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We propose a new measure of variable importance in high-dimensional regression based on the change in the LASSO solution path when one covariate is left out. The proposed procedure provides a novel way to calculate variable importance and conduct variable screening. In addition, our procedure allows for the construction of  p -values for testing whether each coefficient is equal to zero as well as for testing hypotheses involving multiple regression coefficients simultaneously; bootstrap techniques are used to construct the null distribution. For low-dimensional linear models, our method can achieve higher power than the  t -test. Extensive simulations are provided to show the effectiveness of our method. In the high-dimensional setting, our proposed solution path based test achieves greater power than some other recently developed high-dimensional inference methods. We extend our method to logistic regression and demonstrate in simulation that our leave-one-covariate-out solution path tests can provide accurate  p -values.

Inference for sparse linear regression based on the leave-one-covariate-out solution path