A link-free sparse group variable selection method for single-index model

For regression problems with grouped covariates, we adapt the idea of sparse group lasso (SGL) [10 J. Friedman, T. Hastie, and R. Tibshirani, A note on the group lasso and a sparse group lasso, Tech. Rep., Statistics Department, Stanford University, 2010. [Google Scholar]] to the framework of the sufficient dimension reduction. Assuming that the regression falls into a single-index structure, we propose a method called the sparse group sufficient dimension reduction to conduct group and within-group variable selections simultaneously without assuming a specific link function. Simulation studies show that our method is comparable to the SGL under the regular linear model setting and outperforms SGL with higher true positive rates and substantially lower false positive rates when the regression function is nonlinear. One immediate application of our method is to the gene pathway data analysis where genes naturally fall into groups (pathways). An analysis of a glioblastoma microarray data is included for illustration of our method.