Cox Proportional Hazards Model
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A Cox Proportional Hazards Model is a proportional hazards model with a hazard function of [math]\displaystyle{ \lambda(t|X) = \lambda_0(t)\exp(\beta_1X_1 + \cdots + \beta_pX_p) = \lambda_0(t)\exp(\beta^\prime X). }[/math]
- Context:
- It can range from being a Multivariate Cox Model to being a Univariate Cox Model.
- It can yield regression coefficients that can be interpreted as the effect of each feature on the risk of an event.
- It can fail to capture important non-linearities and interactions between features which may influence risk.
- See: ...
References
2013
- http://en.wikipedia.org/wiki/Proportional_hazards_models#Introduction
- Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s) without any consideration of the hazard function. This approach to survival data is called application of the Cox proportional hazards model,[1] sometimes abbreviated to Cox model or to proportional hazards model.
- ↑ Cox, David R (1972). "Regression Models and Life-Tables". Journal of the Royal Statistical Society. Series B (Methodological) 34 (2): 187–220. JSTOR 2985181. Template:MR
- http://en.wikipedia.org/wiki/Proportional_hazards_models#The_partial_likelihood
- Let [math]\displaystyle{ Y_i }[/math]denote the observed time (either censoring time or event time) for subject i, and let Ci be the indicator that the time corresponds to an event (i.e. if Ci = 1 the event occurred and if Ci = 0 the time is a censoring time). The hazard function for the Cox proportional hazard model has the form ::[math]\displaystyle{
\lambda(t|X) = \lambda_0(t)\exp(\beta_1X_1 + \cdots + \beta_pX_p) = \lambda_0(t)\exp(\beta^\prime X).
}[/math] This expression gives the hazard at time t for an individual with covariate vector (explanatory variables) X. Based on this hazard function, a partial likelihood can be constructed from the datasets as ::[math]\displaystyle{ L(\beta) = \prod_{i:C_i=1}\frac{\theta_i}{\sum_{j:Y_j\ge Y_i}\theta_j},
}[/math] where θj = exp(β′Xj) and X1, ..., Xn are the covariate vectors for the n independently sampled individuals in the dataset (treated here as column vectors).
The corresponding log partial likelihood is ::[math]\displaystyle{ \ell(\beta) = \sum_{i:C_i=1} \left(\beta^\prime X_i - \log \sum_{j:Y_j\ge Y_i}\theta_j\right). }[/math] This function can be maximized over β to produce maximum partial likelihood estimates of the model parameters.
- Let [math]\displaystyle{ Y_i }[/math]denote the observed time (either censoring time or event time) for subject i, and let Ci be the indicator that the time corresponds to an event (i.e. if Ci = 1 the event occurred and if Ci = 0 the time is a censoring time). The hazard function for the Cox proportional hazard model has the form ::[math]\displaystyle{
\lambda(t|X) = \lambda_0(t)\exp(\beta_1X_1 + \cdots + \beta_pX_p) = \lambda_0(t)\exp(\beta^\prime X).
}[/math] This expression gives the hazard at time t for an individual with covariate vector (explanatory variables) X. Based on this hazard function, a partial likelihood can be constructed from the datasets as ::[math]\displaystyle{ L(\beta) = \prod_{i:C_i=1}\frac{\theta_i}{\sum_{j:Y_j\ge Y_i}\theta_j},
}[/math] where θj = exp(β′Xj) and X1, ..., Xn are the covariate vectors for the n independently sampled individuals in the dataset (treated here as column vectors).
1989
- (Yin & Wei, 1989) ⇒ D. Y. Lin and L. J. Wei. (1989). “The Robust Inference for the Cox Proportional Hazards Model.” In: Journal of the American Statistical Association, 84(408). http://www.jstor.org/stable/2290085
1972
- (Cox, 1972) ⇒ David R Cox. (1972). “Regression Models and Life-tables.” In: Journal of the Royal Statistical Society. Series B (Methodological).