Integrative weighted group lasso and generalized local quadratic approximation

Longitudinal clinical outcomes are often collected in genomic studies, where selection methods accounting for dynamic effects of biomarkers are desirable. Biomarker effects can be modeled by nonparametric B-splines and selected by group lasso. A novel weight function is proposed based on the extremum of the biomarker effects over time for the penalty. In addition to the common practice treating weights as adaptive functions depending on some first-stage estimates, an integrative group lasso which treats the loss, penalty and weight functions as an integrative whole is proposed, where parameters in all three are jointly estimated in one step. Generalized local quadratic approximations are developed to optimize the integrative group lasso whose guidelines are applicable in a wide range of non-convex optimization problems. The integrative version has theoretical advantages as it requires weaker assumptions in achieving consistency and sparsistency. Both adaptive and integrative procedures show larger areas under the ROC curves as well as smaller biases and mean square prediction errors over unweighted group lasso in simulation studies. Finally, the proposed method is illustrated on the GWAS from the Epidemiology and Intervention of Diabetes Complication trial. To accommodate more candidate markers, 23 chromosomes are analyzed separately with common tuning parameters.