The Partial Likelihood
Let Yi 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
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
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
This function can be maximized over β to produce maximum partial likelihood estimates of the model parameters.
The partial score function is
and the Hessian matrix of the partial log likelihood is
Using this score function and Hessian matrix, the partial likelihood can be maximized using the Newton-Raphson algorithm. The inverse of the Hessian matrix, evaluated at the estimate of β, can be used as an approximate variance-covariance matrix for the estimate, and used to produce approximate standard errors for the regression coefficients.
Read more about this topic: Proportional Hazards Models
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