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A significance test for the lasso
Open accessAbstract
In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an $\operatorname{Exp}(1)$ asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix $X$. On the other hand, our proof for a general step in the lasso path places further technical assumptions on $X$ and the generative model, but still allows for the important high-dimensional case $p>n$, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables. Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a $\chi^{2}_{1}$ distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than $\chi^{2}_{1}$ under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter $\lambda$ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the $\ell_{1}$ penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties—adaptivity and shrinkage—and its null distribution is tractable and asymptotically $\operatorname{Exp}(1)$.
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“Unlike stepwise variable selection procedures, LARS does not over-inflate effect size estimates during variable selection,14 nor bias hypothesis tests of those effect sizes.19”
From A structured approach to hypotheses involving continuous exposures over the life course (2016) · cites this paper as [19]
In the citing paper
“However, stepwise regression would introduce biases that LARS is known to overcome,19 and we have previously discussed how the grouped lasso and elastic net methods would result in the identification of needlessly complicated hypotheses.13”
From A structured approach to hypotheses involving continuous exposures over the life course (2016) · cites this paper as [19]
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Metadata
- Authors
- Lockhart, R., Taylor, J., Tibshirani, R. J., & Tibshirani, R.
- Affiliations
- Simon Fraser University (CA)
- Carnegie Mellon University (US)
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2014-04-01
- Volume
- 42
- Issue
- 2
- Pages
- 413-468
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- PMC
- PMC4285373
- OpenAlex
- W2172185584
- Cited by
- 541
- References
- 64
- Citations by year
- 2026: 6 · 2025: 20 · 2024: 21 · 2023: 22 · 2022: 43 · 2021: 40 · 2020: 59 · 2019: 68 · 2018: 67 · 2017: 51 · 2016: 60 · 2015: 50 · 2014: 23
- FWCI
- 64.76
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- The Annals of Statistics (publishedVersion)
- PubMed (publishedVersion)
- Europe PMC (PubMed Central) (acceptedVersion)
- PubMed Central (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- KiltHub Repository (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- National Institutes of Health
- Natural Sciences and Engineering Research Council of Canada
- National Institute of Biomedical Imaging and Bioengineering (R01EB001988)
- Air Force Office of Scientific Research
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Computer Science › Signal Processing › Sparse and Compressive Sensing Techniques
- Keywords
- LASSO, significance testing, sparse linear regression, shrinkage, high-dimensional regression, nested models
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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