Higher Criticism Tuned Regression For Weak And Sparse Signals

Abstract: Here we propose a novel searching scheme for a tuning parameter in high-dimensional penalized regression methods to address variable selection and modeling when sample sizes are limited compared to the data dimensions. Our method is motivated by high-throughput biological data such as genome-wide association studies (GWAS) and epigenome-wide association studies (EWAS). We propose a new estimate of the regularization parameter $\lambda$ in penalized regression methods based on an estimated lower bound of the proportion of false null hypotheses with confidence $(1 - \alpha)$. The bound is estimated by applying the empirical null distribution of the higher criticism statistic, a second-level significance test constructed by dependent $p$-values using a multi-split regression and aggregation method. A tuning parameter estimate in penalized regression, $\lambda$, corresponds with the lower bound of the proportion of false null hypotheses. Different penalized regression methods with varied signal sparsity and strength are compared in the multi-split method setting. We demonstrate the performance of our method using both simulation experiments and the applications of real data on (1) lipid-trait genetics from the Action to Control Cardiovascular Risk in Diabetes (ACCORD) clinical trial and (2) epigenetic analysis evaluating smoking's influence in differential methylation in the Agricultural Lung Health Study. The proposed algorithm is included in the HCTR package, available at https://cran.r-project.org/web/packages/HCTR/index.html.