimization schema to fit the model: $$\begin{equation*} \omega_{new} = \omega_{old} - \nabla^2 l(\omega )^{-1} \cdot \vec{\nabla l(\omega )} \end{equation*}$$

with:

• $H_j = \left\{ i; T_i = t_j \text{ and } E_i = 1 \right \}$, ${m_j} = \left|H_j \right|.$
• $\theta _{i}= \exp\left(X_{i} \cdot \omega \right)$
• $\phi _{j,s ,m_j}=\sum _{i:T_{i}\geq t_{j}}\theta _{i}-{\frac {s}{m_j}}\sum _{i\in H_{j}}\theta _{i}$
• $z_{j,s ,m_j}=\sum _{i:T_{i}\geq t_{j}}\theta _{i}X_{i}-{\frac {s }{m_j}}\sum _{i\in H_{j}}\theta _{i}X_{i}.$
• ${Z}_{j,s ,m_j}=\sum _{i:T_{i}\geq t_{j}}\theta _{i}X_{i}X_{i}^{T }-{\frac {s }{m_j}}\sum _{i\in H_{j}}\theta _{i}X_{i}X_{i}^{T}$

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DeepSurv/NonLinear CoxPH model

Hazard function's formula

The NonLinear CoxPH model was popularized by Katzman et al. in DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network by allowing the use of Neural Networks within the original design. Here the hazard function $h(t, x_i)$ can be written as
$$\begin{equation*} h(t, \vec{x_i}) = \lambda_0(t)\Psi(\vec{x_i}) \end{equation*}$$

with:

• $\Psi(\vec{x_i}) = \exp(\psi(\vec{x_i}))$, where $\psi$ is a non-linear risk function.

Building the model

We are still using the Efron's partial likelihood to take ties into account, but here the hazard function is $h(t, \vec{x_i}) = \lambda_0(t)\Psi(\vec{x_i})$. Thus, the log partial likelihood is $$\begin{equation*} l (\omega )=\sum _{j}\left(\sum _{i\in H_{j}}\log(\Psi(\vec{x_i})) -\sum _{s =0}^{m_j-1}\log \left(\sum _{i:T_{i}\geq t_{j}}\Psi(\vec{x_i})-{\frac {s }{m_j}}\sum _{i\in H_{j}}\Psi(\vec{x_i})\right)\right). \end{equation*}$$

As the Hessian matrix will be too complicated to calculate, we will use PyTorch to compute the gradient and perform a First-Order optimization.

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References

• Wikipedia - Proportional hazards model
• Cox, David R. "Regression models and life‐tables." Journal of the Royal Statistical Society: Series B (Methodological) 34.2 (1972): 187-202.
• Katzman, Jared, et al. "DeepSurv: Personalized treatment recommender system using A Cox proportional hazards deep neural network." arXiv preprint arXiv:1606.00931 (2016).