Selection adjusted confidence intervals with more power to determine the sign

Asaf Weinstein, William Fithian, Yoav Benjamini

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

In many current large-scale problems, confidence intervals (CIs) are constructed only for the parameters that are large, as indicated by their estimators, ignoring the smaller parameters. Such selective inference poses a problem to the usual marginal CIs that no longer offer the right level of coverage, not even on the average over the selected parameters. We address this problem by developing three methods to construct short and valid CIs for the location parameter of a symmetric unimodal distribution, while conditioning on its estimator being larger than some constant threshold. In two of these methods, the CI is further required to offer early sign determination, that is, to avoid including parameters of both signs for relatively small values of the estimator. One of the two, the Conditional Quasi-Conventional CI, offers a good balance between length and sign determination while protecting from the effect of selection. The CI is not symmetric, extending more toward 0 than away from it, nor is it of constant shape. However, when the estimator is far away from the threshold, the proposed CI tends to the usual marginal one. In spite of its complexity, it is specified by closed form expressions, up to a small set of constants that are each the solution of a single variable equation. When multiple testing procedures are used to control the false discovery rate or other error rates, the resulting threshold for selecting may be data dependent. We show that conditioning the above CIs on the data-dependent threshold still offers false coverage-statement rate (FCR) for many widely used testing procedures. For these reasons, the conditional CIs for the parameters selected this way are an attractive alternative to the available general FCR adjusted intervals.We demonstrate the use of the method in the analysis of some 14,000 correlations between hormone change and brain activity change in response to the subjects being exposed to stressful movie clips. Supplementary materials for this article are available online.

Original languageAmerican English
Pages (from-to)165-176
Number of pages12
JournalJournal of the American Statistical Association
Volume108
Issue number501
DOIs
StatePublished - 2013
Externally publishedYes

Keywords

  • Conditional confidence intervals
  • False coverage rate
  • Selective inference

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