Empirical Likelihood Test for High Dimensional Generalized Linear Models

Abstract

Technological advances allow scientists to collect high dimensional data sets in which the number of variables is much larger than the sample size. A representative example is genomics. Consequently, due to their loss of accuracy or power, many classic statistical methods are being challenged when analyzing such data. In this chapter, we propose an empirical likelihood (EL) method to test regression coefficients in high dimensional generalized linear models. The EL test has an asymptotic chi-squared distribution with two degrees of freedom under the null hypothesis, and this result is independent of the number of covariates. Moreover, we extend the proposed method to test a part of the regression coefficients in the presence of nuisance parameters. Simulation studies show that the EL tests have a good control of the type-I error rate under moderate sample sizes and are more powerful than the direct competitor under the alternative hypothesis under most scenarios. The proposed tests are employed to analyze the association between rheumatoid arthritis (RA) and single nucleotide polymorphisms (SNPs) on chromosome 6. The resulted p-value is 0.019, indicating that chromosome 6 has an influence on RA. With the partial test and logistic modeling, we also find that the SNPs eliminated by the sure independence screening and Lasso methods have no significant influence on RA.