Cox Model

Form of the Model

Let z = z1, z2, …, zp be a p × 1 vector of covariates or risk factors and let h(t|z) be the hazard function, which depends on the covariates z. The general form of the proportional hazard model is

where ψ(z) is a function of the covariates, and h0(t)is the underlying baseline hazard. h0(t) can be interpreted as the hazard of death in the absence of an effect of the covariates or when the covariates assume a reference value (e.g., absence of an exposure factor or membership to a placebo or a standard therapy group).

The most common form of the function ψ(z)is

so that the hazard, given the covariates, can be expressed as

In this model, the predictors (z1, z2, …, zp) are assumed to act additively on ln h(t|z). β = β1, β2, …, βp is a p × 1 vector of unknown parameters that relate the covariates z to the hazard.

While in parametric survival models we make an assumption about the distribution of the baseline hazard h0(t), in the Cox proportional hazard model we make no assumption about the shape of the underlying hazard. The only parametric part of the model is the one that contains information about the predictors, ψ(z). For this reason, the Cox model is called a semi-parametric model. Since in most applications we are interested more in the relationship between the predictors and the survival, rather than the shape of the hazard, the Cox proportional hazard model is suitable for most applied problems. Note that the model does not have an intercept as the baseline hazard h0(t) accounts for that. The Cox model can be used to estimate the survival function s(t) for fixed values of the covariates.

The Partial Likelihood and Parameter Estimation

As with other regression models, estimation of the regression parameters for the Cox model proceed by maximizing the likelihood function.

Although the full likelihood for the survival data with covariates can be expressed as the product of two

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