High-dimensional Cox models: the choice of penalty as part of the model building process

The Cox proportional hazards regression model is the most popular approach to model covariate information for survival times. In this context, the development of high-dimensional models where the number of covariates is much larger than the number of observations (p>>n) is an ongoing challenge. A practicable approach is to use ridge penalized Cox regression in such situations. Beside focussing on finding the best prediction rule, one is often interested in determining a subset of covariates that are the most important ones for prognosis. This could be a gene set in the biostatistical analysis of microarray data. Covariate selection can then, for example, be done by L(1)-penalized Cox regression using the lasso (Tibshirani (1997). Statistics in Medicine 16, 385-395). Several approaches beyond the lasso, that incorporate covariate selection, have been developed in recent years. This includes modifications of the lasso as well as nonconvex variants such as smoothly clipped absolute deviation (SCAD) (Fan and Li (2001). Journal of the American Statistical Association 96, 1348-1360; Fan and Li (2002). The Annals of Statistics 30, 74-99). The purpose of this article is to implement them practically into the model building process when analyzing high-dimensional data with the Cox proportional hazards model. To evaluate penalized regression models beyond the lasso, we included SCAD variants and the adaptive lasso (Zou (2006). Journal of the American Statistical Association 101, 1418-1429). We compare them with "standard" applications such as ridge regression, the lasso, and the elastic net. Predictive accuracy, features of variable selection, and estimation bias will be studied to assess the practical use of these methods. We observed that the performance of SCAD and adaptive lasso is highly dependent on nontrivial preselection procedures. A practical solution to this problem does not yet exist. Since there is high risk of missing relevant covariates when using SCAD or adaptive lasso applied after an inappropriate initial selection step, we recommend to stay with lasso or the elastic net in actual data applications. But with respect to the promising results for truly sparse models, we see some advantage of SCAD and adaptive lasso, if better preselection procedures would be available. This requires further methodological research.