Tests for High-Dimensional Regression Coefficients With Factorial Designs

We propose simultaneous tests for coefficients in high-dimensional linear regression models with factorial designs. The proposed tests are designed for the large p, small n situations where the conventional F-test is no longer applicable. We derive the asymptotic distribution of the proposed test statistic under the high-dimensional null hypothesis and various scenarios of the alternatives, which allow power evaluations. We also evaluate the power of the F-test for models of moderate dimension. The proposed tests are employed to analyze a microarray data on Yorkshire Gilts to find significant gene ontology terms which are significantly associated with the thyroid hormone after accounting for the designs of the experiment.