Testing for the presence of significant covariates through conditional marginal regression

Researchers sometimes have a priori information on the relative importance of predictors that can be used to screen out covariates. An important question is whether any of the discarded covariates have predictive power when the most relevant predictors are included in the model. We consider testing whether any discarded covariate is significant conditional on some pre-chosen covariates. We propose a maximum-type test statistic and show that it has a nonstandard asymptotic distribution, giving rise to the conditional adaptive resampling test. To accommodate signals of unknown sparsity, we develop a hybrid test statistic, which is a weighted average of maximum- and sum-type statistics. We prove the consistency of the test procedure under general assumptions and illustrate how it can be used as a stopping rule in forward regression. We show, through simulation, that the proposed method provides adequate control of the familywise error rate with competitive power for both sparse and dense signals, even in high-dimensional cases, and we demonstrate its advantages in cases where the covariates are heavily correlated. We illustrate the application of our method by analysing an expression quantitative trait locus dataset.