Gradient lasso for Cox proportional hazards model

Motivation: There has been an increasing interest in expressing a survival phenotype (e.g. time to cancer recurrence or death) or its distribution in terms of a subset of the expression data of a subset of genes. Due to high dimensionality of gene expression data, however, there is a serious problem of collinearity in fitting a prediction model, e.g. Cox's proportional hazards model. To avoid the collinearity problem, several methods based on penalized Cox proportional hazards models have been proposed. However, those methods suffer from severe computational problems, such as slow or even failed convergence, because of high-dimensional matrix inversions required for model fitting. We propose to implement the penalized Cox regression with a lasso penalty via the gradient lasso algorithm that yields faster convergence to the global optimum than do other algorithms. Moreover the gradient lasso algorithm is guaranteed to converge to the optimum under mild regularity conditions. Hence, our gradient lasso algorithm can be a useful tool in developing a prediction model based on high-dimensional covariates including gene expression data. Results: Results from simulation studies showed that the prediction model by gradient lasso recovers the prognostic genes. Also results from diffuse large B-cell lymphoma datasets and Norway/Stanford breast cancer dataset indicate that our method is very competitive compared with popular existing methods by Park and Hastie and Goeman in its computational time, prediction and selectivity.