Researchers sometimes have a priori information on 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 of the discarded covariates is significant conditional on some pre-chosen covariates. We propose a maximum-type test statistic, and show that it has a non-standard asymptotic distribution, giving rise to the conditional adaptive resampling test. To accommodate signals of unknown sparsity, we propose a hybrid test statistic, 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 it provides adequate control of family-wise error rate with competitive power for both sparse and dense signals, even in high-dimensional cases, and establish its advantages when covariates are heavily correlated. We illustrate our proposal using an expression quantitative trait locus dataset.
