LASSO-Based Approaches for Joint Tests of Genetic Main Effects and
Gene–Environment Interactions
Jie Zheng1, Dabeeru C. Rao2, Gang Shi3*
1Clinical
Research Center, The First Affiliated Hospital of X’an Jiaotong University, 227 West Yanta
Road, Xi’an, Shaanxi, 710061, China, jie_stat33@yahoo.com
2Division
of Biostatistics, Washington University School of Medicine, 660 South Euclid Avenue,
Campus Box 8067, Saint Louis, Missouri, 63110, USA, rao@wubios.wustl.edu
3State
Key Laboratory of Integrated Services Networks, Xidian University, 2 South Taibai Road,
Xi’an, Shaanxi, 710071, China, gshi@xidian.edu.cn
*Corresponding
author
Abstract
The least absolute shrinkage and selection operator (LASSO) regression, the regression
coefficients of which are under an L1 norm constraint, has the so-called shrinkage property and
yields a parsimonious model. It is an appealing approach for variable selection, especially for
high-dimensional problems, and has attracted considerable interest in genetic studies. The
LASSO regression and its variations have been applied to screening common and rare genetic
variants (Shi et al., 2011; Zhou et al., 2010), testing single-nucleotide polymorphisms (SNPs)
(Ayers et al., 2011) and haplotypes (Biswas and Lin, 2012; Lin et al., 2012), detecting SNPSNP
interactions (D'Angelo et al., 2009; Wu et al, 2010) and haplotypehaplotype interactions (Li et
al., 2010), jointly examining genetic main and dominant effects (Sabourin et al., 2015),
estimating gene–gene and gene–environment interactions simultaneously (Tanck et al., 2006),
analyzing main effects of SNPs and their epistatic interactions jointly (Yang et al., 2010). In this
work, we investigated application of the LASSO regression to the joint testing of genetic main
effects and gene–environment interactions considering multiple SNPs simultaneously. Testing
genetic main effects together with interaction effects is known to be more powerful than either
of the marginal tests, if an interaction exists. A recent large-scale, genome-wide association
study of blood pressure to assess the pervasiveness of gene–age interactions employed the two
degrees-of-freedom joint test and identified twenty significant loci, of which nine demonstrated
nominal evidence of interactions with age and five would have been missed by a
main-effects-only analysis. We generalized the single-marker two degrees-of-freedom joint test
to a multiple-marker version using the LASSO regression. Specifically, we considered two
LASSO-based approaches. One analyzed the genetic main effects of multiple SNPs and their
interactions with an environmental factor. SNPs with a nonzero regression coefficient for either
a main effect or interaction were considered significant. The other employed the group LASSO
regression, which shrinks each SNP with its interaction in a group-wise manner. SNPs in the final
model were deemed significant (for both main effects and interactions). The two
degrees-of-freedom linear regression that tests the main effects and interactions jointly was
used as the benchmark test for comparisons. The statistical power for testing SNPs with various
main and interaction effect sizes was evaluated, and type I errors of the three methods were
examined. Bayesian information criterion (BIC), Mallow's Cp, and Stine's Sp were compared with
respect to their ability to select constraint parameters of the two LASSO approaches. Based on
simulation studies, we showed that BIC and Mallow's Cp tend to generate over-fitted models
resulting in large type I errors for both the LASSO methods. Choosing constraint parameters
based on Stine's Sp demonstrated acceptable empirical type I errors. Since Stine's Sp balances
the model fitness and number of parameters and does not control type I errors directly, the two
LASSO methods showed varying empirical false positive rates for different sample sizes. Group
LASSO displayed lower type I errors and statistical power than did the two degrees-of-freedom
regression test when sample sizes were relatively small and higher type I errors and statistical
power when sample sizes were larger. The first LASSO method showed higher type I errors and
statistical power than did the group LASSO and had a similar pattern of type I errors for different
sample sizes. For computational efficiency, the first approach was much faster than the second
when using the least-angle regression solver.
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