Cox proportional hazards model

Cox Proportional Hazards Model

The Cox proportional hazards model is a statistical technique used in the analysis of survival data. It is a type of regression analysis that is particularly useful for investigating the association between the survival time of subjects and one or more predictor variables. The model is named after the British statistician David Cox, who introduced it in 1972.

Overview

The Cox model is a semiparametric model, meaning it makes no assumptions about the baseline hazard function, which can vary over time. Instead, it assumes that the effect of the predictor variables on the hazard is multiplicative and constant over time. This is known as the proportional hazards assumption.

Model Specification

The Cox proportional hazards model can be expressed as:

<math> h(t|X) = h_0(t) \, e^{\beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p} </math>

where:

• <math> h(t|X) </math> is the hazard function at time <math> t </math> given the covariates <math> X </math>.
• <math> h_0(t) </math> is the baseline hazard function.
• <math> \beta_1, \beta_2, \ldots, \beta_p </math> are the coefficients for the covariates <math> X_1, X_2, \ldots, X_p </math>.

Assumptions

The key assumption of the Cox model is the proportional hazards assumption, which states that the hazard ratios between individuals are constant over time. This implies that the effect of the covariates is multiplicative with respect to the hazard function.

Estimation

The parameters of the Cox model are estimated using the method of partial likelihood. This approach allows for the estimation of the regression coefficients without needing to specify the baseline hazard function.

Applications

The Cox proportional hazards model is widely used in the field of biostatistics and epidemiology for analyzing survival data. It is particularly useful in clinical trials and medical research for assessing the impact of treatments or risk factors on survival time.

Limitations

While the Cox model is powerful, it has limitations. The proportional hazards assumption may not hold in all datasets, and the model does not handle time-varying covariates naturally. Extensions of the Cox model, such as stratified Cox models and time-dependent covariates, can address some of these limitations.

See Also

• Survival analysis
• Kaplan-Meier estimator
• Hazard ratio

References

• Cox, D. R. (1972). "Regression Models and Life-Tables". Journal of the Royal Statistical Society. Series B (Methodological). 34 (2): 187–220.
• Kleinbaum, D. G., & Klein, M. (2012). Survival Analysis: A Self-Learning Text. Springer.

External Links

• Wikipedia: Proportional hazards model