Abstract: Significance testing for high-dimensional generalized linear models (GLMs) has become increasingly important in various applications. However, existing methods are mainly based on a sum of the squares of the elements of the score vector and are only powerful under certain alternative hypotheses. In practice, the density of the true association pattern under an alternative hypothesis dictates whether existing tests are powerful. We propose an adaptive test on a high-dimensional parameter of a GLM (in the presence of a low-dimensional nuisance parameter) that maintains high power across a wide range of scenarios. To evaluate its p-value, its asymptotic null distribution is derived. We conduct simulations to demonstrate the superior performance of the proposed test. In addition, we apply it and other existing tests to an Alzheimer's Disease Neuroimaging Initiative data set to detect possible associations between Alzheimer's disease and gene pathways that have a large number of single nucleotide polymorphisms (SNPs). We implemented the proposed method in the R package GLMaSPU, which is publicly available on GitHub and CRAN.

Key words and phrases: Adaptive tests, generalized linear models, high-dimensional testing, power.