Proportional Hazards Models - The Partial Likelihood

The Partial Likelihood

Let Y_i denote the observed time (either censoring time or event time) for subject i, and let C_i be the indicator that the time corresponds to an event (i.e. if C_i = 1 the event occurred and if C_i = 0 the time is a censoring time). The hazard function for the Cox proportional hazard model has the form

$\lambda(t|X) = \lambda_0(t)\exp(\beta_1X_1 + \cdots + \beta_pX_p) = \lambda_0(t)\exp(\beta^\prime X).$

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

$L(\beta) = \prod_{i:C_i=1}\frac{\theta_i}{\sum_{j:Y_j\ge Y_i}\theta_j},$

where θ_j = exp(β′X_j) and X₁, ..., X_n are the covariate vectors for the n independently sampled individuals in the dataset (treated here as column vectors).

The corresponding log partial likelihood is

$\ell(\beta) = \sum_{i:C_i=1} \left(\beta^\prime X_i - \log \sum_{j:Y_j\ge Y_i}\theta_j\right).$

This function can be maximized over β to produce maximum partial likelihood estimates of the model parameters.

The partial score function is

$\ell^\prime(\beta) = \sum_{i:C_i=1} \left(X_i - \frac{\sum_{j:Y_j\ge Y_i}\theta_jX_j}{\sum_{j:Y_j\ge Y_i}\theta_j}\right),$

and the Hessian matrix of the partial log likelihood is

$\ell^{\prime\prime}(\beta) = -\sum_{i:C_i=1} \left(\frac{\sum_{j:Y_j\ge Y_i}\theta_jX_jX_j^\prime}{\sum_{j:Y_j\ge Y_i}\theta_j} - \frac{\sum_{j:Y_j\ge Y_i}\theta_jX_j\times \sum_{j:Y_j\ge Y_i}\theta_jX_j^\prime}{^2}\right).$

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.

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