Cox Proportional Hazards Model

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The Cox proportional hazards model is a semi-parametric model commonly used in survival analysis. It is a regression model that aims to model the hazard function of survival time \(T\), which represents the probability of an event occurring at time \(t\) given that the event has not occurred before time \(t\), that is, \[ \lambda(t)=\lim_{\Delta t\rightarrow 0}\frac{P(t\leq T<t+\Delta t|T\geq t)}{\Delta t}. \]

Assumptions

The Cox proportional hazards model assumes that the hazard function follows the form: \[ \lambda(t|X) = \lambda_0(t) \exp(X^T\beta), \] where, \(\lambda_0(t)\) represents the baseline hazard function, and \(\beta\) is the vector of regression coefficients.

Estimation

The estimation of the Cox proportional hazards model is based on the partial likelihood function. The partial likelihood function is defined as: \[ L(\beta) = \prod_{i=1}^{n}\left[\frac{\exp(X_i^T\beta)}{\sum_{j\in R(t_i)} \exp(X_j^T\beta)}\right]^{\delta_i}, \] where, \(R(t_i)\) represents the set of individuals at risk at time \(t_i\), and \(\delta_i\) is an indicator variable that takes the value 1 if the event occurs for individual \(i\) and 0 otherwise.

The corresponding log partial likelihood function of \(\beta\) is given by: \[ \ell(\beta)=\sum_{i=1}^{n}\delta_i\left[X_i^T\beta-\log\left(\sum_{j\in R(t_i)}\exp(X_j^T\beta)\right)\right] \] and the cumulative baseline hazard function \(\Lambda_0(t)\) is given by: \[ \Lambda_0(t)=\frac{\sum_{i=1}^{n}\exp(X_i^T\beta)I(t_i\leq t)}{\sum_{i=1}^{n} I(t_i\leq t)}, \] where, \(I(t_i\leq t)\) is an indicator variable that takes the value 1 if \(t_i\leq t\) and 0 otherwise.

Proof

The likelihood function is defined as: \[ L(\beta)=\prod_{i=1}^{n}\lambda(t_i|X_i)^{\delta_i}S(t_i|X_i), \] where \(S(t_i|X_i)\) represents the survival function.

Assuming the baseline hazard function is a piecewise constant function, i.e., \(\lambda_0(t)=\lambda_k\) for \(t\in[t_k,t_{k+1})\), the likelihood function can be written as: \[ L(\beta)=\prod_{i=1}^{n}\lambda_0(t_i)\exp(X_i^T\beta)^{\delta_i}\exp\left(-\int_0^{t_i}\lambda_0(u)\exp(X_i^T\beta)\,\mathrm{d}u\right) \]