  In the analysis of survival outcome supplemented with both clinical
information and high-dimensional gene expression data, use of the traditional
Cox proportional hazards model (1972) fails to meet some emerging needs in
biomedical research. First, the number of covariates is generally much larger
the sample size. Secondly, predicting an outcome based on individual gene
expression is inadequate because multiple biological processes and functional
pathways regulate the expression associated with a gene. Another challenge is
that the Cox model assumes that populations are homogenous, implying that all
individuals have the same risk of death, which is rarely true due to unmeasured
risk factors among populations. In this paper we propose group LASSO with
gamma-distributed frailty for variable selection in Cox regression by extending
previous scholarship to account for heterogeneity among group structures
related to exposure and susceptibility. The consistency property of the
proposed method is established. This method is appropriate for addressing a
wide variety of research questions from genetics to air pollution. Simulated
analysis shows promising performance by group LASSO compared with other
methods, including group SCAD and group MCP. Future directions include
expanding the use of frailty with adaptive group LASSO and sparse group LASS.
