# Exact Post-Selection Inference for Sequential Regression Procedures

*Ryan J. Tibshirani*,
*Jonathan Taylor*,
*Richard Lockhart* and
*Robert Tibshirani*

*Journal of the American Statistical Association*, 2016, vol. 111, issue 514, 600-620

**Abstract:**
We propose new inference tools for forward stepwise regression, least angle regression, and the lasso. Assuming a Gaussian model for the observation vector y, we first describe a general scheme to perform valid inference after any selection event that can be characterized as y falling into a polyhedral set. This framework allows us to derive conditional (post-selection) hypothesis tests at any step of forward stepwise or least angle regression, or any step along the lasso regularization path, because, as it turns out, selection events for these procedures can be expressed as polyhedral constraints on y. The p-values associated with these tests are exactly uniform under the null distribution, in finite samples, yielding exact Type I error control. The tests can also be inverted to produce confidence intervals for appropriate underlying regression parameters. The R package selectiveInference, freely available on the CRAN repository, implements the new inference tools described in this article. Supplementary materials for this article are available online.

**Date:** 2016

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