A pliable lasso for the Cox model

We introduce a pliable lasso method for estimation of interaction effects in the Cox proportional hazards model framework. The pliable lasso is a linear model that includes interactions between covariates X and a set of modifying variables Z and assumes sparsity of the main effects and interaction effects. The hierarchical penalty excludes interaction effects when the corresponding main effects are zero: this avoids overfitting and an explosion of model complexity. We extend this method to the Cox model for survival data, incorporating modifiers that are either fixed or varying in time into the partial likelihood. For example, this allows modeling of survival times that differ based on interactions of genes with age, gender, or other demographic information. The optimization is done by blockwise coordinate descent on a second order approximation of the objective.