Fitting survival data with penalized Poisson regression

Cox's proportional hazards model is the most common way to analyze survival data. The model can be extended in the presence of collinearity to include a ridge penalty, or in cases where a very large number of coefficients (e. g. with microarray data) has to be estimated. To maximize the penalized likelihood, optimal weights of the ridge penalty have to be obtained. However, there is no definite rule for choosing the penalty weight. One approach suggests maximization of the weights by maximizing the leave-one-out cross validated partial likelihood, however this is time consuming and computationally expensive, especially in large datasets. We suggest modelling survival data through a Poisson model. Using this approach, the log-likelihood of a Poisson model is maximized by standard iterative weighted least squares. We will illustrate this simple approach, which includes smoothing of the hazard function and move on to include a ridge term in the likelihood. We will then maximize the likelihood by considering tools from generalized mixed linear models. We will show that the optimal value of the penalty is found simply by computing the hat matrix of the system of linear equations and dividing its trace by a product of the estimated coefficients.