Exact Post-Selection Inference for Sequential Regression Procedures

By Robert Tibshirani, Jonathan E. Taylor, Richard A. Lockhart. 2014

464 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

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Citations in the corpus

Selective inference with a randomized response (2015, methodology): “Many examples of nonrandomized selective inference can be expressed in this way [cf. Fithian et al. (2015), Lee et al. (2016), Tibshirani et al. (2016)].”
Handbook of Big Data (2016, background): “Our running example was to estimate the eﬀect of early ART initiation (within 1 month of diagnosis) on 5-year mortality risk among HIV+ adults in Sub-Saharan Africa.”
Automatic Database Management System Tuning Through Large-scale Machine Learning (2017, background): “There is also both practical and theoretical work backing its effectiveness as a consistent feature selection algorithm [56, 57, 64, 9].”
Selecting the number of principal components: estimation of the true rank of a noisy matrix (2014, methodology): “In its construction, the ICSV test utilizes a post-selection inference framework as in the work of Tibshirani et al. (2014) in the same manner as the ordinary CSV test.”
Selective Sequential Model Selection (2015, methodology): “Tibshirani et al. (2014) study methods for generating exact p-values at each step of these path algorithms, and their methods provide a starting point for our proposals.”
Distribution-Free Predictive Inference for Regression (2016, background): “In the second class, post-selection approaches, the focus is on covering coefficients in the best linear approximation to μ given a subset of selected covariates (see, e.g., Berk et al. 2013; Fithian, Sun, and Taylor 2014; Tian and Taylor 2017, 2018; Lee et al. 2016; Tibshirani et al. 2016).”
Uniformly valid confidence intervals post-model-selection (2016, methodology): “e sets post-model-selection, and are constructed for speciﬁc model selection procedures and for targets of inference similar to those considered in the present article, have been recently obtained by Tibshirani et al. (2014), Lee and Taylor (2014), Fithian et al. (2015), Lee et al. (2016). Tibshiran”
Uniform asymptotic inference and the bootstrap after model selection (2015, methodology): “The proof of this fact is not difficult, and requires only a slight modification of the arguments in Taylor et al. (2014), given in Appendix A.”