Tests for High Dimensional Generalized Linear Models

We consider testing regression coefficients in high dimensional generalized linear models. An investigation of the test of Goeman et al. (2011) is conducted, which reveals that if the inverse of the link function is unbounded, the high dimensionality in the covariates can impose adverse impacts on the power of the test. We propose a test formation which can avoid the adverse impact of the high dimensionality. When the inverse of the link function is bounded such as the logistic or probit regression, the proposed test is as good as Goeman et al. (2011)'s test. The proposed tests provide p-values for testing significance for gene-sets as demonstrated in a case study on an acute lymphoblastic leukemia dataset.