Exact Post-Selection Inference for Sequential Regression Procedures

TL;DR

This paper introduces exact post-selection inference methods for popular regression procedures like forward stepwise, LAR, and lasso, enabling valid hypothesis testing and confidence intervals after model selection under Gaussian assumptions.

Contribution

It develops a general framework for valid post-selection inference based on polyhedral constraints, applicable to multiple regression procedures, with implementation in an R package.

Findings

Exact p-values with uniform null distribution

Finite-sample valid confidence intervals

Framework applicable to various regression steps

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 paper.