Abstract
Various forms of penalty functions have been developed for regularized estimation and variable selection. Screening approaches are often used to reduce the number of covariate before penalized estimation. However, in certain problems, the number of covariates remains large after screening. For example, in genome-wide association (GWA) studies, the purpose is to identify Single Nucleotide Polymorphisms (SNPs) that are associated with certain traits, and typically there are millions of SNPs and thousands of samples. Because of the strong correlation of nearby SNPs, screening can only reduce the number of SNPs from millions to tens of thousands and the variable selection problem remains very challenging. Several penalty functions have been proposed for such high dimensional data. However, it is unclear which class of penalty functions is the appropriate choice for a particular application. In this paper, we conduct a theoretical analysis to relate the ranges of tuning parameters of various penalty functions with the dimensionality of the problem and the minimum effect size. We exemplify our theoretical results in several penalty functions. The results suggest that a class of penalty functions that bridges L0 and L1 penalties requires less restrictive conditions on dimensionality and minimum effect sizes in order to attain the two fundamental goals of penalized estimation: to penalize all the noise to be zero and to obtain unbiased estimation of the true signals. The penalties such as SICA and Log belong to this class, but they have not been used often in applications. The simulation and real data analysis using GWAS data suggest the promising applicability of such class of penalties.
Highlights
In genome-wide association (GWA) studies, the goal is to identify the genetic factors such as single nucleotide polymorphisms (SNPs) that are associated with diseases
The methods with folded concave penalties may not be desirable in terms of computational efficiency, they may lead to nice statistical properties in high dimensional setting (Fan and Li, 2001)
To investigate the applicability of the nonconvex penalty functions in challenging high dimensional settings such as genomic studies, we conducted a theoretical analysis on the roles of tuning parameters with respect to the dimension of the problem and minimum effect size
Summary
In genome-wide association (GWA) studies, the goal is to identify the genetic factors such as single nucleotide polymorphisms (SNPs) that are associated with diseases. Regularized estimation procedures can be applied for simultaneous selection of important variables (SNPs) and estimation of their effects for high dimensional data in GWA studies. Previous work has provided recommendations regarding the choice of tuning parameters, but there is no systematic asymptotic studies on the roles of multiple tuning parameters To address those issues, we will relate the choice of tuning parameters to the difficulty of the variable selection problem, namely the minimum effect size and the dimensions, i.e., the number of important and unimportant covariates. The results suggest that a class of penalty functions that bridges L0 and L1 penalties such as Log and SICA requires less restrictive conditions on dimensionality and minimum effect sizes, while achieving the two fundamental goals of penalized estimation. Those empirical results support the idea that the class of penalty functions that bridges L0 and L1 holds promise for genomic studies
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