Sparse Estimation of Cox Proportional Hazards Models via Approximated Information Criteria

We propose a new sparse estimation method for Cox (1972) proportional hazards models by optimizing an approximated information criterion. The main idea involves approximation of the inline image norm with a continuous or smooth unit dent function. The proposed method bridges the best subset selection and regularization by borrowing strength from both. It mimics the best subset selection using a penalized likelihood approach yet with no need of a tuning parameter. We further reformulate the problem with a reparameterization step so that it reduces to one unconstrained nonconvex yet smooth programming problem, which can be solved efficiently as in computing the maximum partial likelihood estimator (MPLE). Furthermore, the reparameterization tactic yields an additional advantage in terms of circumventing postselection inference. The oracle property of the proposed method is established. Both simulated experiments and empirical examples are provided for assessment and illustration

Variable selection for Cox Proportional Hazards Models via Subtle Uprooting

Cox proportional hazards model (Cox PH model) is heavily used in survival analysis to assess the importance of various covariates on the survival times of individuals or objects through the hazard function. This study suggests a new variables selection method for Cox PH models, under the title \u27Subtle uprooting\u27, that does variable selection and model estimation for Cox proportional hazards (PH) models simultaneously. There are subset selection methods and shrinkage selection methods suggested in the context of Cox PH model. However the subset selection methods become infeasible in higher dimensions and the available shrinkage methods need tuning of parameters making the approach expensive and time consuming. Most attractive feature of the suggested method against available methods is that it does not require tuning of parameters anymore. Subtle uprooting uses hyperbolic tangent function as the penalty function based on its appropriate properties such as being a unit dent function, convenience of deriving deriva- tives and close relationship with the logit function. The procedure includes three steps. First, it approximates the cardilaity using surrogate penalty function. Then uprooting and thresholding are used to enhance the shrinkage. In the simulation, subtle uprooting, best subset selection, SCAD, LASSO and adaptive LASSO methods are used for simulated data sets and comparisons are made between the methods based on the model error (ME), overfiting, underfitting and correct selection percentages. Furthermore performance of the methods are studied for different sample sizes, censoring rates and input signals (strong and weak). It is found that subtle uprooting outperforms SCAD, LASSO and adaptive LASSO methods under strong signals and higher sample sizes. Subtle uprooting, best subset selection, SCAD, LASSO and adaptive LASSO methods are applied to the PBC data set. It is found that subtle uprooting estimates are significantly closer to the best subset selection results