Annals of Statistics

Valid post-selection inference

Richard Berk, Lawrence Brown, Andreas Buja, Kai Zhang, and Linda Zhao

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It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid “post-selection inference” by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing “simultaneity insurance” for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than full Scheffé protection. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.

Article information

Ann. Statist., Volume 41, Number 2 (2013), 802-837.

First available in Project Euclid: 29 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62J15: Paired and multiple comparisons

Linear regression model selection multiple comparison family-wise error high-dimensional inference sphere packing


Berk, Richard; Brown, Lawrence; Buja, Andreas; Zhang, Kai; Zhao, Linda. Valid post-selection inference. Ann. Statist. 41 (2013), no. 2, 802--837. doi:10.1214/12-AOS1077.

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Supplemental materials

  • Supplementary material: Supplement to “Valid post-selection inference”. The online supplement contains the following sections: B.1 The Full Model Interpretation of Parameters (as a contrast to the sub-model interpretation adopted in this article). B.2 “Omitted Variables Bias” (which is not bias in the sense of this article). B.3 Proof of Corollary 4.2 (strong error control). B.4 Alternative PoSI Guarantees (conditional on selection). B.5 PoSI P-Value Adjustment for Model Selection. B.6 The PoSI Process [the PoSI problem in terms of a $(j,\mathrm{M})$-indexed process]. B.7 Figures (illustrating PoSI polytopes and results of a simulation for exchangeable designs).